gerber–shiu risk theory · gerber–shiu risk theory is not widely used. one might be more...

Andreas E. Kyprianou

Gerber–Shiu Risk Theory

Draft

June 10, 2013

Springer

Preface

These notes were developed whilst giving a graduate lecturecourse (Nachdiplomvor-lesung) in the fall of 2012 at the Forschungsinstitut fur Mathematik (FIM), ETHZurich, Switzerland. The same course was given in the spring of 2013 at Centro deInvestigacion en Matematicas (CIMAT), Guanajuato, Mexico and, simultaneouslyby video-link, at the Instituto de Mathematicas, UNAM in Mexico City. The titleof these lecture notes may come as surprise to some readers as, to date, the termGerber–Shiu Risk Theoryis not widely used. One might be more tempted to simplyuse the titleRuin theory for Cramer-Lundberg modelsinstead. However, my ob-jective here is to focus on the recent interaction between a large body of researchliterature, spear-headed by Hans Gerber and Elias Shiu, concerning ever more so-phisticated questions around the event of ruin for the classical Cramer-Lundbergsurplus process, and the parallel evolution of the fluctuation theory of Levy pro-cesses. The fusion of these two fields has provided economiesto proofs of olderresults, as well as pushing much further and faster classical theory into what onemight describe asexotic ruin theory. The latter may be considered as the study ofruinous scenarios which involve perturbations to the surplus coming from dividendor tax payments that have a historical path dependence. These notes keep to theCramer–Lundberg setting. However, the text has been written in a form that appealsto straightforward and accessible proofs, which take advantage, as much as possi-ble, of the fact that Cramer–Lundberg processes have stationary and independentincrements and no upward jumps.

I would like to thank Paul Embrechts for the invitation to spend six months atthe FIM and the opportunity to develop and present this material. Similarly I wouldlike to thank Maria-Emilia Caballero, Juan Carlos Pardo andVictor Rivero for theinvitation to give the same course in CIMAT and UNAM simultaneously. I wouldalso like to thank all attendees in Zurich, Guanajuato and Mexico City for their com-ments (especially Jean Bertoin, Leif Doring, Juan Carlos Pardo and Victor Rivero).Whilst I was in Switzerland I had the opportunity to meet and discuss some of thematerial in this book with Hans Gerber and Hansjorg Albrecher for which I wouldalso like to express my gratitude. Springer produced three anonymous referee’s re-ports whose useful comments I am also grateful for. Finally Iwould like to thank

v

vi Preface

Nick Bingham and Erik Baurdoux who have diligently attendedto my use of theEnglish language.

Guanajuato, Mexico Andreas E. KyprianouMay, 2013

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.1 The Cramer–Lundberg process . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 11.2 The classical problem of ruin . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 31.3 Gerber–Shiu expected discounted penalty functions . . .. . . . . . . . . . . 41.4 Exotic Gerber–Shiu theory . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 51.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 7

2 The Wald martingale and the maximum . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Laplace exponent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 92.2 First exponential martingale . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 112.3 Esscher transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 122.4 Distribution of the maximum . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 142.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 15

3 The Kella-Whitt martingale and the minimum . . . . . . . . . . . . . . . . . . . . 173.1 The Cramer–Lundberg process reflected in its supremum .. . . . . . . . 173.2 A useful Poisson integral . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 183.3 Second exponential martingale . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 213.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 223.5 Distribution of the minimum . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 243.6 The long term behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 243.7 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 25

4 Scale functions and ruin probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1 Scale functions and the probability of ruin . . . . . . . . . . .. . . . . . . . . . . 274.2 Connection with the Pollaczek–Khintchine formula . . . .. . . . . . . . . . 304.3 Gambler’s ruin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 334.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 35

vii

viii Contents

5 The Gerber–Shiu measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.1 Decomposing paths at the minimum . . . . . . . . . . . . . . . . . . . .. . . . . . . 375.2 Resolvent densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 385.3 More on Poisson integrals . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 405.4 Gerber–Shiu measure and gambler’s ruin . . . . . . . . . . . . . .. . . . . . . . . 415.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 42

6 Reflection strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.1 Perpetuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 466.2 Decomposing paths at the maximum . . . . . . . . . . . . . . . . . . . .. . . . . . 476.3 Derivative of the scale function . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 516.4 Present value of dividends paid until ruin . . . . . . . . . . . .. . . . . . . . . . . 526.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 54

7 Perturbation-at-maximum strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.1 Re-hung excursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 577.2 Marked Poisson process revisited . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 597.3 Gambler’s ruin for the perturbed process . . . . . . . . . . . . .. . . . . . . . . . 617.4 Continuous ruin with heavy perturbation . . . . . . . . . . . . .. . . . . . . . . . 637.5 Expected present value of tax at ruin. . . . . . . . . . . . . . . . .. . . . . . . . . . 647.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 65

8 Refraction strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678.1 Pathwise existence and uniqueness . . . . . . . . . . . . . . . . . .. . . . . . . . . . 678.2 Gambler’s ruin and resolvent density . . . . . . . . . . . . . . . .. . . . . . . . . . 708.3 Resolvent density with ruin . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 758.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 78

9 Concluding discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799.1 Mixed-exponential claims . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 799.2 Spectrally negative Levy processes . . . . . . . . . . . . . . . .. . . . . . . . . . . . 819.3 Analytic properties of scale functions . . . . . . . . . . . . . .. . . . . . . . . . . . 849.4 Engineered scale functions . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 869.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 89

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 91

Chapter 1Introduction

In this brief introductory chapter, we shall outline the basic context of these lecturenotes. In particular, we shall explain what we understand byso-calledGerber–Shiutheory and the role that it has played in classical ruin theory.

1.1 The Cramer–Lundberg process

The beginning of ruin theory is based around a very basic model for the evolutionof the wealth, orsurplus, of an insurance company, known as theCramer–Lundbergprocess. In the classical model, the insurance company is assumed tocollect pre-miums at a constant ratec > 0, whereas claims arrive successively according tothe times of a Poisson process, henceforth denoted byN = Nt : t ≥ 0, with rateλ > 0. These claims, indexed in order of appearanceξi : i = 1,2, · · ·, are inde-pendent and identically distributed1 with common distributionF , which is concen-trated on(0,∞). The dynamics of the Cramer–Lundberg process are described byX = Xt : t ≥ 0, where

Xt = ct−Nt

∑i=1

ξi , t ≥ 0, (1.1)

is the gain between time 0 andt. Here, we use standard notation in that a sum ofthe form∑0

i=1 · is understood to be equal to zero. We assume thatX is defined on afiltered probability space(Ω ,F,F ,P), whereF := Ft : t ≥ 0 is the natural filtra-tion generated byX. When the initial surplus of our insurance company is valuedatx> 0, we may consider the evolution of the surplus to follow the dynamics ofx+XunderP.

The Cramer–Lundberg process,(X,P), is nothing more than the difference of alinear trend and a compound Poisson process with positive jumps. Accordingly, it

1 Henceforth written i.i.d. for short.

1

2 1 Introduction

is easy to verify that it conforms to the definition of a so-called Levy process, givenbelow.

Definition 1.1. A processX = Xt : t ≥ 0 with law P is said to be aLevy processif it possesses the following properties:

(i) The paths ofX areP-almost surely right-continuous with left limits.(ii) P(X0 = 0) = 1.(iii) For 0≤ s≤ t, Xt −Xs is equal in distribution toXt−s.(iv) For all n∈ N and 0≤ s1 ≤ t1 ≤ s2 ≤ t2 ≤ ·· · ≤ sn ≤ tn < ∞, the increments

Xti −Xsi , i = 1, · · · ,n, are independent.

Whilst our computations in this text will largely remain within the confines ofthe Cramer–Lundberg model, we shall, as much as possible, appeal to mathematicalreasoning which is handed down from the general theory of Levy process. Specif-ically, our analysis will predominantly appeal tomartingale theoryas well asex-cursion theory. The latter of these two concerns the decomposition of the path of Xinto a sequence of sojourns from its running maximum or, indeed, from its runningminimum.

Many of the arguments we give will apply, either directly or with minor modifi-cation, into the setting of generalspectrally negative Levy processes. These are Levyprocesses which do not experience positive jumps and which do not have monotonepaths. In the forthcoming chapters, we have deliberately stepped back from treatingthe case of general spectrally negative Levy processes in order to keep the presen-tation as mathematically light as possible. Nonetheless, many of the arguments wegive are robust enough to apply to the case of general spectrally negative Levy pro-cesses, either verbatim or with minor modification. At the very end, in Chapter 9,we will spend a little time discussing the connection with the general spectrallynegative setting.

As a Levy processes, it is well understood that Cramer–Lundberg processes arestrong Markov2 and, henceforth, we shall prefer to work with the probabilitiesPx :x∈ R, where, thanks to spatial homogeneity, forx∈ R, (X,Px) is equal in law tox+X underP. For convenience, we shall always prefer to writeP instead ofP0.

Recall thatτ is an stopping time with respect toF if and only if, for all t ≥ 0,τ ≤ t ∈Ft . Moreover, for each stopping time,τ, we associate the sigma-algebra

Fτ := A∈F : A∩τ ≤ t ∈Ft for all t ≥ 0.

(Note, it is a simple exercise to verify thatFτ is a sigma-algebra.) The standard wayof expressing the strong Markov property for a one-dimensional process such asXis as follows: For any Borel setB, onτ < ∞,

P(Xτ+s∈ B|Fτ) = P(Xτ+s∈ B|σ(Xτ)) = h(Xτ ,s),

2 We assume that the reader is familiar with the basic theory ofMarkov processes. In particular,the use of the (strong) Markov property.

1.2 The classical problem of ruin 3

whereh(x,s) = Px(Xs∈B). On account of the fact thatX has stationary independentincrements, we may also state the strong Markov property in aslightly refined form.

Theorem 1.2.Suppose thatτ is a stopping time. Onτ < ∞, define the processX = Xt : t ≥ 0, where

Xt = Xτ+t −Xτ , t ≥ 0.

Then, on the eventτ < ∞, the processX is independent ofFτ and has the samelaw as X.

1.2 The classical problem of ruin

Financial ruin in the Cramer–Lundberg model (or justruin for short) will occur ifthe surplus of the insurance company drops below zero. Sincethis will happen withprobability one ifP(lim inf t↑∞ Xt = −∞) = 1, in order to avoid this situation, it isusual to impose an additional assumption that

P(limt↑∞

Xt = ∞) = 1. (1.2)

Write µ =∫(0,∞) xF(dx) for the common mean of the i.i.d. claim sizesξi : i =

1,2, · · ·. A sufficient condition to guarantee (1.2) is that

c−λ µ > 0, (1.3)

the so-calledsecurity loading condition.To see why this guarantees (1.2), notethat the Strong Law of Large Numbers for Poisson processes, which states thatlimt↑∞ Nt/t = λ a.s., and the obvious fact that limt↑∞ Nt = ∞ a.s. imply that

limt↑∞

Xt

t= lim

t↑∞

(xt+c− Nt

t∑Nt

i=1 ξi

Nt

)= E(X1) = c−λ µ > 0 a.s., (1.4)

from which (1.2) follows. We shall see later that the positive security loading in(1.3) is also a necessary condition for (1.2). Note that (1.3) also implies thatµ < ∞.

Under the security loading condition, it follows that ruin will occur only withprobability less than one. The most basic question that one can therefore ask undersuch circumstances is: What is the probability of ruin when the initial surplus isequal tox> 0? This involves giving an expression forPx(τ−0 < ∞), where3

τ−0 := inft > 0 : Xt < 0.

ThePollaczek–Khintchine formuladoes just this.

3 Throughout this text, we use the standard definition inf /0 := ∞.

4 1 Introduction

Theorem 1.3 (Pollaczek–Khintchine formula).4 Suppose thatλ µ/c< 1. For allx≥ 0,

1−Px(τ−0 < ∞) = (1−ρ)∑k≥0

ρkη∗k (x), (1.5)

whereρ = λ µ/c,

η(x) =1µ

∫ x

0[1−F(y)]dy, x≥ 0,

and, for k≥ 0, we understandη∗k to be the k-fold convolution ofη with the specialunderstanding thatη∗0(dx) = δ0(dx).

It is not our intention to dwell on the Pollaczek–Khintchineformula at this pointin time, although we shall re-derive it in due course later inthis text. This classicalresult is a cornerstone of what is known asinsurance risk theory. Its connection torenewal theory is the inspiration behind a whole body of research literature address-ing more elaborate questions concerning the ruin problem. Our aim in this text is togive an overview of the state of the art in this respect. Amongst the large number ofnames active in this field, one may note, in particular, the many original and variedcontributions of Hans Gerber and Elias Shiu. In recognitionof their foundationalwork spanning several decades, we accordingly refer to the collective results thatwe present here asGerber-Shiu risk theory.

1.3 Gerber–Shiu expected discounted penalty functions

Following Theorem 1.3, an obvious direction in which to turnone’s attention islooking at the joint distribution ofτ−0 , −Xτ−0

andXτ−0 −. These three quantities can

otherwise be called thetime of ruin, the deficit at ruin and the surplus prior toruin. In their well-cited paper of 19985, Gerber and Shiu introduce the so-calledexpected discounted penalty functionas follows. Suppose thatf : (0,∞)2→ [0,∞) isany bounded, measurable function. Then the associated expected discounted penaltyfunction with force of interestq≥ 0, when the initial surplus is equal tox≥ 0, isgiven by

GSf (x,q) := Ex

[e−qτ−0 f (−Xτ−0

,Xτ−0 −)1(τ−0 <∞)

].

Ultimately, we are really interested in, what we call here, theGerber–Shiu measure,namely the exponentially discounted joint law of the pair(−Xτ−0

,Xτ−0 −), denoted by

K(q)(x,dy,dz) := Ex

[e−qτ−0 ;−Xτ−0

∈ dy, Xτ−0 −∈ dz

], x,y,z≥ 0. (1.6)

For the Gerber–Shiu measure, one notes the simple relation

4 The Pollaczek-Khintchine formula is also known as the Beekman convolution formula in someactuarial literature.5 See the historical remarks at the end of this chapter.

1.4 Exotic Gerber–Shiu theory 5

GSf (x,q) =∫

[0,∞)

∫

[0,∞)f (y,z)K(q)(x,dy,dz).

The expected discounted penalty function is a well-studiedobject and there aremany different ways to develop the expression on the right-hand side of (1.6). Weshall show later in this text how the Gerber–Shiu measure canbe written in termsof so-calledscale functions. Scale functions are a natural family of functions withwhich one may develop many of the identities around the eventof ruin, which con-cern the way in which ruin occurs in a variety of different scenarios. We shall spendquite some time discussing the recent theory of scale functions later on in this text.

1.4 Exotic Gerber–Shiu theory

Again inspired by foundational work of Gerber and Shiu, and indeed many others,we shall also look at variants of the classical ruin problem in the setting that theCramer–Lundberg process undergoes perturbations in its trajectory. These perturba-tions will represent pay-outs corresponding to dividend ortaxation payments. Threespecific cases that will interest us are the following.

Reflection strategies:An adaptation of the classical ruin problem, introduced byBruno de Finetti in 1957, is to consider the continuous payment of dividends fromthe surplus process to (hypothetical) share holders. Naturally, for a given stream ofdividend payments, this will continuously change the aggregate value of the surplusprocess and the problem of ruin will look quite different. Indeed, the event of ruinwill be highly dependent on the choice of dividend payments.One may formulatethe problem of finding an optimal way of paying out dividends such as to maximisethe expected present value of the total income of the shareholders, under force ofinterestq≥ 0, from time zero until ruin. The optimisation is made over anappro-priate class of dividend strategies. Mathematically speaking, de Finetti’s dividendproblemamounts to solving a control problem which we reproduce here.

Let ξ = ξt : t ≥ 0, with ξ0 = 0, be a dividend strategy consisting of a left-continuous, non-negative, non-decreasing process, whichisF-adapted. The quantityξt represents the cumulative dividends paid out up to timet ≥ 0 by the insurancecompany whose surplus is modelled byX. The aggregate, orcontrolled, value of thesurplus process, when taking account of the dividend strategy ξ , is thusUξ = Uξ

t :

t ≥ 0, whereUξt = Xt − ξt , t ≥ 0. An additional constraint onξ is thatξt+− ξt ≤

maxUξt ,0 for t ≥ 0 (i.e. lump sum dividend payments are always smaller than the

available reserves).Let Ξ be the family of dividend strategies, as outlined in the previous paragraph,

and, for eachξ ∈ Ξ , write σξ = inft > 0 : Uξt < 0 for the time at which ruin

occurs for the controlled risk process. The expected present value, with discountingat rateq≥ 0, associated to the dividend policyξ , when the risk process has initialcapitalx≥ 0, is given by

6 1 Introduction

vξ (x) = Ex

(∫ σ ξ

0e−qtdξt

).

De Finetti’s dividend problem consists of solving the stochastic control problem

v∗(x) := supξ∈Ξ

vξ (x), x≥ 0. (1.7)

That is to say, if it exists, to establish a strategy,ξ ∗ ∈ Ξ , such thatv∗ = vξ ∗ .We shall refrain from giving a complete account of this problem, other than

to say that under certain conditions on the jump distribution, F, of the Cramer–Lundberg process,X, the optimal strategy consists of a so-calledreflection strategy,also known as abarrier strategy. Specifically, there exists ana ∈ [0,∞) such that

ξ ∗t = (a∨Xt)−a t > 0.

In that case, theξ ∗-controlled risk process is identical, ont > 0, to the processa−Yt : t ≥ 0 underPx, where

Yt = (a∨Xt)−Xt , t ≥ 0,

andXt = sups≤t Xs is the running supremum of the surplus process.

Refraction strategies:An adaptation of the optimal control problem deals with thecase that optimality is sought in a subclass ofΞ , sayΞδ . Specifically,Ξδ denotesthe set of dividend strategiesξ ∈ Ξ such that

ξt =

∫ t

0αsds, t ≥ 0,

whereαt : t ≥ 0 is uniformly bounded by some constant, sayδ > 0. In otherwords, Ξδ consists of dividend strategies which are absolutely continuous, withuniformly bounded density.

Again, we refrain from going into the details of the solutionto (1.7), other thanto say that, under certain conditions, the optimal strategy, ξ δ = ξ δ

t : t ≥ 0 in Ξδturns out to satisify

ξ δt = δ

∫ t

01(Zs>b)ds, t ≥ 0,

for someb ≥ 0, whereZ = Zt : t ≥ 0 is the controlled surplus processX− ξ δ .The pair(Z,ξ δ ) cannot be expressed autonomously and we are forced to work withthe stochastic differential equation

Zt = Xt − δ∫ t

01(Zs>b)ds, t ≥ 0. (1.8)

For reasons that we shall elaborate on later, the process in (1.8) is called arefractedprocess.

1.5 Comments 7

Perturbation-at-maximum strategies:Another way of perturbing the path of ourCramer–Lundberg process is by forcing payments from the surplus at times thatit attains new maxima. This may be be interpreted, for example, as tax payments.To this end, consider the process

Ut = Xt −∫

(0,t]γ(Xu)dXu, t ≥ 0, (1.9)

whereγ : [0,∞)→ [0,∞) satisfies appropriate conditions.We distinguish two regimes,light perturbationandheavy perturbationregimes.

The first corresponds to the case thatγ : [0,∞)→ [0,1) and the second to the casethatγ : [0,∞)→ (1,∞). The light perturbation regime has a similar flavour to payingdividends at a weaker rate than a reflection strategy, and maybe seen as a taxationon new levels of wealth. In contrast, the heavy perturbationregime is equivalent topaying dividends at a much stronger rate than a reflection strategy. (The connectionwith tax payments is arguably lost.)

For each of the three scenarios described above, reflection,refraction and perturbation-at-maximum, questions concerning the way in which ruin occurs remain just as per-tinent as for the case of the Cramer–Lundberg process alone. In addition, we arealso interested in the distribution of the present value of payments made out of thesurplus process until ruin. For example in the case of a reflection strategy with bar-rier a≥ 0 and force of interest equal toq≥ 0, this boils down to understanding thedistribution of ∫ σa

0e−qtdXt ,

whereσa = inft > 0 : Xt −Xt > a.

1.5 Comments

For more on the classical derivation of the standard model ofa surplus process, seeLundberg (1903), Cramer (1994a) and Cramer (1994b). Asmussen and Albrecher(2010) serves as an encyclopaedic reference for all mattersconcerning ruing theory.See also, for example, the books of Embrechts et al. (1997) and Dickson (1999), toname but a few standard texts on the classical theory. A classical derivation of thePollaczek–Khintchine formula (also known in the actuarialliterature asBeekman’sconvolution formula) can be found in Chapter XII of Feller (1971).

Within the setting of the classical Cramer–Lundberg model, Gerber and Shiu(1998) introduced the expected discounted penalty function. See also Gerber and Shiu(1997). It has been widely studied since, with too many references to list here. Thespecial issue in volume 46 of the journalInsurance: Mathematics and Economicscontains a selection of papers focused on the Gerber–Shiu expected discountedpenalty function, with many further references therein.

8 1 Introduction

An adaptation of the classical ruin problem was introduced within the setting ofa discrete-time surplus process by de Finetti (1957). There, dividends are paid outto share holders up to the moment of ruin, resulting in a discrete-time analogue ofthe control problem (1.7). This control problem was considered in the framework ofCramer–Lundberg processes by Gerber (1969) and Gerber (1972) and then, after alarge gap, by Azcue and Muler (2005). Schmidli (2008) gives an extensive accountof (1.7) and variants thereof. A string of articles, each onesuccessively improv-ing on its predecessor, develops the solution to (1.7) in thesetting that the surplusprocess is modelled by a spectrally negative Levy process;see Avram et al. (2007),Loeffen (2008), Kyprianou et al. (2010) and Loeffen and Renaud (2010). The vari-ant of (1.7) resulting in refraction strategies was studiedby Jeanblanc and Shiryaev(1995) and Asmussen and Taksar (1997) in the diffusive setting, Gerber and Shiu(2006b) in the Cramer–Lundberg setting and Kyprianou et al. (2012) in the settingof spectrally negative Levy processes.

In the setting of the classical Cramer–Lundberg model, Albrecher and Hipp(2007) introduced the idea of tax payments as in (1.9), for the case thatγ is aconstant in(0,1). This model was quickly generalised by Albrecher et al. (2008),Kyprianou and Zhou (2009), Albrecher et al. (2011) and Kyprianou and Ott (2012).

Chapter 2The Wald martingale and the maximum

In this chapter, we shall introduce the first of our two key martingales and considertwo immediate applications. In the first application, we will use the martingale toconstruct a change of measure with respect toP and thereby consider the dynamicsof X under the new law. In the second application, we shall use themartingale tostudy the law of the processX = Xt : t ≥ 0, where we recall that

Xt = sups≤t

Xs, t ≥ 0. (2.1)

In particular, we shall discover that the position of the trajectory ofX, when sam-pled at an independent and exponentially distributed time,is again exponentiallydistributed.

2.1 Laplace exponent

A key quantity in the forthcoming analysis is theLaplace exponentof the Cramer–Lundberg process, whose definition is contained in the following lemma.

Lemma 2.1.For all θ ≥ 0 and t≥ 0, we have

E(eθXt ) = expψ(θ )t,

where the Laplace exponentψ satisfies

ψ(θ ) := cθ −λ∫

(0,∞)(1−e−θx)F(dx). (2.2)

Proof. Given the definition (1.1), one easily sees that it suffices toprove that

E

(e−θ ∑Nt

i=1ξi

)= exp

−λ t

∫

(0,∞)(1−e−θx)F(dx)

, (2.3)

9

10 2 The Wald martingale and the maximum

for θ , t ≥ 0. To establish (2.3), we make use of the fact thatNt is independent ofξi : i ≥ 1 and Poisson distributed with rateλ t. More precisely,

E

(e−θ ∑Nt

i=1 ξi

)=

∞

∑n=0

E

(e−θ ∑n

i=1ξi

)e−λ t (λ t)n

n!

=∞

∑n=0

[E(e−θξ1)]ne−λ t (λ t)n

n!

= exp−λ t(1−E(e−θξ1))

= exp

−λ t

∫

(0,∞)(1−e−θx)F(dx)

,

for all θ , t ≥ 0. Note that the integral in the final equality is trivially finite on accountof the fact thatF is a probability distribution.

As we shall see, the Laplace exponent (2.2) is used as a way of identifying certaincharacteristics of Cramer–Lundberg processes. To this end, let us start by looking atthe shape of (2.2). Straightforward differentiation, withthe help of the DominatedConvergence Theorem, tells us that, for allθ > 0,

ψ ′′(θ ) = λ∫

(0,∞)x2e−θxF(dx)> 0,

which in turn implies thatψ is strictly convex on(0,∞). Integration by parts allowsus to write

ψ(θ ) = cθ −λ θ∫

(0,∞)e−θxF(x)dx, θ ≥ 0, (2.4)

whereF(x) := 1−F(x), x≥ 0. Moreover, this representation allows us to deducethat

limθ→∞

ψ(θ )θ

= c

and

ψ ′(0+)= limθ→0

ψ(θ )θ

= c−λ µ ,

where we recall thatµ :=∫(0,∞)xF(dx) ∈ (0,∞] and the left-hand side is the right

derivative ofψ at the origin. In particular, we see that

E(X1) = limθ→0

E(X1eθX1) = ψ ′(0+) ∈ [−∞,∞),

where we have again used the Dominated Convergence Theorem to justify the firstequality above. The security loading condition (1.3) can thus otherwise be expressedsimply asψ ′(0+)> 0.

A quantity which will also repeatedly appear in our computations is the rightinverse ofψ . That is,

2.2 First exponential martingale 11

ψ′(0+) ∈ [−∞, 0) ψ′(0+) ∈ [0, ∞)ψ ψ

q q

F(q) F(q)

F(0)

Fig. 2.1 Two examples ofψ, the Laplace exponent of a Cramer–Lundberg process, correspondingto the casesψ ′(0+)< 0 andψ ′(0+) ≥ 0, respectively.

Φ(q) := supθ ≥ 0 : ψ(θ ) = q, (2.5)

for q≥ 0. Thanks to the strict convexity ofψ , we can say that there is exactly onesolution to the equationψ(θ ) = q, whenq> 0, and at most two whenq= 0. Thenumber of solutions in the latter of these two cases depends on the value ofψ ′(0+).Indeed, whenψ ′(0+) ≥ 0, thenθ = 0 is the only solution toψ(θ ) = 0. Whenψ ′(0+)< 0, there are two solutions, one atθ = 0 and a second solution, in(0,∞),which, by definition, gives the value ofΦ(0); see Fig. 2.1.

2.2 First exponential martingale

For eachβ > 0, define the processE (β ) = Et(β ) : t ≥ 0, where

Et(β ) := eβ Xt−ψ(β )t , t ≥ 0. (2.6)

Theorem 2.2.Fix β > 0. The processE (β ) is aP-martingale with respect toF.

Proof. Note that the processE (β ) is F-adapted. With this in hand, it suffices tocheck that, for allβ > 0 ands, t ≥ 0, E[Et+s(β )|Ft ] = Et(β ). On account of posi-tivity, this would immediately show thatE[|Et(β )|]< ∞, for all t ≥ 0, which is alsorequired forE (β ) to be a martingale.

Thanks to stationary and independent increments,F -adaptedness as well asLemma 2.1, for allβ ,s, t ≥ 0,


E[Et+s(β )|Ft ] = Et(β )E[

eβ (Xt+s−Xt )−ψ(β )s∣∣∣Ft

]

= Et(β )E[eβ Xs]e−ψ(β )s

= Et(β )

and the proof is complete.

This martingale is known as theWald martingale. See Sect. 2.5 for further histo-rial details.

2.3 Esscher transform

Fix β > 0 andx∈R. NormalisingE (β ) by its expectation, we may use the resultingmean-one martingale to perform a change of measure on(X,Px) via

dPβx

dPx

∣∣∣∣∣Ft

=Et(β )E0(β )

= eβ (Xt−x)−ψ(β )t , t ≥ 0. (2.7)

In the special case thatx= 0, we shall writePβ in place ofPβ0 . Since the processX

underPx may be written asx+X underP, it is not difficult to see that the changeof measure on(X,Px) corresponds to the analogous change of measure on(X,P).Also known as theEsscher transform, (2.7) alters the law ofX and it is important tounderstand the dynamics ofX underPβ .

Theorem 2.3.Fix β > 0. The process(X,Pβ ) is equal in law to a Cramer–Lundbergprocess with premium ratec and claims that arrive at rateλm(β ) with common dis-tribution e−β xF(dx)/m(β ) on (0,∞), where m(β ) =

∫(0,∞)e−β xF(dx). Said another

way, the process(X,Pβ ) is equal in law to Xβ , where Xβ := Xβt : t ≥ 0 is a

Cramer–Lundberg process with Laplace exponent

ψβ (θ ) := ψ(θ +β )−ψ(β ) θ ≥ 0.

Proof. For all 0≤ s≤ t ≤ u< ∞, θ ≥ 0 andA∈Fs, with the help of stationary andindependent increments of(X,P), we have that

Eβ[1Aeθ(Xu−Xt )

]= E

[1Aeβ Xt−ψ(β )te(θ+β )(Xu−Xt )

]e−ψ(β )(u−t)

= E

[1Aeβ Xt−ψ(β )t

]E

[e(θ+β )Xu−t

]e−ψ(β )(u−t)

= E

[1Aeβ Xs−ψ(β )s

]Eeψ(β+θ)(u−t)e−ψ(β )(u−t)

= Pβ (A)eψβ (θ)(u−t), (2.8)

where in the second equality we have conditioned onFt and in the third equality wehave conditioned onFs and used the martingale property ofE (β ). It now follows

2.3 Esscher transform 13

from (2.8) that, for all 0≤ v≤ s≤ t ≤ u< ∞ andθ1,θ2 ≥ 0,

Eβ[eθ1(Xs−Xv)eθ2(Xu−Xt )

]= eψβ (θ1)(s−v)+ψβ (θ2)(u−t).

Using a straightforward argument by induction, again using(2.8), we also havethat, for alln∈ N, 0≤ s1≤ t1≤ ·· · ≤ sn ≤ tn < ∞ andθ1, · · · ,θn ≥ 0,

Eβ

[n

∏j=1

eθ j (Xt j−Xsj )

]=

n

∏j=1

eψβ (θ j )(t j−sj ). (2.9)

Moreover, a brief computation shows that

ψβ (θ ) = cθ −λm(β )∫

(0,∞)(1−e−θx)

e−β x

m(β )F(dx), θ ≥ 0.

Coupled with (2.9), this shows that(X,Pβ ) has stationary and independent incre-ments, which are equal in law to those of a Cramer–Lundberg process with premiumratec, arrival rate of claimsλm(β ) and distribution of claims e−β xF(dx)/m(β ).Since the measuresPβ andP are equivalent onFt , for all t ≥ 0, then the prop-erty thatX has paths that are almost surely right-continuous with leftlimits and nopositive jumps on[0, t] carries over to the measurePβ .

The Esscher transform may also be formulated at stopping times.

Corollary 2.4. Under the conditions of Theorem 2.3, ifτ is anF-stopping time, then

dPβ

dP

∣∣∣∣∣Fτ

= Eτ(β ) onτ < ∞.

Said another way, for all A∈Fτ , we have

Pβ (A, τ < ∞) = E(1(A,τ<∞)Eτ(β )).

Proof. By definition, if A∈Fτ , thenA∩τ ≤ t ∈Ft . Hence

Pβ (A∩τ ≤ t) = E(Et(β )1(A,τ≤t))

= E(1(A,τ≤t)E(Et(β )|Fτ ))

= E(Eτ (β )1(A,τ≤t)),

where in the third equality we have used the strong Markov property as well as themartingale property forE (β ). Now taking limits ast ↑∞, the result follows with thehelp of the Monotone Convergence Theorem.


2.4 Distribution of the maximum

We want to use the Esscher transform to characterise the law of the first passagetimes

τ+x := inft > 0 : Xt > x,for x≥ 0, and subsequently the law of the running maximum when sampled at anindependent and exponentially distributed time. Note thatthe stopping timeτ+x maybe infinite in value, depending on the long-term behaviour ofthe processX. Accord-ingly, in the theorem below, whereτ−0 appears in an exponent, we will understande−∞ := 0.

Theorem 2.5.For x,q≥ 0,

E(e−qτ+x ) = e−Φ(q)x,

where we recall thatΦ(q) is given by (2.5).

Proof. Fix q > 0. Using the fact thatX has no positive jumps, it must follow thatXτ+x = x onτ+x < ∞. With the help of the strong Markov property we have that

E(eΦ(q)Xt−qt|Fτ+x )

= 1(τ+x ≥t)eΦ(q)Xt−qt +1(τ+x <t)e

Φ(q)x−qτ+x E(eΦ(q)(Xt−Xτ+x

)−q(t−τ+x )|Fτ+x ),

= eΦ(q)X

t∧τ+x−q(t∧τ+x )

,

where, in the final equality, we have used the fact thatE(Et(Φ(q))) = 1 for all t ≥ 0.Taking expectations again, we have

E(eΦ(q)X

t∧τ+x−q(t∧τ+x )

) = 1.

Noting that the expression in the latter expectation is bounded above by eΦ(q)x, anapplication of dominated convergence yields

E(eΦ(q)x−qτ+x ) = 1,

which is equivalent to the statement of the theorem.

We recover the promised distributional information about the maximum process(3.1) in the next corollary. In its statement, we understandan exponential randomvariable with rate 0 to be infinite in value with probability one.

Corollary 2.6. Fix q≥ 0 and leteq be an exponentially distributed random variablewith rate q, which is independent of X. ThenXeq is exponentially distributed withparameterΦ(q).

Proof. First suppose thatq> 0. The result is an easy consequence of the fact that

2.5 Comments 15

P(Xeq > x) = P(τ+x < eq) = E

(∫ ∞

0qe−qt1(τ+x <t)

)= E(e−qτ+x ),

together with the conclusion of Theorem 2.5. For the remaining case thatq= 0, weneed to take limits asq ↓ 0 in the previous conclusion. Note that we may alwayswrite eq as equal in distribution toq−1e1. Hence, thanks to monotonicity ofX, theMonotone Convergence Theorem for probabilities and the continuity of Φ, we have

P(X∞ > x) = limq↓0

P(Xq−1e1> x) = lim

q↓0e−Φ(q)x = e−Φ(0)x,

and the proof is complete.

2.5 Comments

The idea oftilting a distribution by exponentially weighting its probabilitydistri-bution function was introduced by Esscher (1932). This idealends itself well tochanges of measure in the theory of stochastic processes, inparticular for Levyprocesses. The Wald martingale can be traced back to the workof Wald (1944)and Wald (1945). The associated Esscher transform is analogous to the exponentialmartingale for Brownian motion and the role that it plays in the classical Cameron-Martin-Girsanov change of measure. Indeed, the theory presented here may be ex-tended to the general class of spectrally negative Levy processes, which includesboth Cramer–Lundberg processes as well as Brownian motion. See for exampleChapter 3 of Kyprianou (2013). The Esscher transform plays aprominent role infundamental mathematical finance as well as insurance mathematics; see for ex-ample the discussion in the paper of Gerber and Shiu (1994) and references therin.The style of reasoning in the proof of Theorem 2.5 is inspiredby the classical com-putations of Wald (1944) for random walks, see also Bingham (1975) and Gerber(1990).

Chapter 3The Kella-Whitt martingale and the minimum

We move now to the second of our two key martingales. In a similar spirit to theprevious chapter, we shall use the martingale to study the law of the processX =Xt : t ≥ 0, where

Xt := infs≤t

Xs, t ≥ 0. (3.1)

As with the case ofX, we shall characterise the law ofX when sampled at an in-dependent and exponentially distributed time. Unlike the case ofX however, thiswill not turn out be exponentially distributed. In order to carry out the necessaryanalysis, we will need to pass through two sections of preparatory material.

3.1 The Cramer–Lundberg process reflected in its supremum

Fix x≥ 0. Define the processYx = Yxt : t ≥ 0, where

Yxt := (x∨Xt)−Xt , t ≥ 0.

Lemma 3.1.For each x≥ 0, Yx is a Markov process.

Proof. Let Xu = Xt+s−Xt , s, t ≥ 0, and recall thatXs : s≥ 0 is independent ofFt

with the same law as(X,P). Note that, fort,s≥ 0,

(x∨Xt+s)−Xt+s =

(x∨Xt ∨ sup

u∈[t,t+s]Xu

)−Xt− Xs

=

[(x∨Xt −Xt)∨

(sup

u∈[t,t+s]Xu−Xt

)]− Xs

=

[Yx

t ∨ supu∈[0,s]

Xu

]− Xs.

17

18 3 The Kella-Whitt martingale and the minimum

From the right-hand side above, it is clear that the law ofYxt+s depends only on

Yxu : u≤ t through the value ofYx

t . HenceYxt : t ≥ 0 is a Markov process.

Remark 3.2.Note that the argument given in the proof above shows that, ifτ is anystopping time with respect toF, then, onτ < ∞,

(x∨Xτ+s)−Xτ+s=

[Yx

τ ∨ supu∈[0,s]

Xu

]− Xs,

where now,Xs := Xτ+s−Xτ , s≥ 0, is independent ofFτ and has the same law as(X,P). In other words,

Yxτ+s = Yz

s such thatz=Yxτ ,

whereYz is independent ofFτ and equal in law toYz.

Remark 3.3.It is also possible to argue, in the style of the proof of Lemma3.1, that,for eachy≤ 0, the processZy, given by

Zyt := Xt − (y∧Xt), t ≥ 0,

is also a Markov process. In fact, one can go much further and show that, forx≥ 0,y≤ 0, the quadruplet(Yx,Zy,X,N) is also Markovian. Specifically, for eachs, t ≥ 0,

(Yxt+s,Z

yt+s,Xt+s,Nt+s) = (Yu

s , Zvs,w+ Xs,n+ Ns)

such thatu= Yxt ,v = Zy

t ,w= Xt andn= Nt , where(Yus , Z

vs, Xs, Ns) : s≥ 0 is in-

dependent ofFt and equal in law to(Yu,Zv,X,N) underP. Again, one also easilyreplacest by anF-stopping time in the above observation, as in the previous remark.

3.2 A useful Poisson integral

In the next section, we will come across some functionals of the driving PoissonprocessN = Nt : t ≥ 0. Specifically, we will be interested in expected sums of theform

E

[Nt

∑i=1

f (YxTi−, ξi)

], x, t ≥ 0,

where f : [0,∞)× (0,∞)→ [0,∞) is measurable,Ti : i ≥ 1 are the arrival times inthe processN and recall thatξi : i ≥ 1 are the i.i.d. subsequent claim sizes ofXwith common distributionF. We need the following result.

Theorem 3.4 (Compensation formula).For all non-negative, bounded, measur-able f and x, t ≥ 0,

E

[Nt

∑i=1

f (YxTi−, ξi)

]= λ

∫ t

0

∫

(0,∞)E[ f (Yx

s , u)]F(du)ds. (3.2)

3.2 A useful Poisson integral 19

Proof. With the help of Fubini’s Theorem, we can write

E

[∞

∑i=1

1(Ti≤t) f (YxTi−, ξi)

]=

∞

∑i=1

E[1(Ti≤t) f (Yx

Ti−, ξi)]. (3.3)

Note thatTi = inft > 0 : Nt = i and hence eachTi is a stopping time. Note alsothat, for eachi ≥ 1, the termsf (Yx

Ti−) are each measurable in the sigma-algebraHi := σ(Ns : s≤ Ti,ξ j : j = 1, · · · , i −1). Breaking each of the expectationsin the sum on the right-hand side of (3.3), by first conditioning onHi and thenapplying Fubini’s Theorem again, it follows that

∞

∑i=1

∫

(0,∞)E[1(Ti≤t) f (Yx

Ti−, u)]F(du) =

∫

(0,∞)E

[Nt

∑i=1

f (YxTi−, u)

]F(du).

The proof is therefore complete as soon as we show that, for all x, t ≥ 0 andu> 0,

E

[Nt

∑i=1

f (YxTi−, u)

]= λ

∫ t

0E[ f (Yx

s , u)]ds. (3.4)

To this end, define, forx, t ≥ 0 andu > 0, ηu(x, t) = E[∑Nti=1 f (Yx

Ti−, u)]. With thehelp of the Markov property for(Yx,N) and as well as the stationary independentincrements ofN, we have1

ηu(x, t + s)−ηu(x, t) = E

[E

[Nt+s

∑i=Nt+1

f (YxTi−, u)

∣∣∣∣∣Ft

]]

= E

E[

Ns

∑i=1

f (YzTi−

, u)

]∣∣∣∣∣z=Yx

t

= E [ηu(Yxt ,s)] ,

where the process(Yzs , Ns) : s≥ 0 is independent ofFt and equal in law to(Yz,N)

underP. Note here thatTi : i ≥ 1 are the arrival times of the processN. Next, notethat, for allx,s≥ 0, we have thats−1η(x,s) is bounded bys−1CE(Ns) = λC, whereC= supy≥0 f (y)<∞. Hence, with the help of the Dominated Convergence Theorem,our objective now is to compute the right-derivative ofηu(x, t) by evaluating thelimit

lims↓0

ηu(x, t + s)−ηu(x, t)s

= E

[lims↓0

1s

ηu(Yxt ,s)

]. (3.5)

Note that, for allx,s≥ 0,

1 We use the standard protocol that∑ni=n+1 · is defined to be an empty sum.


1s

ηu(x,s) =1sE[ f (Yx

T1−, u)1(Ns=1)]+1sE

[Ns

∑i≥1

f (YxTi−, u)1(Ns≥2)

]. (3.6)

Moreover, recall that, forv≤ s,

P(T1 ∈ dv,Ns = 1) = P(T1 ∈ dv,T2 > s)

= P(T1 ∈ dv,T2−T1 > s− v)

= λe−λ vdv×e−λ (s−v)

= λe−λ sdv.

Hence, for the first term on the right-hand side of (3.6) we have

lims↓0

1sE[ f (Yx

T1−, u)1(Ns=1)] = lims↓0

λe−λ s

s

∫ s

0f ((x∨cv)−cv, u)dv= λ f (x,u).

For the second term on the right-hand side of (3.6), we also have

lims↓0

1sE

[Ns

∑i≥1

f (YxTi−, u)1(Ns≥2)

]≤C lim

s↓01sE[Ns1(Ns≥2)] = lim

s↓01s[λs(1−e−λ s)] = 0.

Returning to (3.6), it follows that, for allx≥ 0, lims↓0s−1ηu(x,s) = λ f (x,u) andhence, from (3.5), we have that

∂∂ t

ηu(x, t+) = λE[ f (Yxt , u)].

A similar argument, looking at the differenceηu(x, t)−ηu(x, t− s), for x≥ 0 andt > s> 0, also shows that the left derivative satisfies

∂∂ t

ηu(x, t−) = λE[ f (Yxt , u)].

It follows thatηu(x, t) is differentiable int on (0,∞) and hence, sinceηu(x,0) = 0,

ηu(x, t) = λ∫ t

0E[ f (Yx

s , u)]ds,

which establishes (3.4) and completes the proof.

Remark 3.5.It is a straightforward exercise to deduce from Theorem 3.4 that thecompensated process

Nt

∑i=1

f (YxTi−,ξi)−λ

∫ t

0

∫

(0,∞)E( f (Yx

s ), u)F(du)ds, t ≥ 0,

is a martingale.

3.3 Second exponential martingale 21

3.3 Second exponential martingale

We are now ready to introduce our second exponential martingale, also known astheKella-Whittmartingale. See Sect. 3.7 for historical remarks regardingits name.

Theorem 3.6.For θ > 0 and x≥ 0,

Mxt := ψ(θ )

∫ t

0e−θYx

s ds+1−e−θYxt −θ (x∨Xt), t ≥ 0 (3.7)

is aP-martingale with respect toF.

Proof. Let us start by showing that

E[|Mxt |]< ∞,

for all t,x ≥ 0. To this end, suppose thate1 is an independent and exponentiallydistributed random variable. We know from Corollary 2.6 that Xe1 is exponentiallydistributed with parameterΦ(1). In particular, this implies that

E[Xe1] =

∫ ∞

0e−t

E[Xt ]dt < ∞.

SinceX is an increasing process, it follows thatE[Xt ]< ∞, for all t ≥ 0.Using the triangle inequality for each of the terms inMx

t , we may now estimate

E[|Mxt |]≤ ψ(θ )t +2+θE[Xt ]< ∞,

for x, t ≥ 0.Next, use the Markov property ofYx to write, forx,s, t ≥ 0,

x∨Xt+s =Yxt+s+Xt+s = Yz

s |z=Yxt+ Xs+Xt = (z∨ Xs)|z=Yx

t+Xt ,

whereXs= Xt+s−Xt andYz := (z∨ X)− X is independent ofYxu : u≤ t and equal

in law toYz. Using this decomposition, it is straightforward to show that

E[Mxt+s|Ft ] = ψ(θ )

∫ t

0e−θYx

u du+1−θXt

+ E

[ψ(θ )

∫ s

0e−θYz

u du−e−θYzs −θ (z∨Xs)

]∣∣∣∣z=Yx

t

.

The proof is thus complete as soon as we show that, for allz,s≥ 0,

E

[ψ(θ )

∫ s

0e−θYz

u du−e−θYzs −θ (z∨Xs)

]=−e−θz−θz.

In order to achieve this goal, we shall develop the left-handside above using theso-calledchain rule for right-continuous functions of bounded variation. We have


that

e−θYzs = e−θz−θ

∫

(0,s]e−θYz

u d(Yzu)

c+Ns

∑i=1

[e−θYzTi −e−θYz

Ti− ], (3.8)

for z,s≥ 0, where(Yzu)

c is the continuous part ofYz. Note that

∫

(0,s]e−θYz

u d(Yzu)

c =∫

(0,s]e−θYz

u d(z∨Xu)−c∫ s

0e−θYz

u du

=∫

(0,s]1(Yz

u=0)e−θYz

u d(z∨Xu)−c∫ s

0e−θYz

u du

= (z∨Xs)− z−c∫ s

0e−θYz

u du,

where in the second equality we have used the factYzu = 0 on the set of times that

the processz∨Xu increments. We may now take expectations in (3.8) to deduce that

E

[e−θYz

s +θ (z∨Xs)]

= e−zθ +θz+E

[cθ∫ s

0e−θYz

u du+Ns

∑i=1

e−θYzTi−(e−θξi −1)

]

= e−zθ +θz+cθE[∫ s

0e−θYz

u du

]+λ

∫

(0,∞)(e−θx−1)F(dx)E

[∫ s

0e−θYz

u du

]

= e−zθ +θz+ψ(θ )E[∫ s

0e−θYz

u du

],

where we have applied Theorem 3.4 in the second equality. Theproof is now com-plete.

3.4 Duality

For our main application of the Kella-Whitt martingale, we need to address one ad-ditional property of the Cramer-Lundberg process, which follows as a consequenceof the fact that it is also a Levy process. This property concerns the issue ofduality.

Lemma 3.7 (Duality Lemma). For each fixed t> 0, define the time-reversed pro-cess

X(t−s)−−Xt : 0≤ s≤ tand the dual process,

−Xs : 0≤ s≤ t.Then the two processes have the same law underP.

Proof. Define the processRs= Xt−X(t−s)− for 0≤ s≤ t. UnderP, we haveR0 = 0almost surely, ast is a jump time with probability zero. As can be seen from Fig. 3.1,

3.4 Duality 23

Fig. 3.1 A realisation of the trajectory ofXs : 0≤ s≤ t and ofX(t−s)−−Xt : 0≤ s≤ t, respec-tively.

the paths ofR are obtained from those ofX by a rotation about 180, with an ad-justment of the continuity at the jump times, so that its paths are almost surelyright-continuous with left limits. The stationary independent increments ofX implydirectly that the same is true ofR. This putsR in the class of Levy processes. More-over, for each 0≤ s≤ t, the distribution ofRs is identical to that ofXs. It followsthat

E(eλ Rs) = eψ(λ )s,

for all 0≤ s≤ t < ∞ andλ ≥ 0. HenceRhas the same law asX.

One interesting feature that follows as a consequence of theDuality Lemma isthe relationship between the running supremum, the runninginfimum, the processreflected in its supremum and the process reflected in its infimum. The last fourobjects are, respectively,

Xt = sup0≤s≤t

Xs, Xt = inf0≤s≤t

Xs

Xt −Xt : t ≥ 0 and Xt −Xt : t ≥ 0.

Lemma 3.8.For each fixed t> 0, the pairs(Xt ,Xt −Xt) and (Xt −Xt ,−Xt) havethe same distribution underP.

Proof. DefineRs = Xt −X(t−s)− for 0≤ s≤ t, as in the previous proof, and writeRt = inf0≤s≤t Rs. Using right-continuity and left limits of paths we may deduce that

(Xt ,Xt −Xt) = (Rt −Rt ,−Rt)

almost surely. Now appealing to the Duality Lemma we have that Rs : 0≤ s≤ tis equal in law toXs : 0≤ s≤ t underP and the result follows.


3.5 Distribution of the minimum

We are now able to deliver the promised result concerning thelaw of the minimum.

Theorem 3.9.Let Xt = inf0≤u≤t Xu and suppose thateq is an exponentially dis-tributed random variable with parameter q> 0, which is independent of the processX. Then, forθ > 0,

E(eθXeq

)=

q(θ −Φ(q))Φ(q)(ψ(θ )−q)

, (3.9)

where the right-hand side is understood in the asymptotic sense whenθ =Φ(q), i.e.q/Φ(q)ψ ′(Φ(q)).

Proof. Let us first consider the case thatθ ,q > 0 andθ 6= Φ(q). Let Yt = Y0t =

Xt −Xt . By an application of Fubini’s theorem together with Lemma 3.8,

E

[∫ eq

0e−θYs ds

]=∫ ∞

0e−qs

E(e−θYs

)ds=

1qE(e−θYeq

)=

1qE(eθXeq

).

From Theorem 3.6, we have thatE(M0

eq

)= E(M0

0) = 0 and hence we obtain

ψ(θ )−qq

E(eθXeq

)= θ E

(Xeq

)−1.

Recall from Corollary 2.6 thatXeq is exponentially distributed with parameterΦ(q)and henceE

(Xeq

)= 1/Φ(q). It follows that

ψ(θ )−qq

E(eθXeq

)=

θ −Φ(q)Φ(q)

. (3.10)

For the case thatq> 0 andθ = Φ(q), the result follows from the case thatθ 6=Φ(q)by taking limits asθ →Φ(q).

3.6 The long term behaviour

Let us conclude this chapter by returning to earlier remarksmade in Sect. 1.2regarding the long term behaviour of the Cramer-Lundberg process. Recall thatψ ′(0+) = c− λ µ , wherec is the premium rate,λ is the rate of claim arrivalsand µ is their common mean. It is clear from (1.4) that, whenψ ′(0+) > 0, wehave limt→∞ Xt = ∞ and whenψ ′(0+) < 0, limt→∞ Xt = −∞. For the remainingcase, whenψ ′(0+) = 0, the Strong Law of Large Numbers is not as informative.We can, however, use our previous results on the law of the maximum and min-imum of X to determine the long term behaviour ofX. Specifically, the lemmabelow shows that, whenψ ′(0+) = 0, the processX oscillatesin the sense thatlimsupt→∞ Xt =− lim inf t→∞ Xt = ∞.

3.7 Comments 25

Lemma 3.10.We have that

(i) X∞ and−X∞ are either infinite almost surely or finite almost surely,(ii) X∞ = ∞ if and only if ψ ′(0+)≥ 0,(iii) X∞ =−∞ if and only if ψ ′(0+)≤ 0.

Proof. On account of the strict convexityψ , we have thatΦ(0) > 0 if and only ifψ ′(0+)< 0. Hence,

limq↓0

qΦ(q)

=

0 if ψ ′(0+)≤ 0ψ ′(0+) if ψ ′(0+)> 0.

In the case thatψ ′(0+) ≥ 0 above, i.e. the caseΦ(0) = 0, to compute the limit,we make use of the fact thatq/Φ(q) can otherwise be writtenψ(Φ(q))/Φ(q). Bytaking the limit asq tends to zero in the identity (3.9), we now have that

E(eθX∞

)=

0 if ψ ′(0+)≤ 0ψ ′(0+)θ/ψ(θ ) if ψ ′(0+)> 0.

(3.11)

In the first of the two cases above, it is clear thatP(−X∞ = ∞) = 1. In the secondcase, taking limits asθ ↑ ∞, one sees thatP(−X∞ = ∞) = 0.

Next, recall from Corollary 2.6 thatX∞ is exponentially distributed with param-eterΦ(0). In particular,X∞ is almost surely infinite whenψ ′(0+) ≥ 0 and almostsurely finite whenψ ′(0+)< 0.

Putting this information together, the statements (i)–(iii) are easily recovered.

3.7 Comments

The fact that the processYx is a Markov process for eachx≥ 0 is well known fromqueuing theory, where the processYx can be seen as the workload in anM/G/1queue (when the initial workload at time zero is x). Theorem 3.4 is an exampleof the so-called compensation formula which can be stated for general Poisson in-tegrals. See for example Chapter XII.1 of Revuz and Yor (2004). The Kella-Whittmartingale was first introduced in Kella and Whitt (1992) in the setting of a generalLevy process. It closely related to the so-called Kennedy and Azema-Yor, martin-gales, both of which have previously been studied in the setting of Brownian motion.The Duality Lemma is also well known for (and in fact originates from) the theoryof random walks and is justified using an almost identical proof. See for exampleChapter XII of Feller (1971).

Chapter 4Scale functions and ruin probabilities

The two main results from the previous chapters, concerningthe law of the max-imum and minimum of the Cramer–Lundberg process, can now beput to use inorder to establish our first results concerning the classical ruin problem. We shallintroduce the so-calledscale functions, which will prove to be indispensable, bothin this chapter and later, when describing various distributional features of the ruinproblem.

4.1 Scale functions and the probability of ruin

For a given Cramer–Lundberg processX with Laplace exponentψ , we want todefine a family of functions, indexed byq≥ 0, which we shall denote byW(q) :R→ [0,∞). For allq≥ 0 we shall setW(q)(x) = 0 for x< 0. The next theorem willalso serve as a definition forW(q) on [0,∞).

Theorem 4.1.For all q ≥ 0 there exists a function W(q) on [0,∞) defined to be theunique non-decreasing, right-continuous function whose Laplace transform is givenby ∫ ∞

0e−β xW(q)(x)dx=

1ψ(β )−q

, β > Φ(q). (4.1)

For convenience, we shall always writeW in place ofW(0). Typically, we shallrefer to the functionsW(q) asq-scale functions, but we shall also refer toW as justthescale function.

Proof (of Theorem 4.1).First assume thatψ ′(0+) > 0. With a pre-emptive choiceof notation, we shall define the function

W(x) =1

ψ ′(0+)Px(X∞ ≥ 0), x∈ R. (4.2)

27

28 4 Scale functions and ruin probabilities

ClearlyW(x) = 0 for x < 0 and it is non-decreasing and right-continuous since itis also proportional to the distribution functionP(−X∞ ≤ x). Integration by partsshows that, on the one hand,

∫ ∞

0e−β xW(x)dx =

1ψ ′(0+)

∫ ∞

0e−β x

P(−X∞ ≤ x)dx

=1

ψ ′(0+)β

∫

[0,∞)e−β x

P(−X∞ ∈ dx)

=1

ψ ′(0+)βE

(eβ X∞

). (4.3)

On the other hand, recalling (3.11), we also have that

P

(eβ X∞

)=

ψ ′(0+)βψ(β )

, β ≥ 0.

When combined with (4.3), this gives us (4.1) for the caseq= 0 andψ ′(0+)> 0.Next we deal with the case thatq > 0 or thatq = 0 and ψ ′(0+) < 0. To this

end, again making use of a pre-emptive choice of notation, let us define the non-decreasing and right-continuous function

W(q)(x) = eΦ(q)xWΦ(q)(x), x≥ 0, (4.4)

whereWΦ(q) plays the role ofW, but now for the process(X,PΦ(q)). Note in partic-

ular that, by Theorem 2.3,(X,PΦ(q)) has Laplace exponent

ψΦ(q)(θ ) = ψ(θ +Φ(q))−q, θ ≥ 0. (4.5)

Henceψ ′Φ(q)(0+)= ψ ′(Φ(q))> 0, which ensures thatWΦ(q) is well defined by theprevious part of the proof. Taking Laplace transforms, we have, forβ > Φ(q),

∫ ∞

0e−β xW(q)(x)dx =

∫ ∞

0e−(β−Φ(q))xWΦ(q)(x)dx

=1

ψΦ(q)(β −Φ(q))

=1

ψ(β )−q,

thus completing the proof for the case thatq> 0 or thatq= 0 andψ ′(0+)< 0.Finally, we deal with the case thatq = 0 andψ ′(0+) = 0. SinceWΦ(q)(x) is

an increasing function, we may also treat it as a distribution function of a measurewhich, with as an abuse of notation, we also callWΦ(q). Integrating by parts thusgives us, forβ > 0,

∫

[0,∞)e−β xWΦ(q)(dx) =

βψΦ(q)(β )

. (4.6)

4.1 Scale functions and the probability of ruin 29

Note that the assumptionψ ′(0+)= 0 implies thatΦ(0) = 0 and hence, forθ ≥ 0,

limq↓0

ψΦ(q)(θ ) = limq↓0

[ψ(θ +Φ(q))−q] = ψ(θ ).

One may appeal to (4.6) and the Extended Continuity Theorem for Laplace trans-forms, see for example Theorem XIII.1.2a of Feller (1971), to deduce that, since

limq↓0

∫

[0,∞)e−β xWΦ(q)(dx) =

βψ(β )

,

there exists a measureW∗ such thatW∗(x) :=W∗[0,x] = limq↓0WΦ(q)(x) and

∫

[0,∞)e−β xW∗(dx) =

βψ(β )

.

Integration by parts shows thatW∗ satisfies

∫ ∞

0e−β xW∗(x)dx=

1ψ(β )

,

for β > 0, as required. It is clear from its definition thatW :=W∗ is non-decreasing,right-continuous and satisfies (4.1).

With the definition of scale functions in hand, we can return to the problem ofruin. The following corollary follows as a simple consequence of Laplace inversionof the identity in Theorem 3.9, taking account of (4.1).

Corollary 4.2. For x,q> 0,

P(−Xeq∈ dx) =

qΦ (q)

W(q)(dx)−qW(q)(x)dx. (4.7)

In the above formula, thanks to (4.4), the functionW(q) is increasing and hencethe measureW(q)(dx), x≥ 0, makes sense (albeit being another abuse of notation).Formula (4.7) can also be stated whenq= 0, providingψ ′(0+)> 0. In that case, aswe have seen before, the termq/Φ(q) should be understood, in the limiting sense,as equal toψ ′(0+).

We complete this section with our main result about ruin probabilities, usingscale functions. To this end, let us define the functions

Z(q)(x) = 1+q∫ x

0W(q)(y)dy, x∈ R,

for q≥ 0.

Theorem 4.3 (Time to ruin).Recall that

τ−0 := inft > 0 : Xt < 0


and thate−∞ := 0. For any x∈ R and q≥ 0,

Ex(e−qτ−0 ) = Z(q)(x)− qΦ (q)

W(q)(x) , (4.8)

where we understand q/Φ (q) in the limiting sense for q= 0, so that

Px(τ−0 < ∞) =

1−ψ ′(0+)W(x) if ψ ′(0+)≥ 01 if ψ ′(0+)< 0.

(4.9)

Proof. Appealing to (4.7), we have, forx≥ 0,

Ex(e−qτ−0 ) = Px(eq > τ−0 )

= Px(Xeq< 0)

= P(−Xeq> x)

= 1−P(−Xeq≤ x)

= 1+q∫ x

0W(q)(y)dy− q

Φ (q)W(q)(x)

= Z(q)(x)− qΦ (q)

W(q)(x). (4.10)

Note that, sinceZ(q)(x) = 1 andW(q)(x) = 0 for all x ∈ (−∞,0), the statement isvalid for all x∈R. The proof is now complete for the case thatq> 0.

In order to deal with the caseq= 0, recall our previous trick in writing

limq↓0

q/Φ (q) = limq↓0

ψ(Φ(q))/Φ(q).

If ψ ′(0+)≥ 0, i.e. the process drifts to infinity or oscillates, thenΦ(0) = 0 and thelimit is equal toψ ′(0+). Otherwise, whenΦ(0) > 0, the aforementioned limit iszero. The proof is thus completed by taking the limit inq in (4.8).

The last part of the above theorem can also be recovered directly from the defi-nition of W in the case thatψ ′(0+) > 0, see (4.2). Moreover, given the discussionin Sect. 3.6, the probability of ruin whenψ ′(0+)≤ 0 is obviously 1.

4.2 Connection with the Pollaczek–Khintchine formula

In Theorem 1.3, we gave the classical Pollaczek-Khintchineformula for the proba-bility of ruin in the case thatψ ′(0+)> 0. Compared with (4.9), it is not immediatelyobvious how these two formulae relate to one another. Let us therefore spend a littletime to make the connection between the two, first with an analytical explanationand then with a probabilistic explanation.

4.2 Connection with the Pollaczek–Khintchine formula 31

Analytical explanation.Let us start by noting that, just as in formula (4.6), wecan integrate the Laplace transform ofW by parts to show that

∫

[0,∞)e−β xW(dx) =

βψ(β )

, β > 0.

Alternatively, this also follows from the definition (4.2) and the expression for theLaplace transform of−X∞, given in (3.11). Next, note thatψ ′(0+) = c−λ µ > 0implies thatµ < ∞ and that

ρ := λ µ/c< 1.

This inequality also implies that

λ µc

∫ ∞

0e−β x 1

µF(x)dx< 1.

Hence, recalling the representation ofψ given in (2.4), we can write, forβ > 0,

βψ(β )

=1c

1

1− λ µc

∫ ∞0 e−β x 1

µ F(x)dx,

=1c

∞

∑k=0

ρk(∫ ∞

0e−β x 1

µF(x)dx

)k

, (4.11)

where we recall thatF(x) = 1−F(x). Next note that

η(dx) :=1µ

F(x)dx, x≥ 0,

is a probability measure. For eachk≥ 0, denote byη∗k(dx), x≥ 0, its k-fold con-volution, where we understandη∗0(dx) := δ0(dx), x≥ 0, the Dirac delta measurewhich places an atom at zero. Since, forβ > 0 andk≥ 0,

∫

[0,∞)e−β xη∗k(dx) =

(∫ ∞

0e−β x 1

µF(x)dx

)k

,

we may apply Laplace inversion to the right-hand side of (4.11) and conclude that,for x≥ 0,

W(dx) =1c

∞

∑k=0

ρkη∗k(dx).

Said another way, we have, forx≥ 0,

W(x) =1c

∞

∑k=0

ρkη∗k(x). (4.12)

Returning to the formula in (4.9), whenψ ′(0+) = c−λ µ > 0, we now see that


1−Px(τ−0 < ∞) = (1−ρ)∞

∑k=0

ρkη∗k(x), (4.13)

as stated in Theorem 1.3.

Probabilistic explanation. The Pollaczek-Khintchine formula can also be recov-ered by looking at the successive minima of the processX. To this end, let us setΘ0 = 0 and sequentially define, for allk≥ 1 such thatΘk−1 < ∞,

Θk = inft >Θk−1 : Xt < XΘk−1,

with the usual understanding that inf /0= ∞. As long as they are finite, the timesΘk

are the times of successive new minima.The strong Markov property implies that, for eachk≥ 1 such thatΘk−1 < ∞,

the pair(Θk−Θk−1,XΘk−XΘk−1) is independent ofFΘk−1 and equal in law to thepair(τ−0 ,Xτ−0

), where we understandXΘk :=∞ whenΘk =∞ and, similarly,Xτ−0:=∞

whenτ−0 = ∞. Fork≥ 1, define onΘk−1 < ∞

∆k =−(XΘk−XΘk−1).

The eventτ−0 = ∞ underPx, x≥ 0, corresponds to the event that

ν−1

∑n=1

∆n≤ x

,

whereν = mink≥ 1 : Θk = ∞.1 See Fig 4.1. By the strong Markov property, theindex ν is the time of first success in an independent sequence of Bernoulli trialswith probability “success” 1− ρ := P(Θ1 = ∞) = P(τ−0 = ∞). In other words,νis geometrically distributed. Moreover,ν is independent of the outcome of each ofaforesaid independent trials, each of which “fail”, delivering a random value whichis distributed according to the measureη(dx) := P(∆1 ∈ dx) = P(−Xτ−0

∈ dx|τ−0 <

∞), x> 0.In conclusion, we see that

1−Px(τ−0 < ∞) = (1− ρ) ∑k≥0

ρkη∗k(x), x≥ 0. (4.14)

Comparing the formulae (4.13) and (4.14) whenx = 0, we see thatP(τ−0 < ∞) =ρ = ρ, and hence it follows thatη = η .

The following corollary falls straight out of the above comparison.

Corollary 4.4. If ψ ′(0+)> 0, then

1 We use the standard convention that∑0n=1 · := 0.

4.3 Gambler’s ruin 33

x

Fig. 4.1 A path of the Cramer–Lundberg process which drifts to∞ before passing below 0. Thehorizontal line segments mark the successive minima. The vertical distance between each horizon-tal line segment represent the quantities∆n.

P(τ−0 < ∞) =λ µc

and P(−Xτ−0≤ x|τ−0 < ∞) =

1µ

∫ x

0F(y)dy, x≥ 0.

4.3 Gambler’s ruin

A slightly more elaborate version of the ruin problem is to consider the event that acertain surplus level, saya≥ 0, can be achieved before ruin. This is also known asthegambler’s ruin problem.

Theorem 4.5.For all q≥ 0, a> 0 and x≤ a,

Ex

(e−qτ+a 1τ+a <τ−0

)=

W(q)(x)

W(q)(a). (4.15)

Proof. First we deal with the case thatq = 0 andψ ′(0+) > 0 as in the previousproof. Since we have identifiedW(x) = Px(X∞ ≥ 0)/ψ ′(0+), a simple argument,using the law of total probability and the strong Markov property, now yields, forx∈ [0,a],

Px(X∞ ≥ 0)

= Ex

(Px

(X∞ ≥ 0|Fτ+a

))

= Ex

(1(τ+a <τ−0 )Pa (X∞ ≥ 0)

)+Ex

(1(τ+a >τ−0 )PXτ−0

(X∞ ≥ 0)

). (4.16)

The first term on the right-hand side of (4.16) is equal to

Pa(X∞ ≥ 0) Px(τ+a < τ−0

).


The second term on the right-hand side of (4.16) turns out to be equal to zero. To seewhy, note thatXτ−0

< 0 and the claim follows by virtue of the fact thatPx(X∞ ≥ 0) =0 for all x< 0. We may now deduce that

Px(τ+a < τ−0

)=

W(x)W(a)

, x∈ [0,a], (4.17)

and clearly the same equality holds even whenx < 0 as both left- and right-handside are identically equal to zero.

Next, we deal with the caseq > 0. Making use of the Esscher transform andrecalling thatXτ+a = a, we have that

Ex

(e−qτ+a 1(τ+a <τ−0 )

)= e−Φ(q)(a−x)

Ex

(e

Φ(q)(Xτ+a−x)−qτ+a 1(τ+a <τ−0 )

)

= e−Φ(q)(a−x)P

Φ(q)x

(τ+a < τ−0

)

= e−Φ(q)(a−x)WΦ(q)(x)

WΦ(q)(a)

=W(q)(x)

W(q)(a).

Finally, to deal with the case thatq = 0 andψ ′(0+) ≤ 0, one needs only totake limits asq ↓ 0 in the above identity, making use of monotone convergence onthe left-hand side and continuity inq on the right-hand side, using the ContinuityTheorem for Laplace transforms.

We can also consider the converse event that ruin occurs prior to achieving adesired surplus ofa≥ 0.

Theorem 4.6.For any x≤ a and q≥ 0,

Ex

(e−qτ−0 1(τ−0 <τ+a )

)= Z(q)(x)−Z(q)(a)

W(q)(x)

W(q)(a). (4.18)

Proof. Fix q> 0. We have forx≥ 0,

Ex(e−qτ−0 1(τ−0 <τ+a )) = Ex(e−qτ−0 )−Ex(e−qτ−0 1(τ+a <τ−0 )).

Applying the strong Markov property atτ+a and using the fact thatXτ+a = a onτ+a < ∞, we also have that

Ex(e−qτ−0 1(τ+a <τ−0 )) = Ex(e−qτ+a 1(τ+a <τ−0 ))Ea(e−qτ−0 ).

Appealing to (4.8) and (4.15), we now have that

4.4 Comments 35

Ex(e−qτ−0 1(τ−0 <τ+a )) = Z(q)(x)− q

Φ(q)W(q)(x)

−W(q)(x)

W(q)(a)

(Z(q)(a)− q

Φ(q)W(q)(a)

),

and the required result follows in the case thatq> 0. The case thatq= 0 is againdealt with by taking limits asq ↓ 0.

4.4 Comments

The name ‘scale function’ forW was first used by Bertoin (1992) to reflect theanalogous role it plays in (4.15) to scale functions for diffusions. The gambler’s ruinproblem (also known as the two-sided exit problem) has a longhistory, starting withthe early work of Zolotarev (1964) and Takacs (1966), followed by Rogers (1990),all of whom dealt with (4.15) in the caseq= 0. The case thatq> 0 was dealt with byGerber (1972), Korolyuk (1975a), and later by Bertoin (1997). A recent summaryof the theory of scale functions and its applications can be found in Cohen et al.(2012).

Chapter 5The Gerber–Shiu measure

Having introduced scale functions, we are now ready to look at the Gerber–Shiumeasure in detail. In this chapter, we shall develop an idea from the previous chapter,involving Bernoulli trials of excursions from the minimum,to provide an identityfor the expected total occupation time of the Cramer–Lundberg process in Borel setsuntil ruin. This identity will then play a key role in identifying an expression for theGerber–Shiu measure. In fact, the analysis we give will workequally well in thecontext of the gambler’s ruin problem.

5.1 Decomposing paths at the minimum

The main objective in this section is to prove the following decoupling for the path ofthe Cramer–Lundbergprocess when sampled over an independent and exponentiallydistributed random time,eq, with rateq> 0.

Theorem 5.1.For all q > 0, the pair of random variables Xeq −Xeqand Xeq

areindependent.

Proof. The proof makes use of the path decomposition we previously employed forthe probabilistic explanation of the Pollaczek-Khintchine formula. Recall that, inthat setting, we sequentially definedΘ0 = 0 and, for allk≥ 1 such thatΘk−1 < ∞,

Θk = inft >Θk−1 : Xt < XΘk−1.

Moreover, on the eventΘk−1 < ∞ we defined∆k = −(XΘk −XΘk−1). Formallyspeaking, the timesΘ1,Θ2, · · · describe the right end points ofexcursions from theminimum. The latter may be thought of as the sequence of segments of the trajectoryof X given byXΘk−1+t : t ∈ (0,Θk] for all k≥ 1 such thatΘk−1 < ∞.

Although the discussion in the context of the Pollaczek-Khintchine formula fo-cused exclusively on the case thatψ ′(0+) > 0, the timesΘk are still well definedwhenψ ′(0+)≤ 0. In fact, in this regime, sinceX∞ = −∞ almost surely, it followsthatΘk < ∞ almost surely for allk≥ 1.

37

38 5 The Gerber–Shiu measure

Now suppose thate(k)q : k ≥ 1 is a sequence of independent and identicallydistributed random variables with parameterq> 0. Let

ℓ := mink≥ 1 :Θk−Θk−1 > e(k)q

be the index of the first excursion from the minimum which, in duration, exceeds thecorrespondingly indexed exponential random variable in the sequencee(k)q : k≥ 1.By the lack of memory property, we can now identify the pair(Xeq−Xeq

,−Xeq) as

equal in law to the pair(

XΘℓ−1+e(ℓ)q

−XΘℓ−1,ℓ−1

∑j=1

∆ j

). (5.1)

Appealing again to the concept of Bernoulli trials, it is clear that both the randomvariableℓ and theℓ-th excursion from the minimum will be independent from theprecedingℓ−1 excursions from the minimum. In particular, this implies that thepair in (5.1) is independent, and hence so is the pair(Xeq−Xeq

,−Xeq).

Note that the above proof allows us to say a little more in the description of(Xeq−Xeq

,−Xeq), thanks to the representation (5.1). Indeed,Xeq−Xeq

is equal in

law to Xeq conditional oneq < τ−0 . Moreover, each of the∆ j in the sum are i.i.d.and equal in distribution to−Xτ−0

conditional onτ−0 < eq, andℓ is independent

and geometrically distributed with parameterP(eq < τ−0 ).

5.2 Resolvent densities

As an intermediate step to deriving a closed-form expression for the Gerber–Shiumeasure, we are interested in computing the so-calledq-resolvent measurefor theCramer–Lundberg process,X, killed on exiting [0,∞). Said another way, we areinterested in characterising the measure

U (q)(a,x,dy) :=∫ ∞

0e−qt

Px(Xt ∈ dy, t < τ [0,a])dt, y∈ [0,a],

wherea> 0, q≥ 0 andτ [0,a] = τ+a ∧ τ−0 .

If, for eachx ∈ [0,a], a density ofU (q)(a,x,dy) exists with respect to Lebesguemeasure, then we call it theresolvent densityand denote it byu(q)(a,x,y). (Notethat this density can only be defined Lebesgue almost everywhere.) It turns out that,for each Cramer–Lundberg process, not only does a potential density exist, but wecan write it in semi-explicit terms with the help of scale functions. Note, in thestatement of the result, it is implicitly understood thatW(q)(z) is identically zero forz< 0.

5.2 Resolvent densities 39

Theorem 5.2.For each q≥ 0 and a> 0, the density u(q)(a,x,y) exists and is equalto

u(q)(a,x,y) =W(q)(x)W(q)(a− y)

W(q)(a)−W(q)(x− y), x,y∈ [0,a], (5.2)

Lebsgue almost everywhere.

Proof. Define, for allx,y≥ 0 andq> 0,

R(q)(x,dy) =∫ ∞

0e−qt

Px(Xt ∈ dy, t < τ−0 )dt.

This is theq-resolvent measure for the processX when killed on exiting[0,∞). Notethat, for the same parameter regimes ofx,y andq, we can also write

R(q)(x,dy) =1qPx(Xeq ∈ dy,Xeq

≥ 0),

where, as usual,eq is an independent, exponentially distributed random variablewith parameterq> 0. Appealing to Theorem 5.1, we have that

R(q)(x,dy) =1qP(x+(Xeq−Xeq

)+Xeq∈ dy,−Xeq

≤ x)

=1q

∫

[x−y,x]P(−Xeq

∈ dz)P(Xeq−Xeq∈ dy− x+ z).

Note that the delimiter on the integral appears by virtue of the fact that, on the onehand−Xeq

≤ x and, on the other hand, ifx+(Xeq−Xeq)+Xeq

≤ y, then−Xeq≥ x−

y. By duality,Xeq−Xeqis equal in distribution toXeq, which itself is exponentially

distributed with parameterΦ(q). In addition, the law of−Xeqhas been identified in

Corollary 4.2. Using these facts we may write, forq,x,y≥ 0,

R(q)(x,dy) =

∫

[x−y,x]

(1

Φ(q)W(q)(dz)−W(q)(z)dz

)Φ(q)e−Φ(q)(y−x+z)

dy.

In particular, this shows that, forq,x,y≥ 0, there exists a density, sayr(q)(x,y), forthe measureR(q)(x,dy). Now note that

d[e−Φ(q)zW(q)(z)] = e−Φ(q)z[W(q)(dz)−Φ(q)W(q)(z)dz],

and, hence, straightforward integration gives us

r(q)(x,y) = e−Φ(q)yW(q)(x)−W(q)(x− y), x,y,q≥ 0.

Finally, we may use the expression forr(q) to compute the potential densityu(q).First note that, with the help of the strong Markov property,


qU(q)(a,x,dy) = Px(Xeq ∈ dy,Xeq≥ 0,Xeq ≤ a)

= Px(Xeq ∈ dy,Xeq≥ 0)

−Px(Xeq ∈ dy,Xeq≥ 0,Xeq > a)

= Px(Xeq ∈ dy,Xeq≥ 0)

−Px(Xτ [0,a] = a,τ [0,a] < eq)Pa(Xeq ∈ dy,Xeq≥ 0).

The first and third of the three probabilities on the right-hand side above have beencomputed in the previous paragraph, the second probabilityis equal to

Ex(e−qτ+a ;τ+a < τ−0 ) =

W(q)(x)

W(q)(a).

In conclusion, we have that, forq≥ 0 andx∈ [0,a], U (q)(a,x,dy) has a density

r(q)(x,y)−W(q)(x)

W(q)(a)r(q)(a,y), y∈ [0,a],

which, after a short amount of algebra, can be shown to be equal to the right-handside of (5.2).

To complete the proof whenq = 0, one may take limits in (5.2), noting thatthe measureU (q)(a,x,dy) is monotone increasing inq and that the scale functionis continuous inq (again, thanks to the Extended Continuity Theorem for Laplacetransforms).

The above proof contains the following corollary for theq-resolvent measureR(q)(x,dy).

Corollary 5.3. Fix q≥ 0. The q-resolvent measure for X killed on exiting[0,∞) hasdensity given by

r(q)(x,y) = e−Φ(q)yW(q)(x)−W(q)(x− y),

for x,y≥ 0.

5.3 More on Poisson integrals

Before putting together the results in the previous sectionto derive an expression forthe Gerber–Shiu measure, we need to briefly return to the issue of Poisson integrals.One can improve upon the result in Theorem 3.4 with a little more work.

Theorem 5.4.Suppose that f: R× [0,∞)× (−∞,0]× (0,∞)→ R is bounded andmeasurable. Then, for all t≥ 0,

5.4 Gerber–Shiu measure and gambler’s ruin 41

Ex

(Nt

∑i=1

f (Ti ,XTi−, XTi−, XTi−, ξi)

)= λEx

(∫ t

0

∫

(0,∞)f (s,Xs, Xs, Xs, u)F(du)ds

).

In particular, by taking limits on both sides of the above equality as t↑∞, thanks tomonotonicity, the same result holds when both t and Nt are replaced by∞.

One can reconstruct the proof of this theorem by following the main steps ofTheorem 3.4. In that case, it will be convenient to first show that, in the spirit ofTheorem 3.1, the triplet(Xt , Xt , Xt) : t ≥ 0 is a Markov process.

5.4 Gerber–Shiu measure and gambler’s ruin

We now have all the tools we need to provide a characterisation of the Gerber–Shiumeasure (1.6) in terms of scale functions. In fact, we shall establish an identity fora slightly more general measure. With a slight abuse of notation, for a > 0 andx∈ [0,a], define

K(q)(a,x,dy,dz)=Ex

[e−qτ−0 ;−Xτ−0

∈ dy, Xτ−0 −∈ dz, τ−0 < τ+a

], z∈ [0,a],y≥ 0.

Note that the Gerber–Shiu measure, previously calledK(q)(x,dy,dz), satisfies

K(q)(x,dy,dz) = K(q)(∞,x,dy,dz),

for y,z≥ 0. Here is the main result of this chapter.

Theorem 5.5 (Gerber–Shiu measure).Fix q≥ 0 and a> 0. Then

K(q)(a,x,dy,dz) = λ

W(q)(x)W(q)(a− z)−W(q)(a)W(q)(x− z)

W(q)(a)

F(z+dy)dz,

for x,z∈ [0,a] and y≥ 0. Moreover,

K(q)(x,dy,dz) = λ

e−Φ(q)yW(q)(x)−W(q)(x− y)

F(z+dy)dz, (5.3)

for y,z≥ 0.

Proof. Fix q≥ 0 andx∈ [0,a]. For the first identity, it suffices to show that, for allbounded continuousf : (0,∞)× [0,a] :→ [0,∞),

Ex

[e−qτ−0 f (−Xτ−0

,Xτ−0 −);τ−0 < τ+a

]

= λ∫ a

0

∫

(0,∞)f (y,z)u(q)(a,x,z)F(z+dy)dz. (5.4)

To this end, note that


τ−0 < τ+a =∞⋃

i=1

XTi < 0, XTi− ≤ a, XTi− ≥ 0,

where the union is taken over disjoint events. It follows with the help of Theorem5.4 that

Ex

[e−qτ−0 f (−Xτ−0

,Xτ−0 −);τ−0 < τ+a

]

= Ex

[∞

∑i=1

1(XTi−−ξi<0)1(XTi−≤a)1(XTi−≥0)e−qTi f (−XTi−+ ξi, XTi−)

]

= λ∫

(0,∞)Ex

[∫ ∞

01(u>Xt−)1(Xt−≤a)1(Xt−≥0)e

−qt f (−Xt−+u, Xt−)dt

]F(du)

= λ∫

(0,∞)Ex

[∫ ∞

01(u>Xt−)1(t<τ [0,a])e

−qt f (−Xt−+u, Xt−)dt

]F(du)

= λ∫

(0,∞)

∫

[0,a]

∫ ∞

01(u>z)e

−qt f (u− z, z)Px(Xt ∈ dz, t < τ [0,a])dtF(du). (5.5)

Recalling the definition ofU (q)(a,x,dz), it follows that

Ex

[e−qτ−0 f (−Xτ−0

,Xτ−0 −);τ−0 < τ+a

]

= λ∫

(0,∞)

∫ a

01(u>z) f (u− z, z)u(q)(a,x,z)dzF(du).

The right-hand side above is equal to (5.4), after a straightforward application ofFubini’s Theorem and a change of variables.

For the second part of the theorem, use monotonicity ina on the left- and right-hand side of (5.5) and take limits asa↑∞. The consequence of this is that we recoverthe identity

Ex

[e−qτ−0 f (−Xτ−0

,Xτ−0 −);τ−0 < ∞

]= λ

∫ ∞

0

∫

(0,∞)f (y,z)r(q)(x,z)F(z+dy)dz,

for all x,q≥ 0, from which (5.3) follows.

5.5 Comments

In the broader context, Theorem 5.1 is a simple example of oneof the severalstatements that concerns the so-calledWiener-Hopf factorisationfor Levy pro-cesses. See Chapter VI of Bertoin (1996) or Chapter 6 of Kyprianou (2013). Themethod of analysing the path of the Cramer–Lundberg process (and indeed anyrandom walk) through a sequence of excursions from the minimum was largely

5.5 Comments 43

popularised by Feller. See for example Chapter XII of Feller(1971). Many of thecomputations concerning resolvent densities are taken directly from Bertoin (1997)who deals with general spectrally negative Levy processes. However, older liter-ature dealing with the current setting also exists; see for example Suprun (1976),Korolyuk (1974), Korolyuk (1975a), Korolyuk (1975b), Korolyuk et al. (1976) andKorolyuk and Borovskich (1981).

Chapter 6Reflection strategies

Let us now return to the first of the three cases in which we perturb the path ofthe Cramer–Lundberg process through the payments of dividends. Recall that areflection (or barrier) strategy consists of paying dividends out of the surplus insuch a way that, for a fixed barriera > 0, any excess of the surplus above thislevel is instantaneously paid out. The cumulative dividendstream is thus given byLt := (a∨Xt)−a, for t ≥ 0. The resulting trajectory satisfies the dynamics

Ut := Xt −Lt = Xt +a− (a∨Xt), t ≥ 0,

with probabilitiesPx : x∈ [0,a]. The present value of dividends paid until ruin isthus given by ∫

[0,ς)e−qtdLt ,

whereς = inft > 0 :Ut < 0.It is more convenient to both measure and start the reflected process from the

thresholda. In which casea−Ut : t ≥ 0, underPa, is equal in law toY0t : t ≥ 0,

underP, where we recall thatYxt := (x∨Xt)−Xt , x, t ≥ 0. Moreover, from this point

of view, the dividends that are paid out underPa are equal in law to the processXt : t ≥ 0 underP and the time of ruin corresponds to

σa = inft > 0 :Y0t > a.

The key object of interest in this chapter, the present valueof the dividends paiduntil ruin, will boils down to the study of the quantity

∫

[0,σa)e−qtdXt (6.1)

underP, whereq≥ 0 is the force of interest.

45

46 6 Reflection strategies

6.1 Perpetuities

Suppose thatN = Nt : t ≥ 0 is a Poisson process with rateα > 0, ζi : i ≥ 0 isa sequence of i.i.d. positive random variables with common distribution functionGandb≥ 0 is a constant. The process

ℓt := bt+Nt

∑i=1

ζi , t ≥ 0,

is a compound Poisson process with positive jumps and positive drift. Then, in thespirit of (2.3), we can compute its Laplace exponent as follows:

E(e−θℓt ) = e−Λ(θ)t , t,θ ≥ 0,

whereΛ(θ ) = bθ +α

∫

(0,∞)(1−e−θx)G(dx).

The key mathematical object that will help us to analyse the present value ofdividends paid until ruin is the so-calledperpetuity

∫ ep

0e−qℓt dt, (6.2)

whereq≥ 0 and, as usual,ep is an independent exponential random variable withratep≥ 0. We have the following main result which characterises itsmoments.

Theorem 6.1.For all n ∈ N,

E

[(∫ ep

0e−qℓt dt

)n]= n!

n

∏k=1

1p+Λ(qk)

.

Proof. Define, fort ≥ 0,

Jt =

∫ ∞

te−qℓu1(u<ep)du.

Our objective is to compute, forn∈ N,

Ψn := E(Jn0) .

To this end, note that

ddt

Jnt =−nJn−1

t e−qℓt 1(t<ep),

and, hence, we obtain

Jn0− Jn

t = n∫ t

0e−qℓu1(u<ep)J

n−1u du. (6.3)

6.2 Decomposing paths at the maximum 47

Using thatℓt : t ≥ 0 has stationary and independent increments together with thelack of memory property, we have

Jt = e−qℓt 1(t<ep)J∗0,

whereJ∗0 is independent ofℓu : u≤ t and has the same distribution asJ0. In con-clusion, taking expectations in (6.3), we find that

Ψn

(1−E(e−nqℓt1(t<ep))

)= nΨn−1

∫ t

0E(e−nqℓu1(u<ep))du. (6.4)

SinceE

(e−nqℓt 1(t<ep)

)= exp−(p+Λ(nq))t,

it follows thatΨn =

np+Λ(nq)

Ψn−1.

Iterating gives the result.

Remark 6.2.It is worth noting that Theorem 6.1 and its proof are still valid whenwe replace the processℓ by a general subordinator, meaning a non-decreasing Levyprocess.

6.2 Decomposing paths at the maximum

Let us now look at the reflected processXt −Xt : t ≥ 0 and consider how thetime it spends in the state zero, as well as the excursions it makes from the statezero, will reveal yet another path decomposition, again incorporating the idea of“Bernoulli trials”.

For convenience, writeYt in place ofY0t , t ≥ 0. DefineS0 = 0 and

S∗0 = inft > 0 :Yt > 0.

Then continue recursively, so that, fork∈N, onSk−1 < ∞,

Sk = inft > S∗k−1 : Yt = 0 andS∗k = inft > Sk : Yt > 0

and setχk = XS∗k−1

, ζk = Sk−S∗k−1 andhk = sups∈[S∗k−1,Sk]

Ys.

Finally, defineυa = mink≥ 1 : hk > a.

In words, fork≥ 1, the intervals[Sk−1,S∗k−1) are the successive periods that theprocessY spends at the origin. Equivalently, they are the intervals of time that theprocessX is increasing. By the lack of memory property, each of these periods,


S∗k−1−Sk−1, are independent and exponentially distributed with the arrival rateλ .The intervals[S∗k−1,Sk) are the successive periods that the processY undertakes anexcursion from the origin. Equivalently they are the intervals of time that the processX does not increase (and henceX makes an excursion away fromX). The triplets(χk,ζk,hk) represent the height ofX immediately prior to thek-th excursion, thelength of thek-th excursion and the height ofk-th excursion, respectively. Moreover,υa is the index of the first excursion from the maximum which exceeds a heighta.Appealing again to the logic of Bernoulli trials,υa is geometrically distributed withparameter

ra =∫

(0,∞)F(dx)P−x(τ−−a < τ+0 ).

Note that, appealing to spatial homogeneity,ra is the probability that an excursionfrom the maximum, which has initial distance below the previous maximum dis-tributed according toF , reaches a distancea below the previous maximum before itreturns to the the level of the previous maximum, thereby completing the excursion.In other words,ra is the probability that an excursion exceedsa in height.

X

Y

a

h4

S0S3

χ2

χ3

χ1

t

S∗0 S∗1S1S2 S∗3

h3

h2

h1

γt

S∗2

Fig. 6.1 Decomposing the path ofX into excursions from its maximum.

Now note thatX increases if and only ifY is zero. Moreover, when it increases,it does so at a ratec. In other words,

6.2 Decomposing paths at the maximum 49

Xt = c∫ t

01(Ys=0)ds. (6.5)

With this in mind, we see that the sequence of positive randomvariables

XS∗k−XSk : k= 0, · · · ,υa−1= c(S∗k−Sk) : k= 0, · · · ,υa−1

is nothing more than an independent geometric number of independent exponentialrandom variables, each with rateλ/c. Therefore, recalling that the sum of an inde-pendent geometrically distributed number of i.i.d. exponential random variables isagain exponentially distributed, the maximum height reached byX at the momentthat it first drops a distancea below its previous maximum,

XS∗υa−1= c

υa−1

∑k=0

(S∗k−Sk),

is exponentially distributed with ratepa := λ ra/c. In fact, recalling that

χk = XS∗k−1= c

k−1

∑j=0

(S∗j −Sj),

the triplets(χk,ζk,hk) : k= 1, · · · ,υa

are the times of arrival,χk, and the marks,(ζk,hk), of a marked Poisson process upuntil the first markhk exceedsa in value, where the arrival rate isλ/c. The marksare distributed according to the joint law

H(dy,dz) =∫

(0,∞)F(dx)P−x(τ+0 ∈ dy,−Xτ+0

∈ dz)1(z≥x), y,z≥ 0.

To see why, note, as before, that each excursion begins with ajump below the pre-vious maximum, distributed according toF. The excursion length is therefore thetime it takes to reach the previous maximum from this initialposition. Moreover,the height of this excursion is also the depth below the previous maximum thatXreaches before returning to the level of the previous maximum. Appealing again tospatial homogeneity, the excursion length and height are respectively equivalent tothe first passage timeτ+0 and to the depth below the origin thatX reaches beforeτ+0 ,i.e.−Xτ+0

, whenX is issued from a position below the origin which is randomisedaccording toF .

Appealing to the Poisson thinning theorem, we can also say that

(χk,ζk,hk) : k= 1, · · · ,υa−1

is equal in law to the times of arrival and the marks of a Poisson processes, sayNa = Nat : t ≥ 0, with arrival rateαa = λ (1− ra)/c and mark distribution on(0,∞)× (0,a]


Ha(dy,dz) =∫

(0,a]F(dx)P−x(τ+0 ∈ dy,−Xτ+0

∈ dz|τ+0 < τ−−a)1(z≥x),

when sampled up to an independent and exponentially distributed random time,epa . The distributionHa(dy,dz) is the joint law of the excursion length and heightconditioned on the excursion height not exceedinga. Indeed, appealing to earlierreasoning, this is equivalent to the law of the pair(τ+0 ,−Xτ+0

) conditioned on the

event−Xτ+0≤ a = τ+0 < τ−−a, whenX0 is has a random position below the

origin, with distance below it distributed according toF . Moreover, recalling thatrais the probability that an excursion has height greater thana, λ (1− ra)/c is the rateof arrival of excursions with height not exceedinga. Similarly,λ ra/c is the rate ofarrival of excursions with height exceedinga.

Fix t > 0 and writeℓt := infs> 0 : Xs > t,

with the usual understanding that inf /0 :=∞. In other words,ℓt is the amount of timeit takes forX to climb to the levelt. We can spilt the time horizon[0, ℓt ] into theintervals of time thatY spends at zero (i.e. the time thatX is climbing) together withthe intervals of time thatY undertakes excursions from the origin (i.e. the intervalsof time thatX is stationary). On the one hand, thanks to the relation (6.5), on theeventℓt < ∞, the time thatY spends at zero untilX reaches the levelt is given by

∫ ℓt

01(Ys=0)ds=

1c

Xℓt =tc.

On the other hand, if we further insist thatt < XS∗υa−1 (which necessarily implies

the eventℓt < ∞), then all of the excursions ofY from zero there are less thanain height. Moreover, still on the eventt < XS∗υa−1

, their number and lengths are

equal in law toNat andζai : i = 1, · · · ,Nat , respectively, where theζai are i.i.d. withcommon distribution given by

Ga(dy) = Ha(dy, [0,a]) =∫

(0,a]F(dx)P−x(τ+0 ∈ dy|τ+0 < τ−−a), y≥ 0.

In conclusion, we have that, ont < XS∗υa−1, the time it takes forX to reach height

t is equal in law to

ℓat := bt+Nat

∑i=1

ζai , t ≥ 0,

on the eventt < epa, whereb= 1/c.Putting all these pieces together, we develop the expression for present value of

dividends paid until ruin (6.1) for the special case thatx= 0. Indeed, by making asimple change of variables,t 7→ ℓs, we have∫

[0,σa)e−qtdXt =

∫ ∞

01(t<S∗υa−1)

e−qtdXt =

∫ ∞

01(u<XS∗υa−1

)e−qℓudu=

∫ epa

0e−qℓau du.

6.3 Derivative of the scale function 51

We therefore see that, when the surplus process starts at thebarriera, equivalentlyx= 0 in (6.1), the present value of dividends until ruin is equalin distribution to theperpetuity (6.2) for appropriate choices ofpa,b,αa andGa, as given above.

If we would be able to say a little more about these quantities, then we will beable to develop an expression for then-th moments of the present value of dividendspaid at ruin, at least when the surplus process starts at the barrier, by using Theorem6.1. That is to say, we would be able to develop further the identity

E

[(∫

[0,σa)e−qtdXt

)n]= n!

n

∏k=1

1pa+Λa(qk)

,

whereΛa(θ ) := bθ +αa∫(0,a](1−e−θx)Ga(dx), θ ≥ 0. Once again, scale functions

come to our assistance.

6.3 Derivative of the scale function

Before we can proceed to develop identities using scale functions, we need to say afew words about differentiability as derivatives of the scale functions will appear inthe forthcoming analysis. Recall that, for eachq≥ 0, the functionW(q) is monotone.

Lemma 6.3.For x≥ 0,

W(q)(dx) =1c

δ0(dx)+(λ +q)c

W(q)(x)dx−(

λc

∫

(0,x]W(q)(x− y)F(dy)

)dx. (6.6)

In particular,

W(q)′+ (x) =

(λ +q)c

W(q)(x)−(

λc

∫

(0,x]W(q)(x− y)F(dy)

), (6.7)

for x > 0, where W(q)′+ (x) is the right derivative of W(q) at x. Moreover, W(q) is

continuously differentiable on(0,∞) if and only if F has no atoms.

Proof. Note also that, where asW(q)(0−) = 0, we have that, forq≥ 0,

W(q)(0) = limβ↑∞

∫ ∞

0βe−β xW(q)(x)dx

= limβ↑∞

βψ(β )−q

= limβ↑∞

1

c−λ∫ ∞

0 e−β xF(x)dx−q/β

=1c.


It follows thatW(q)(dx) has an atom at zero of size 1/c.Integrating (4.1) by parts, we have, on the one hand, that

∫

[0,∞)e−β xW(q)(dx) =

∫

(0,∞)e−β xW(q)(dx)+

1c=

βψ(β )−q

, (6.8)

for β > Φ(q). On the other hand, we also have that

(λ +q)c

∫ ∞

0e−β xW(q)(x)dx− λ

c

∫ ∞

0e−β x

∫

(0,x]W(q)(x− y)F(dy)dx

=1c

(λ +q)ψ(β )−q

− 1c

λψ(β )−q

∫

(0,∞)e−β xF(dx)

=1c

λ∫(0,∞)(1−e−β x)F(dx)+q

ψ(β )−q

=1c

cβ −ψ(β )+qψ(β )−q

=β

ψ(β )−q− 1c, (6.9)

for β > Φ(q). Comparing the transforms in (6.8) and (6.9), it follows that (6.6)holds. In particular, this implies thatW(q) is an absolutely continuous (and hence acontinuous) function on(0,∞).

Thanks to the just-proved fact thatW(q) is a continuous function on(0,∞), in-specting the right-hand side of (6.6), we see that its density with respect to Lebesgue

measure is, in general, right-continuous. The formula forW(q)′+ (x) on (0,∞) in (6.7)

thus follows. Moreover, it is continuous if and only if the convolution is continu-ous. This is equivalent to requiring thatF has no atoms. In that case, on(0,∞), wehave thatW(q) is absolutely continuous with a continuous version of its density, thusmaking it continuously differentiable.

6.4 Present value of dividends paid until ruin

Let us use a completely different technique to examine the first moments of thepresent value of dividends paid until ruin when the initial value of the surplus isa. This will give us an expression from which we can glean the desired informationabout the quantitiespa andΛa as per the discussion at the end of Section 6.2. There-after we can return to the more general problem of then-th moments of the presentvalue of dividends paid until ruin when the initial value of the surplus isx∈ [0,a].

Lemma 6.4.For all q≥ 0,

Ea

[∫

[0,ς)e−qtdLt

]= E

[∫

[0,σa)e−qtdXt

]=

W(q)(a)

W(q)′+ (a)

. (6.10)

6.4 Present value of dividends paid until ruin 53

Proof. The proof works by splitting the integral∫ σa

0 e−qtdXt at the time of the firstjump ofX. Note thatX initially increases at ratec for an independent and exponen-tially distributed period of time, with rateλ , and then undertakes a jump of sizeξ1

downwards. This jump kick-starts an excursion from the maximum. This excursionwill either cause ruin, or its height will remain below the levela andX will return toits previous maximum, during which time no dividends will have been paid and ruinhas not occured. Thereafter, by the strong Markov property,the process we have justdescribed begins again, except that future payments are additionally discounted bythe time that has already lapsed. Let

V(1)a = E

[∫

[0,σa)e−qtdXt

].

Following the description given above, we have, with the help of Theorem 4.5,

V(1)a = E

[∫ eλ

0e−qtcdu

]+E

[e−qeλ E−ξ1

[e−qτ+0 1(τ+0 <τ−−a)]1(ξ1≤a)V

(1)a

]

=c

qE(1−e−qeλ )+E(e−qeλ )

∫

(0,a]

W(q)(a− y)

W(q)(a)F(dy)V(1)

a

=c

λ +q+

λλ +q

∫

(0,a]

W(q)(a− y)

W(q)(a)F(dy)V(1)

a . (6.11)

From (6.7) we have

λλ +q

∫

(0,a]

W(q)(a− y)

W(q)(a)F(dy) = 1− c

λ +q

W(q)′+ (a)

W(q)(a).

Substituting the above convolution into (6.11) and solvingfor V(1)a now gives the

statement of the lemma.

Taking account of the discussion in Sect. 6.2 as well as the conclusion of Theo-rem 6.1, we see that

V(1)a =

1pa+Λa(q)

=W(q)(a)

W(q)′+ (a)

.

It follows that, forn≥ 1,


V(n)a := Ea

[(∫

[0,ς)e−qtdLt

)n]

= E

[(∫

[0,σa)e−qtdXt

)n]

= n!n

∏k=1

1pa+Λa(qk)

= n!n

∏k=1

W(qk)(a)

W(qk)′+ (a)

.

We are almost done. All we need now is to deal with the case thatthe surplusprocess starts from anyx∈ [0,a]. To this end, note that, when the Cramer–Lundbergprocess is issued fromx∈ [0,a], dividends are not paid untilX reaches the thresholda, providing ruin does not occur beforehand. In that case, future dividend paymentsare discounted by the time elapsed until reaching the threshold. Hence, by the strongMarkov property,

Ex

[(∫

[0,ς)e−qtdLt

)n]= Ex

[e−qnτ+a 1(τ+a <τ−0 )

]V(n)a

In conclusion, taking account of Theorem 4.5, we have the following main result forthis chapter.

Theorem 6.5.For q≥ 0 and x∈ [0,a],

Ex

[(∫

[0,ς)e−qtdLt

)n]= n!

W(qn)(x)

W(qn)(a)

n

∏k=1

W(qk)(a)

W(qk)′+ (a)

.

6.5 Comments

As alluded to in the introduction, reflection strategies forCramer–Lundberg pro-cesses emerge naturally from the control problem (1.7). An expression for the ex-pected present value of dividends paid until ruin in terms of(what amounts to) scalefunctions can be found in the work Gerber (1972). See equation (8.9) of that paper.Given that the optimal strategy to (1.7) falls within the class of reflection strategies,one is left with the problem of choosing an optimal value ofa for the threshold.Note from Theorem 6.5 that the expected present value of dividends paid until ruin

with reflection at thresholda is W(q)(x)/W(q)′+ (a) whenU0 = x≤ a. Going back

to the original work of Gerber (1969) and Gerber (1972), it isknown thata should

be chosen so as to minimise the value ofW(q)′+ (a). Amongst other literature on this

control problem, Loeffen (2008) makes an important step in identifying natural as-sumptions on the jump distributionF to ensure thatW(q)′(a) is strictly convex and,hence, can be minimised at a unique value ofa≥ 0.

6.5 Comments 55

Higher moments of the present value of reflection strategieswere first consid-ered in Dickson and Waters (2004) for the case of exponentially distributed jumps.The connection with scale functions was made simultaneously in Renaud and Zhou(2007) and Kyprianou and Palmowski (2007) for the case thatX is a general spec-trally negative Levy process. Albrecher and Gerber (2011)extend their result fur-ther to the case of stationary upward-skip-free Markov processes. The proof wegive here is a hybrid version of the proofs found in Dickson and Waters (2004)and Kyprianou and Palmowski (2007). The review article Bertoin and Yor (2005)gives an interesting overview of perpetuities for Levy processes in general. De-composing the paths of the Cramer–Lundberg process at the maximum and iden-tifying excursions through a marked Poisson process is an idea that goes back toGreenwood and Pitman (1980). Ultimately, however, it is thenatural continuous-time analogue of the discrete analysis that occurs when one decomposes paths atthe minimum, and therefore relates back to Feller’s ideas onthe renewal processof maxima (or indeed minima) for random walks. For further information on thesmoothness of scale functions, the reader is referred to Cohen et al. (2012).

Chapter 7Perturbation-at-maximum strategies

In this chapter, we dig a little deeper in the decomposition discussed in Sect. 6.2.In particular, we bring out in more detail the characterisation of the marked Pois-son process of excursion heights in terms of scale functions. This will help us toanalyse the case where payments are made from the surplus that are proportional toincrements of the maximum.

Recall that ifγ : [0,∞)→ [0,∞), then the surplus process, when perturbed at rateγ with respect to its maximum process, yields an aggregate

Ut = Xt −∫

(0,t]γ(Xu)dXu, t ≥ 0.

We restrict ourselves to the cases mentioned in the introduction: theheavy pertur-bation regime, for which γ : [0,∞)→ (1,∞) and thelight perturbation regime, forwhich γ : [0,∞)→ (0,1). The latter of these two may be seen as the result of tax-ation.The case of a reflection strategy, whenγ(x) = 1(x≥a), sits between these tworegimes.

7.1 Re-hung excursions

Let us start by noting that, for allx≥ 0, underPx,

Ut = At −Yt , t ≥ 0, (7.1)

where we recall thatYt = Xt −Xt and

At = Xt −∫

(0,t]γ(Xu)dXu = γx(Xt), t ≥ 0,

with

γx(s) := s−∫ s

xγ(y)dy= x+

∫ s

x[1− γ(y)]dy, s≥ x.

57

58 7 Perturbation-at-maximum strategies

The last equality shows us that processA= At : t ≥ 0 is strictly increasing (resp.decreasing) whenγ belongs to the light perturbation (resp. heavy perturbation)regime. In essence, monotonicity ofA follows from monotonicity ofX andγx. (Thelatter observation will also be important later as we will use the inverse functionγ−1x , which, accordingly, is well defined in both regimes.) Moreover,A is stationary

in value if and only if the processY is non-zero valued. As a consequence we mayinterpret (7.1) as a path decomposition in which excursionsof X from its maximum(equivalently excursions ofY away from zero) are ‘hung’ off the trajectory ofAduring its stationary periods. See Fig. 7.1 for a visual interpretation of this heuristic.

t

Ut

x

Fig. 7.1 Re-hanging excursions from the processA in the heavy perturbation case.

In the light perturbation regime, sinceA is an increasing process which is station-ary whenever the processY is non-zero valued, we have that

←−U t := sup

s≤tUs = Agt = At ,

wheregt = sups≤ t : Ys = 0. Hence, unless it is assumed that∫ ∞

x(1− γ(s))ds= ∞, (7.2)

in the light perturbation regime, the perturbed processU will have an almost surelyfinite global maximum.

7.2 Marked Poisson process revisited 59

In contrast, in the heavy perturbation regime whenA is a decreasing, similarreasoning shows that −→

U t := sups≥t

Us = Adt = At ,

wheredt = infs> t : Ys= 0. Hence, the processU is always bounded by its initialvaluex.

Henceforth, our analysis of the processU will centre around further analysis ofthe excursions ofX from its maximumX. In particular, their representation througha marked Poisson process, such as we saw in Sect. 6.2, will prove to be extremelyuseful. In the next section, we shall revisit this marked Poisson process and look ata more detailed characterisation of its parameters in termsof scale functions.

7.2 Marked Poisson process revisited

Recall from the previous chapter that we writeYt = Xt−Xt , for t ≥ 0. Moreover, werecursively definedS0 = 0,

S∗0 = inft > 0 :Yt > 0.

and, fork∈N, onSk−1 < ∞,

Sk = inft > S∗k−1 : Yt = 0 andS∗k = inft > Sk : Yt > 0.

Thek-th excursion from the maxima occurs over the time intervals[S∗k−1,Sk], andthe height of the excursion was denoted by

hk = sups∈[S∗k−1,Sk]

Ys.

Now letυ∞ = mink≥ 1 : Sk = ∞,

the index of the first excursion which is infinite in length. Note, by Lemma 3.10,that if ψ ′(0+) ≥ 0, thenυ∞ = ∞ almost surely. Moreover, ifψ ′(0+) < ∞, thenυ∞is geometrically distributed with parameter

r∞ =

∫

(0,∞)F(dx)P−x(τ+0 = ∞).

In both cases, it is clear that we may equivalently define

υ∞ = mink≥ 1 : hk = ∞.

UnderPx, define in a similar manner to the previous chapterχk = XS∗k−1− x,

where, now, we allow the indexk to run from 1 toυ∞. In the spirit of the reasoning


in Sect. 6.2, we also note that

(χk,hk) : k= 1, · · · ,υ∞

is equal to times of arrival and the marks of a marked Poisson process, sayN= Nt :t ≥ 0, with rateλ/c up until the first mark which is infinite in value. Moreover,marks are i.i.d. (and independent ofN) with common distribution

Q(dy) :=∫

(0,∞)F(dx)P−x(−Xτ+0

∈ dy), y∈ [0,∞].

Let us return to the eventτ+a < τ−0 , for a> 0. Note that, for 0≤ x≤ a,

Px(τ+a < τ−0 ) = P(hk ≤ x+ χk for k= 1, · · · ,Na−x).

Fig. 7.2 gives a visual explanation of this equality.

hNa−x

h2

h1

x

a

χ2

Fig. 7.2 Excursion heights as the processX climbs fromx to a.

Classical theory for Poisson processes tells us that, conditional onNa−x = n,the arrival timesχ1, · · · ,χn are equal in law to an ordered i.i.d. sample of uni-formly distributed points on[0,a− x]. Then,

7.3 Gambler’s ruin for the perturbed process 61

P(hk≤ x+ χk for k= 1, · · · ,Na−x)

=∞

∑n=0

e−λc (a−x) 1

n!

(λ (a− x)

c

)n(∫ a−x

0Q(x+ t)

1(a− x)

dt

)n

= exp

λc

∫ a−x

0(Q(x+ t)−1)dt

= exp

−λc

∫ a

xQ(y)dy

,

whereQ(y) = 1−Q(y), for y≥ 0.It was proved earlier thatPx(τ+a < τ−0 ) = W(x)/W(a), whereW is the scale

function associated toX. This leads us to the identity

W(x)W(a)

= exp

−λc

∫ a

xQ(y)dy

. (7.3)

Note that this confirms the conclusion of Lemma 6.3 thatW is almost everywheredifferentiable. Taking account of the above discussion, wecome to rest at the fol-lowing important result.

Theorem 7.1.Recall that W′+ is the right derivative of W on(0,∞). Then, for allx> 0,

λc

Q(x) =W′+(x)W(x)

(7.4)

Later on in this chapter, when using this result, the quantity Q will always appearin the context of a Lebesgue integral. In that case, it suffices to writeW′/W on theright-hand side of (7.4), without needing to refer to the right derivative ofW.

7.3 Gambler’s ruin for the perturbed process

LetT−0 := inft > 0 : Ut < 0,

where we understand, as usual, inf /0 := ∞. We shall also use the earlier introducedstopping time

τ+a = inft > 0 : Xt > a= inft > 0 : Xt > a.

In the light perturbation case, we may write for all valuesb in the range ofγx,

τ+γ−1x (b)

= T+b , (7.5)

whereT+

b = inft > 0 :Ut > b.


Theorem 7.2.Fix x > 0 and assume (7.2) in the case of the light perturbationregime. In the case of the heavy perturbation regime, define

s∗(x) = infs≥ x : γx(s)< 0.

Then, for any q≥ 0, and0≤ x≤ a in the case of light perturbation, resp.0≤ x≤a< s∗(x) in the case of heavy perturbation, we have

Ex[e−qτ+a 1τ+a <T−0

]= exp

(−∫ a

x

W(q)′(γx(s))

W(q)(γx(s))ds

). (7.6)

Before moving to the proof of this result, let us remark that,taking account of theequivalence (7.5) in the light perturbation regime, (7.6) can be more convenientlywritten as

Ex[e−qT+

a 1T+a <T−0

]= exp

(−∫ γ−1

x (a)

x

W(q)′(γx(s))

W(q)(γx(s))ds

).

Proof (of Theorem 7.2).The proof does not distinguish between the two differentregimes of light and heavy perturbation. All that is required in what follows is thatγ−1x (a) < ∞. Thereafter, the proof needs little more than to recycle a number of

existing computations we have already seen.Using the Esscher transform and appealing to similar reasoning found in the

proof of Theorem 7.1, we have

Ex[e−qτ+a 1τ+a <T−0

]

= e−(a−x)Φ(q)P

Φ(q)x (τ+a < T−0 )

= e−(a−x)Φ(q)P

Φ(q)(hk ≤ γx(x+ χk) for k= 1, · · · ,Na−x)

= e−(a−x)Φ(q)∞

∑n=0

e−λc (a−x) 1

n!

(λ (a− x)

c

)n(∫ a−x

0QΦ(q)(γx(x+ t))

1(a− x)

dt

)n

= exp

(−∫ a−x

0Φ(q)+

λc

QΦ(q)(γx(x+ t))dt

), (7.7)

whereQΦ(q) plays the role of the quantityQ, but under the measurePΦ(q) andQΦ(q) = 1−QΦ(q). In particular, recalling the conclusion of Theorems 2.3 and 7.1,we have, for Lebesgue almost everyx≥ 0,

λc

QΦ(q)(x) =W′Φ(q)(x)

WΦ(q)(x).

Moreover, recalling from the definition (4.4) thatW(q)(x) = eΦ(q)xWΦ(q)(x), x≥ 0,we have that, for Lebesgue almost everyx≥ 0,

7.4 Continuous ruin with heavy perturbation 63

W(q)′(x)

W(q)(x)= Φ(q)+

W′Φ(q)(x)

WΦ(q)(x).

Putting the pieces together in (7.7) produces the required identity.

Theorem 7.2 motivates some interesting observations concerning the event ofruin, T−0 < ∞. First, suppose that we are in the heavy perturbation regimeands∗(x)< ∞. In that case

Px(T−0 < ∞)≥ Px(τ+s∗(x) < ∞)∨Px(τ−0 < ∞).

Indeed, on the eventτ+s∗(x) < ∞, we haveXτ+s∗(x)−Xτ+s∗(x)

= 0 and henceUτ+s∗(x)=

Aτ+s∗(x)= γx(s∗(x)) = 0. Moreover, sinceUt ≤ Xt for all t ≥ 0, it follows that

τ−0 < ∞ ⊆ T−0 < ∞. In the event that limsupt↑∞ Xt = ∞ almost surely, wehavePx(τ+s∗(x) < ∞) = 1. Otherwise, it follows thatPx(τ−0 < ∞) = 1. Either way,

Px(T−0 < ∞) = 1.

Remaining in the heavy perturbation regime, suppose thats∗(x) = ∞. Then from(7.6), by taking limits asa ↑∞, we get an expression for the ruin probability,

Px(T−0 < ∞) = 1−exp

(−∫ ∞

x

W′(γx(s))W(γx(s))

ds

). (7.8)

However, the right-hand side above turns out to be equal to 1.Recalling thatW′(x)/W(x) = λQ(x)/c for Lebesgue almost everyx > 0, sinceQ(x) is non-increasing on(0,∞) andγx(s)≤ x for all s≥ 0, the claim follows.

Finally, in the light perturbation regime, where necessarily s∗(x) = ∞, the rea-soning that leads to (7.8) still applies, from which one may deduce that this prob-ability need not be unity. Indeed, suppose that we take the function γ to be sim-ply constant in value, also denoted byγ ∈ (0,1), andψ ′(0+) > 0. In that case,γx(s) = (s− x)(1− γ) + x, and hence, noting that d[logW(x)]/dx = W′(x)/W(x)Lebesgue almost everywhere on(0,∞), after a change of variables, we have

Px(T−0 < ∞) = 1−exp

(− 1(1− γ)

∫ ∞

x

W′(u)W(u)

du

)(7.9)

= 1−(ψ ′(0+)W(x)

)1/(1−γ), (7.10)

for x≥ 0, where we have used, from (4.2), thatW(∞) = 1/ψ ′(0+).

7.4 Continuous ruin with heavy perturbation

Although the perturbed process is almost surely ruined in the heavy perturbationregime, it is interesting to note that, unlike a Cramer–Lundberg process, there are


two different ways to become ruined. The first, i.e. by a jump downwards, is a prop-erty inherited from the underlying process,X. The other way of becoming ruined,which we refer to ascontinuous ruin, is the result of continuously passing the originat the moment in time that an increment inX bringsU along the curveγx just asit intersects the origin. Said another way, continuous ruincorresponds to the eventthat τ+s∗(x) = T−0 , in which case, as remarked upon above,Uτ+s∗(x)

= 0. This can

only happen with positive probability ifs∗(x)< ∞.The following result is a corollary to Theorem 7.2 on accountof the fact that its

proof is identical, albeit that one replacesa by s∗(x).

Corollary 7.3. Fix x> 0, and suppose thatγ : [0,∞)→ (1,∞) such that s∗(x) < ∞.Then, for all q≥ 0,

Ex[e−qT−0 1T−0 =τ+s∗(x)

]= exp

(−∫ s∗(x)

x

W(q)′(γx(s))

W(q)(γx(s))ds

).

If we take the case thatγ(s) is a constant valued in(1,∞), again denoted byγ, then we may simplify the formula in the above corollary. Indeed, we haveγx(s) = x− (s− x)(γ − 1) and hences∗(x) = γx/(γ − 1). Moreover, a straightfor-ward computation, in a similar vein to the computation in (7.10), gives us

Px(continuous ruin) = exp

(− 1(γ−1)

∫ x

0

W′(u)W(u)

du

)=

(1

cW(x)

)1/(γ−1)

,

for x≥ 0, where we have used, from Lemma 6.3, thatW(0) = 1/c.

7.5 Expected present value of tax at ruin

In the spirit of the Gerber–Shiu-type results presented in the previous sections, ourfinal theorem for perturbed processes (with either light or heavy perturbation) con-siders the expected present value of perturbation paid out until ruin. In the case oflight perturbation, this corresponds to the expected present value of tax paid untilruin.

Theorem 7.4.Fix x ≥ 0 and assume (7.2) in the case of the light perturbationregime. Then, for q≥ 0,

Ex

[∫ T−0

0e−quγ(Xu)dXu

]=

∫ s∗(x)

xexp

(−∫ t

x

W(q)′(γx(s))

W(q)(γx(s))ds

)γ(t)dt.

Proof. Appealing to a straightforward change of variables and Fubini’s Theorem,noting in particular thatτ+t = infs> 0 : Xs > t, we have

7.6 Comments 65

Ex

[∫ T−0

0e−quγ(Xu)dXu

]= Ex

[∫ s∗(x)

01(u<T−0 )e

−quγ(Xu)dXu

]

= Ex

[∫ s∗(x)

x1(τ+t <T−0 )e

−qτ+t γ(t)dt

]

=

∫ s∗(x)

xEx

[e−qτ+t 1(τ+t <T−0 )

]γ(t)dt.

The proof is completed by taking advantage of the identity in(7.6).

We again get a simplification of this formula in the case that we take the functionγ(s) to be a constant, for either the light perturbation or heavy perturbation regime.For example whenγ(s) is equal to the constantγ ∈ (0,1), one gets, forx,q≥ 0,

Ex

[∫ T−0

0e−quγ(Xu)dXu

]=

γ1− γ

∫ ∞

x

(W(q)(x)

W(q)(z)

)1/(1−γ)

dz.

7.6 Comments

The representation of the scale function in the form (7.3) islifted from Theo-rem 8 of Chapter VII in Bertoin (1996). This representation and the observationthat excursions are re-hung from the processA form a key part of the analy-sis in Albrecher et al. (2008), Kyprianou and Zhou (2009) andKyprianou and Ott(2012), in increasing degrees of generality, respectively.

Chapter 8Refraction strategies

Let us return to the case of the refracted Cramer–Lundberg process. That is, thesolution to the stochastic differential equation

Zt = Xt − δ∫ t

01(Zs>b)ds, t ≥ 0, (8.1)

also written asdZt = dXt − δ1(Zt>b)dt, t ≥ 0,

for some threshold b≥ 0. We shall charge ourselves with the task of providing iden-tities for the probability of ruin as well as the expected present value of dividendspaid until ruin. As we have seen earlier, for the case of a Cramer–Lundberg pro-cess, it turns out to be convenient to first derive an identityfor the resolvent of therefracted process until first exiting a finite interval. It turns out that all identitiescan be written in terms of two scale functions for two different Cramer–Lundbergprocesses. As one might expect these identities are somewhat more complicated.

8.1 Pathwise existence and uniqueness

Before we can look at functionals which pertain to Gerber-Shiu theory, we are con-fronted with the more pressing issue of whether a solution tothis SDE exists. As thereader might already suspect, problems my occur whenδ ≥ c, as, in that case, whendividends are paid, it is at a higher rate than the premiums collected. We thereforeassume throughout that

0< δ < c.

Theorem 8.1.The SDE (8.1) has a unique pathwise1 solution.

1 We can understand the words “pathwise solution” here to meana solution which is constructeddirectly from the path of the driving processX.

67

68 8 Refraction strategies

Proof. Define the timesT↑n andT↓n recursively as follows. We setT↓0 = 0 and, for

n= 1,2, . . ., on the eventsT↓n−1 < ∞ andT↑n < ∞ respectfully, set

T↑n = inft > T↓n−1 : Xt − δn−1

∑i=1

(T↓i −T↑i )> b,

T↓n = inft > T↑n : Xt − δn−1

∑i=1

(T↓i −T↑i )− δ (t−T↑n )< b.

The difference between the two consecutive timesT↑n andT↓n is strictly positive.Moreover, limn↑∞ T↑n = limn↑∞ T↓n = ∞, almost surely. Now we construct a solutionto (8.1),Z = Zt : t ≥ 0, as follows. The process is issued fromX0 = x and

Zt =

Xt − δ ∑n

i=1(T↓i −T↑i ), for t ∈ [T↓n ,T

↑n+1) andn≥ 0,

Xt − δ ∑n−1i=1 (T

↓i −T↑i )− δ (t−T↑n ), for t ∈ [T↑n ,T

↓n ) andn≥ 1.

Note that forn= 1,2, . . ., on the eventsT↓n−1 < ∞ andT↑n < ∞ respectfully, the

timesT↑n andT↓n , can then be identified as

T↑n = inft > T↓n−1 : Zt > b, T↓n = inft > T↑n : Zt < b.

Hence

Zt = Xt − δ∫ t

01Zs>bds, t ≥ 0,

thereby proving the existence of a pathwise solution to (8.1).

For uniqueness of this solution, suppose thatZ(1)t : t ≥ 0 andZ(2)

t : t ≥ 0 aretwo pathwise solutions to (8.1). Then, writing

∆t = Z(1)t −Z(2)

t =−δ∫ t

0(1Z(1)

s >b−1Z(2)s >b)ds,

it follows from integration by parts that

∆2t =−2δ

∫ t

0∆s(1Z(1)

s >b−1Z(2)s >b)ds.

Thanks to the fact that1x>b is an increasing function, it follows from the aboverepresentation, that, for allt ≥ 0, ∆2

t ≤ 0 and hence∆t = 0 almost surely, therebyproving uniqueness of pathwise solutions to (8.1).

Let us momentarily return to the reason whyZ is referred to as a refracted Levyprocesses. A simple sketch of a realisation of the path ofZ (see for example Fig.8.1) gives the impression that the trajectory ofZ “refracts” each time it passes con-tinuously from(−∞,b] into (b,∞), much as a beam of light does when passing fromone medium to another.

8.1 Pathwise existence and uniqueness 69

Fig. 8.1 A sample path ofZ when the driving Levy process is a Cramer–Lundberg process. Itstrajectory “refracts” as it passes continuously above the horizontal dashed line at level b.

The construction of the unique pathwise solution describedabove clearly showsthatZ is adapted to the natural filtrationF = Ft : t ≥ 0 of X. Conversely, since,for all t ≥ 0, Xt = Zt + δ

∫ t0 1Zs>bds, it is also clear thatX is adapted to the natural

filtration of Z. We can use this observation to reason thatZ is a strong Markovprocess.

To this end, suppose thatT is a stopping time with respect toF. Then define aprocessZ whose dynamics are those ofZt : t ≤ T issued fromx∈ R and, givenFT , on the event thatT < ∞, it continues to evolve on the time horizon[T,∞) asthe unique solution, sayZ, to (8.1) but driven by the Levy processX = XT+s−XT :s≥ 0 and issued fromZT . Note that, by construction, onT < ∞, the dependenceof Zt : t ≥ T on Zt : t ≤ T occurs only through the valueZT = ZT . Note alsothat fort > 0,

ZT+t = Zt

= ZT + Xt− δ∫ t

01Zs>bds

= x+XT− δ∫ T

01Zs>bds+(XT+t −XT)− δ

∫ t

01ZT+s>bds

= x+XT+t − δ∫ T+t

01Zs>bds,

thereby showing thatZ solves (8.1) withZ0 = x. Since (8.1) has a unique pathwisesolution, this solution must beZ and therefore possesses the strong Markov property.


8.2 Gambler’s ruin and resolvent density

Let us now introduce the stopping times forZ,

κ+a := inft > 0 : Zt > a andκ−0 := inft > 0 : Zt < 0,

wherea> 0. We are interested in studying the ruin probability

Px(κ−0 < ∞), (8.2)

as well as the expected present value of dividends paid untilruin,

δEx

(∫ κ−0

0e−qt1Zt>bdt

). (8.3)

Not unlike our treatment of the analogous object for the Cramer–Lundberg process,it turns out to be more convenient to first study the seeminglymore complex two-sided exit problem. To this end, letXδ = Xδ

t : t ≥ 0, whereXδt = Xt − δ t and

denote by Px the law of the processXδ when issued fromx (with Ex as the associatedexpectation operator). For eachq≥ 0, W(q) andZ(q) denote, as usual, theq-scalefunctions associated withX. We shall writeW(q) for theq-scale function associatedwith Xδ . For convenience, we will write

w(q)(x;y) =W(q)(x− y)+ δ1(x≥b)

∫ x

bW

(q)(x− z)W(q)′(z− y)dz,

for x,y ∈ R andq≥ 0. We have two main results concerning the gambler’s ruinproblem, from which more can be said about the quantities (8.2) and (8.3).

Theorem 8.2.For q≥ 0 and0≤ x,b≤ a, we have

Ex

(e−qκ+

a 1κ+a <κ−0

)=

w(q)(x;0)

w(q)(a;0). (8.4)

Theorem 8.3.For q≥ 0 and0≤ x,y,b≤ a,∫ ∞

0e−qt

Px(Zt ∈ dy, t < κ−0 ∧κ+a )dt

= 1y∈[b,a]

w(q)(x;0)

w(q)(a;0)W

(q)(a− y)−W(q)(x− y)

dy

+1y∈[0,b)

w(q)(x;0)

w(q)(a;0)w(q)(a;y)−w(q)(x;y)

dy. (8.5)

Although appealing to relatively straightforward methods, the proofs are quitelong, requiring a little patience.

8.2 Gambler’s ruin and resolvent density 71

Proof (of Theorem 8.2).Write p(x,δ ) = Ex(e−qκ+a 1κ+

a <κ−0 ). Suppose thatx≤ b.

Then, by conditioning onFτ+b, we have

p(x,δ ) = Ex

(e−qτ+b 1τ−0 >τ+b

)p(b,δ ) =

W(q)(x)

W(q)(b)p(b,δ ), (8.6)

where, in the last equality, we have used Theorem 4.5. Suppose now that b≤ x≤ a.Using, Theorem 4.5, the strong Markov property (8.6) and theGerber-Shiu measurefrom Theorem 5.5, we have

p(x,δ )

= Ex

(e−qτ+a 1τ−b >τ+a

)+Ex

(e−qτ−b 1τ−b <τ+a p(Zτ−b

,δ ))

=W(q)(x−b)

W(q)(a−b)+

p(b,δ )W(q)(b)

Ex

(e−qτ−b 1τ−b <τ+a W

(q)(Xδτ−b))

=W(q)(x−b)

W(q)(a−b)+

p(b,δ )W(q)(b)

h(a,b,x), (8.7)

where

h(a,b,x)

=

∫ a−b

0

∫

(y,∞)W(q)(b+ y−θ )

×[W(q)(x−b)W(q)(a−b− y)

W(q)(a−b)−W

(q)(x−b− y)

]F(dθ )dy.

By settingx= b in (8.7) and recalling from Lemma 6.3 thatW(q)(0) = 1/(c− δ ),we can now solve forp(b,δ ). Indeed, we have

p(b,δ ) =W(q)(b)

(c− δ )W(q)(a−b)W(q)(b)

−∫ a−b

0

∫

(y,∞)W(q)(b+ y−θ )W(q)(a−b− y)F(dθ )dy

−1

. (8.8)

Next, we want to simplify the term involving the double integral in the above ex-pression.

To this end, note that whenδ = 0 (the case that there is no refraction), we have,again by Theorem 4.5, that, for allx≥ 0,

p(b,0) = Eb

(e−qτ+a 1τ−0 >τ+a

)=

W(q)(b)

W(q)(a). (8.9)

It follows, by comparing (8.8) (forδ = 0) with (8.9), that


∫ a−b

0

∫

(y,∞)W(q)(b+ y−θ )W(q)(a−b− y)F(dθ )dy

= cW(q)(b)W(q)(a−b)−W(q)(a). (8.10)

As a≥ b is taken arbitrarily, we may take Laplace transforms ina on the interval(b,∞) of both sides of the above expression. Denote byLb the operator which sat-isfiesLb f [λ ] :=

∫ ∞b e−λ x f (x)dx and letλ > Φ(q). For the left-hand side of (8.10),

with the help of Fubini’s Theorem, we get

∫ ∞

be−λ x

∫ ∞

0

∫

(y,∞)W(q)(b+ y+θ )W(q)(x−b− y)dyF(dθ )dx

=e−λ b

ψ(λ )−q

∫ ∞

0

∫

(y,∞)e−λ yW(q)(b+ y−θ )F(dθ )dy.

For the right-hand side of (8.10), we get∫ ∞

be−λ x

(W(q)(x−b)cW(q)(b)−W(q)(x)

)dx

=e−λ b

ψ(λ )−qcW(q)(b)−

∫ ∞

be−λ xW(q)(x)dx,

and so∫ ∞

0

∫

(y,∞)e−λ yW(q)(b+ y−θ )F(dθ )dy

= cW(q)(b)− (ψ(λ )−q)eλ bLbW

(q)[λ ], (8.11)

for λ > Φ(q). Our objective now is to use (8.11) and show that, forq≥ 0 andx≥ b,we have

∫ ∞

0

∫

(y,∞)W(q)(b+ y−θ )W(q)(x−b− y)F(dθ )dy

=−W(q)(x)+ (c− δ )W(q)(b)W(q)(x−b)

−δ∫ x

bW

(q)(x− y)W(q)′(y)dy. (8.12)

We will do this by taking Laplace transforms inx on (b,∞) on both sides of theequality in (8.12). To this end, note that, by (8.11), with the help of Fubini’s Theo-rem, the Laplace transform of the left-hand side of (8.12) equals∫ ∞

be−λ x

∫ ∞

0

∫

(y,∞)W(q)(b+ y−θ )W(q)(x−b− y)F(dθ )dydx

=e−λ b

ψ(λ )− δλ −q

(cW(q)(b)− (ψ(λ )−q)eλ b

LbW(q)[λ ]

), (8.13)

8.2 Gambler’s ruin and resolvent density 73

whereλ > ϕ(q) and, forq≥ 0, ϕ(q) = supθ ≥ 0 : ψ(θ )−cθ = q. (Note thatϕis the right inverse of the Laplace exponent ofXδ .) Since

Lb

(∫ x

bf (x− y)g(y)dy

)[λ ] = (L0 f )[λ ](Lbg)[λ ]

and, forλ > Φ(q),

LbW(q)′[λ ] = λLbW

(q)[λ ]−e−λ bW(q)(b)

(which follows from integration by parts), we have that the Laplace transform of theright-hand side of (8.12) is equal to the right-hand side of (8.13), for all sufficientlylargeλ . Hence (8.12) holds for almost everyx≥ b. Because both sides of (8.12) arecontinuous inx, we finally conclude that (8.12) holds for allx≥ b.

To complete the proof, it suffices to plug (8.12) and the expression forh(a,b,x)into (8.7) and the desired identity follows after straightforward algebra.

In anticipation of the proof of Theorem 8.3, we shall note here a particular iden-tity which follows easily from (8.12). That is, forv≥ u≥m≥ 0,

∫ ∞

0

∫

(z,∞)W(q)(z−θ +m)

×[W(q)(v−m− z)

W(q)(v−m)W

(q)(u−m)−W(q)(u−m− z)

]F(dθ )dz

=−W(q)(u−m)

W(q)(v−m)

(W(q)(v)+ δ

∫ v

mW

(q)(v− z)W(q)′(z)dz

)

+W(q)(u)+ δ∫ u

mW

(q)(u− z)W(q)′(z)dz. (8.14)

Proof (of Theorem 8.3).For BorelB⊆ [0,a] andx,q≥ 0, define

V(q)(x,a,B) =∫ ∞

0e−qt

Px(Zt ∈ B, t < κ−0 ∧κ+a )dt.

Forx≤ b, by the strong Markov property, Theorem 4.5 and Theorem 5.2, we have


V(q)(x,a,B) = Ex

(∫ τ+b

0e−qt1Zt∈B,t<κ+

a ∧κ−0 dt

)

+Ex

(∫ ∞

τ+be−qt1Zt∈B,t<κ+

a ∧κ−0 ,τ+b <τ−0 dt

)

= Ex

(∫ τ+b ∧τ−0

0e−qt1Xt∈Bdt

)

+Ex

(e−qτ+b 1τ+b <τ−0

)V(q)(b,a,B)

=

∫

B

(W(q)(b− y)

W(q)(b)W(q)(x)−W(q)(x− y)

)dy

+W(q)(x)

W(q)(b)V(q)(b,a,B). (8.15)

Moreover, for b≤ x≤ a, we have, using similar arguments,

V(q)(x,a,B)

=∫ ∞

0e−qtPx

(Xδ

t ∈ B∩ [b,a], t < τ−b ∧ τ+a)

dt

+Ex

(1τ−b <τ+a e

−qτ−b V(q)(Xδτ−b,a,B)

)

=∫

B∩[b,a]

(W(q)(a− z)

W(q)(a−b)W

(q)(x−b)−W(q)(x− z)

)dz

+

∫ ∞

0

∫

(−∞,−z)

∫

B

[W(q)(b− y)

W(q)(b)W(q)(z+θ +b)−W(q)(z+θ +b− y)

]dy

+V(q)(b,a,B)

W(q)(b)W(q)(z+θ +b)

×[W(q)(a−b− z)

W(q)(a−b)W

(q)(x−b)−W(q)(x−b− z)

]F(dθ )dz,

where in the first equality we have used the strong Markov property and in thesecond equality we have again used the Gerber-Shiu measure from Theorem 5.5.Next, we shall apply the identity (8.14) twice in order to simplify the expression forV(q)(x,a,B), a≥ x≥ b. We use it once by settingm= b, u= x, v= a and once bysettingm= b− y andu= x− y, v= a− y for y∈ [0,b]. We obtain

8.3 Resolvent density with ruin 75

V(q)(x,a,B)

=∫

B∩[b,a]

(W(q)(a− z)

W(q)(a−b)W

(q)(x−b)−W(q)(x− z)

)dz

+

∫

B∩[0,b)

W(q)(b− y)

W(q)(b)

(−W

(q)(x−b)

W(q)(a−b)w(q)(a;0)+w(q)(x;0)

)

−(−W(q)(x−b)

W(q)(a−b)w(q)(a;y)+w(q)(x;y)

)dy

+V(q)(b,a,B)

W(q)(b)

(−W(q)(x−b)

W(q)(a−b)w(q)(a;0)+w(q)(x;0)

). (8.16)

Settingx = b in (8.16), we get an expression forV(q)(b,a,B) in terms of itself.Solving this and then putting the resulting expression forV(q)(b,a,B) back in (8.15)and (8.16) leads to (8.5), which completes the proof.

8.3 Resolvent density with ruin

The two expressions we are interested in, namely the ruin probability and the ex-pected present value of dividends paid until ruin, can both be extracted from theidentity for the potential measure ofZ on [0,∞),

∫ ∞

0Px(Zt ∈ B, t < κ−0 )dt = lim

a↑∞V(x,a,B),

whereB is any Borel set in[0,∞). Note that the limit is justified by monotone con-vergence. In order to describe this potential measure, let us introduce some morenotation. Recall thatϕ was defined as the right inverse of the Laplace exponent ofXδ , so that

ϕ(q) = supθ ≥ 0 : ψ(θ )− δθ = q.

Corollary 8.4. For x,y,b≥ 0 and q≥ 0,∫ ∞

0e−qt

Px(Zt ∈ dy, t < κ−0 )dt

= 1y∈[b,∞)

w(q)(x;0)

δ∫ ∞

b e−ϕ(q)zW(q)′(z)dze−ϕ(q)y−W

(q)(x− y)

dy

+1y∈[0,b)

∫ ∞b e−ϕ(q)zW(q)′(z− y)dz∫ ∞

b e−ϕ(q)zW(q)′(z)dzw(q)(x;0)−w(q)(x;y)

dy. (8.17)

Proof. Assume thatq> 0. We begin by noting that, from the representation ofW(q)

in (4.4), it is straightforward to deduce that, for allx,q> 0,


lima↑∞

W(q)(a− x)

W(q)(a)= e−ϕ(q)x. (8.18)

Note that, for eachq≥ 0, ϕ(q)≥Φ(q) and hence, appealing to the same represen-tation in (4.4) for bothW(q) andW(q), it also follows that, for allq,x> 0,

lima↑∞

W(q)(a− x)

W(q)(a)= 0. (8.19)

For q> 0, the result we are after is obtained by dividing the numerator and de-nominator of each of the first terms in the curly brackets of (8.5) byW(q)(a) andtaking limits asa↑∞, making use of (8.18) and (8.19). The case thatq=0 is handledby taking limits asq ↓ 0 in (8.17).

Now we are in a position to derive expressions for (8.2) and (8.3).

Corollary 8.5. For x≥ 0, if ψ ′(0+) ≤ δ , thenPx(κ−0 < ∞) = 1. Otherwise, whenψ ′(0+)> δ , we have

Px(κ−0 < ∞)

= 1− ψ ′(0+)− δ1− δW(b)

(W(x)+ δ1(x≥b)

∫ x

bW(x− y)W′(y)dy

). (8.20)

Proof. Let Zt = infs≤t Zs and, as usual,eq denotes an independent and exponentiallydistributed random variable with mean 1/q. Note that, forq> 0,

Ex

(e−qκ−0 1κ−0 <∞

)= 1−Px(Zeq

≥ 0)

= 1−q∫ ∞

0e−qt

Px(Zt ∈ [0,∞), t < κ−0 )dt.

Computing the integral above from (8.17) is relatively straightforward and gives us,for x,b≥ 0 andq> 0,

Ex(e−qκ−0 1κ−0 <∞)

= z(q)(x)− q∫ ∞

b e−ϕ(q)yW(q)(y)dy∫ ∞b e−ϕ(q)yW(q)′(y)dy

w(q)(x;0)

+q1(x≥b)

∫ x

bW

(q)(x− z)dz+q∫ b

0W(q)(x− z)dz

−q∫ x

0W(q)(z)dz−qδ1(x≥b)

∫ x

bW

(q)(x− z)W(q)(z−b)dz, (8.21)

where

z(q)(x) = Z(q)(x)+ δq∫ x

bW

(q)(x− z)W(q)(z)dz, x∈ R,q≥ 0.

8.3 Resolvent density with ruin 77

The details of the computation are left to the reader.Although it is not immediately obvious, it turns out that thelast four terms

in (8.21) sum to precisely zero. Indeed, in the case thatx < b, this observationis straightforward, noting that the indicators preceding the two integrals from bto x are identically zero and the second of these four terms may bereplaced by∫ x

0 W(q)(x− z)dz=∫ x

0 W(q)(z)dz on account of the fact thatW(q) is identically zeroon (−∞,0). In the case thatx≥ b, the last four terms of (8.21) can be easily rear-ranged to be equal toC(x−b), whereC : [0,∞)→ [0,∞) is the continuous function

C(u) = q∫ u

0W

(q)(z)dz−q∫ u

0W(q)(z)dz−qδ

∫ u

0W

(q)(z)W(q)(u− z)dz.

Taking Laplace transforms ofC and using (4.1), we easily verify thatC is identicallyzero.

In conclusion, we have that, forx,b≥ 0 andq> 0,

Ex(e−qκ−0 1κ−0 <∞) = z(q)(x)− q

∫ ∞b e−ϕ(q)yW(q)(y)dy

∫ ∞b e−ϕ(q)yW(q)′(y)dy

w(q)(x;0).

The expression for the ruin probability in (8.20) by taking limits on the left- andright-hand side above asq ↓ 0. On the left-hand side, thanks to monotone conver-gence, the limit is equal toPx(κ−0 < ∞). Computing the limits on the right-hand sideis relatively straightforward, taking account of the fact that

∫ ∞

0e−ϕ(q)zW(q)(z)dz=

1ϕ(q)δ

(8.22)

and the fact that

limq↓0

qϕ(q)

= limq↓0

ψ(ϕ(q))− δϕ(q)ϕ(q)

= 0∨ (ψ ′(0+)− δ ).

The details are again left to the reader.

Corollary 8.6. For x,q,b≥ 0,

Ex

(∫ κ−0

0e−qtδ1Zt>bdt

)= −δ1(x≥b)

∫ x

bW

(q)(z−b)dz

+W(q)(x)+ δ1(x≥b)

∫ xb W

(q)(x− y)W(q)′(y)dy

ϕ(q)∫ ∞

0 e−ϕ(q)yW(q)′(y+b)dy.

Proof. The proof is a simple exercise in integrating the potential measure (8.17)over(b,∞).


8.4 Comments

The terminology “refraction” comes from Gerber and Shiu (2006b). See also Gerber and Shiu(2006a). As indicated in the introduction, refraction strategies emerge as an opti-mal solution to (1.7). Kyprianou et al. (2012) offer a general perspective on howto choose the refraction threshold b optimally for the setting thatX is a spectrallynegative Levy process. Making use of Corollary 8.6, they introduce the function

r(b) = ϕ(q)∫ x

0e−ϕ(q)yW(q)′(y+b)dy, b≥ 0.

Note in particular, from the same corollary, the expected present value of dividendsuntil ruin with refraction threshold b≥ 0 is equal toW(q)(x)/r(b), whenZ0 = x≤ b.With a similar flavour to the case of reflection strategies, Kyprianou et al. (2012)give sufficient conditions under which the optimal threshold b should be chosen tominimise the functionr(b).

The majority of the arguments in this chapter are taken from Kyprianou and Loeffen(2010), whereX is taken to be a general spectrally negative Levy process. The ques-tion of existence and uniqueness for the SDE (8.1) whenX is a general Levy processhas not yet been handled in the literature. In particular, when X has no Gaussiancomponent, although still a simple-looking SDE, (8.1) liesoutside of the realms ofthe standard theory of stochastic differential equations.Further fluctuation identitiesfor refracted spectrally negative Levy processes can be found in Kyprianou et al.(2013).

Chapter 9Concluding discussion

On the one hand, the use of scale functions would appear to have made many ofthe problems we have looked at solvable. On the other hand, one may also questionthe extent to which we have solved the posed problems as our scale functions areonly defined up to a Laplace transform. Therefore we have arguably only provided asolution “up to the inversion of a Laplace transform”. It would be nice to have someconcrete examples of scale functions. It turns out that few concrete examples areknown and they are quite difficult to produce in general. Nonetheless, we shall showthat there is still sufficient analytical structure known for a general scale function tojustify their use, in particular when moving to the bigger class of processes for whichthe surplus process is modelled by a general spectrally negative Levy process.

9.1 Mixed-exponential claims

In general, it is quite hard to construct the scale functionW(q) for a Cramer–Lundberg process. There are a handful of claim distributions, F , for which thereis a reasonable degree of tractability as far as scale functions are concerned. Onesuch example is the case thatF belongs to the class ofmixed-exponential distribu-tions. This family is also known as thehyper-exponential distributions.. In order tospecify these processes, let us define the arrival rate

λ =m

∑j=1

a j/ρ j (9.1)

and the claims distribution by

F(dx) =1λ

(m

∑j=1

a je−ρ j x

)dx, x> 0, (9.2)

79

80 9 Concluding discussion

wherem∈ N, and, for j = 1, · · · ,m, the coefficientsa j andρ j are strictly positive.For convenience, we also assume that theρ j are arranged in increasing order. Thepremium ratec may be taken valued in(0,∞) without restriction.

It is easily verified that the Laplace exponent is given by

ψ(θ ) = cθ −θm

∑j=1

a j

ρ j(ρ j +θ ), θ ≥ 0.

A little thought reveals that the exponentψ is, in fact, well defined as a mappingfromC into C, provided one is prepared to see it as a meromorphic functionwhichhas only singular poles atz=−ρ j . Henceforth, we shall treatψ in the broader senseof a complex-valued function. We know from our previous analysis that, forq> 0,the equationψ(θ ) = q has a unique solutionz= Φ(q) in the half-plane Re(θ )> 0,and we also know that this solution is real. It can be proved that if we look for otherroots of the equationψ(θ ) = q on the complex plane, then there are preciselym ofthem and they are all to be found on the negative part of the real axis. If we writethese solutions asθ = −ζ j , for j = 1, · · · ,m, then it also turns out that they satisfythe interlacing property

0< ζ1 < ρ1 < ζ2 < ρ2 < · · ·< ζm < ρm. (9.3)

The following lemma gives an explicit formula for the scale function when ex-pressed in terms of the rootsζ j and the first derivative of the Laplace exponent.

Lemma 9.1.If X is a Cramer–Lundberg process withλ and F given by (9.1) and(9.2), respectively, then, for all q> 0, the scale function is given by

W(q)(x) =eΦ(q)x

ψ ′(Φ(q))+

m

∑j=1

e−ζ j x

ψ ′(−ζ j), x≥ 0. (9.4)

Proof. We sketch the proof. Knowing that−ζm, · · · ,−ζ1,Φ(q) are all roots ofthe equationψ(θ ) = q in C, the basic idea is to use partial fractions to write

1ψ(θ )−q

=c0

(θ −Φ(q))+

m

∑j=1

c j

(θ + ζ j), θ ∈ C. (9.5)

To determine the constantsc0, note that, for example,

1ψ ′(Φ(q))

= limθ→Φ(q)

(θ −Φ(q))ψ(θ )−q

= c0+m

∑j=1

limθ→Φ(q)

c j(θ −Φ(q))(θ + ζ j)

= c0.

One derivesc1, · · · ,cm similarly. The identity (9.4) now follows by inverting (9.5)in a straightforward way.

9.2 Spectrally negative Levy processes 81

9.2 Spectrally negative Levy processes

One of the advantages working with scale functions is that all of the results, as wellas many of their proofs, are practically identical if we replace the Cramer–Lundbergprocess by a general spectrally negative Levy process. Recall from Chapter 1 thatX is a spectrally negative Levy process if it has stationary independent increments,it has paths that are almost surely right-continuous with left limits, there are nopositive discontinuities in its trajectories and that its paths are not monotone.

A simple example of a spectrally negative Levy process would be the resultingobject we would get from adding a Cramer–Lundberg process to a (scaled) indepen-dent Brownian motion, i.e.

Xt = σBt +ct−Nt

∑i=1

ξi , t ≥ 0, (9.6)

whereBt : t ≥ 0 is a standard independent Brownian motion andσ ≥ 0. Thisprocess is often referred to as aperturbedCramer–Lundberg process.

In general, spectrally negative Levy processes can be characterised through theirLaplace exponent, also denoted byψ , which satisfies

ψ(θ ) :=1t

logE[eθXt ],

and is well defined forθ ≥ 0. The Levy-Khintchine formula, which is normallycited for the characteristic exponent of a Levy process, also identifies the Laplaceexponent in its general form as

ψ(θ ) = aθ +12

σ2θ 2+

∫

(0,∞)(e−θx−1+θx1(x<1))Π(dx), θ ≥ 0, (9.7)

wherea∈ R, σ2 ≥ 0 and the so-calledLevy measureΠ is a (not-necessarily finite)measure concentrated on(0,∞) which satisfies the integrability condition

∫

(0,∞)(1∧x2)Π(dx)< ∞.

Althoughψ looks more complicated than for the case of a Cramer–Lundberg pro-cess, its shape is essentially the same. Indeed, it is not difficult to show, againby differentiation, thatψ is a strictly convex function which satisfiesψ(0) = 0and ψ(∞) = ∞. Moreover, just as with the Cramer–Lundberg process, we haveE(X1) = ψ ′(0+) ∈ [−∞,∞). Accordingly, we may also work with the right inverseof ψ ,

Φ(q) := supθ ≥ 0 : ψ(θ ) = q, q≥ 0.

Just as is the case with Cramer–Lundberg processes, the quantity Φ(0) is strictlypositive if and only ifψ ′(0+) < 0 and otherwise, it is zero. In this respect, Fig.


2.1 could equally depict the Laplace exponent of a general spectrally negative Levyprocess.

We should note that the class of spectrally negative Levy processes contains theclass of Cramer–Lundberg processes. Indeed, this can be seen by takingΠ(dx) =λF(dx) on (0,∞) andσ = 0. In that case, asF is now a finite measure, (9.7) can bewritten

ψ(θ ) =(

a+λ∫

(0,1)xF(dx)

)θ −λ

∫

(0,∞)(1−e−θx)F(dx), θ ≥ 0.

The exclusion of monotone paths from the definition of spectrally negative Levyprocesses would force us to take

c := a+λ∫

(0,1)xF(dx)> 0. (9.8)

Describing the paths of the Levy processX = Xt : t ≥ 0 associated toψ isnot as straightforward as in the case of a Cramer–Lundberg process. It is clear thatthe quadratic termaθ + σ2θ 2/2 is the result of an independent linear Browniancomponentat+σBt : t ≥ 0 in X. The integral term inψ can be written

∫

(0,∞)(e−θx−1+θx1(x<1))Π(dx)

= ∑n≥1

cnθ −λn

∫

(2−n,2−(n−1)](1−e−θx)

Π(dx)λn

−λ0

∫

(0,∞)(1−e−θx)

Π(dx)λ0

, θ ≥ 0, (9.9)

whereλ0 = Π((1,∞)) and, forn≥ 1,

cn =

∫

(2−n,2−(n−1)]xΠ(dx) andλn = Π((2−n,2−(n−1)])

Note that, ifλn = 0 for somen≥ 0, then we should understand the relevant term onthe right-hand side of (9.9) as absent.

The decomposition (9.9) gives us the intuitive understanding that, for the giventriplet (a,σ ,Π), the associated spectrally negative Levy process may be seen asthe independent sum of a linear Brownian component, a seriesof Cramer–Lundbergprocesses and the negative of a compound Poisson process. The special choice ofcn,for n≥ 1, means that each of the Cramer–Lundberg processes have zero mean (infact they are martingales). Moreover then-th Cramer–Lundberg process experiencesjumps whose magnitude falls strictly into the interval(2−n,2−(n−1)]. Meanwhile,the compound Poisson process experiences jumps which are ofmagnitude strictlygreater than 1.

The resulting path of the superimposition of these processes can be quite varied.Indeed, over any finite time horizon, there will be an almost surely infinite (albeit

9.2 Spectrally negative Levy processes 83

countable) number of jumps if and only ifΠ is an infinite measure. Moreover,Xhas paths of bounded variation (almost surely over each finite time horizon) if andonly if

∫(0,1) xΠ(dx) < ∞ andσ = 0. Finally, X is a Cramer–Lundberg process if

and only ifσ = 0 andΠ has finite total mass.

For spectrally negative Levy processes, we may also define scale functionsW(q)(x), q ≥ 0,x ∈ R, in exactly the same way as we did for Cramer–Lundbergprocesses. In particular, forq≥ 0,W(q)(x) = 0 for x< 0 and otherwise, on[0,∞), itis the unique right-continuous increasing function whose Laplace transform satisfies

∫ ∞

0e−θxW(q)(x)dx=

1ψ(θ )−q

, θ > Φ(q). (9.10)

The majority of the main results presented in the previous chapters and indeedmany of their proofs, are still valid when the setting of a Cramer–Lundberg processis replaced by a general spectrally negative Levy process.We are thus brought againto the question of the existence of concrete examples of scale functions.

Not surprisingly, it is also difficult to find closed form examples of scale functionsfor a spectrally negative Levy process that is not a Cramer–Lundberg process. Hereare a couple of related examples however.

A spectrally negativeα-stable process, forα ∈ (1,2), has Levy measure

Π(dx) =kα

x1+α dx, x> 0,

wherekα is a constant that can be chosen appropriately so that

ψ(θ ) = θ α , θ ≥ 0.

Note thatψ ′(0+) = 0 and hence theα-stable process oscillates.Denote by

Eα ,β (x) = ∑n≥0

xn

Γ (nα +β ), x∈R, (9.11)

the two-parameter Mittag-Leffler function. It is characterised by a Laplace-Fouriertransform. Specifically, forλ ∈ R, θ ∈C andℜ(θ )> |λ |1/α , we have

∫ ∞

0e−θxxβ−1

Eα ,β (λxα)dx=θ α−β

θ α −λ. (9.12)

We recognise immediately that

W(q)(x) = xα−1Eα ,α(qxα), q,x≥ 0.

Suppose that we look at theα-stable process under the Esscher transform (2.7).As alluded to above, the main part of this result is still valid for general spectrally


negative Levy processes. In particular, we note that the class of spectrally negativeLevy processes is closed under the Esscher transformation. For eachγ ≥ 0 andα ∈ (1,2), if (X,P) is an α-stable process, then(X,Pγ) is a spectrally negativeLevy process with Laplace exponent

ψγ (θ ) = (θ + γ)α − γα , θ ≥−γ.

This process is also known as atempered stable process. Like the α-stable pro-cess, it has no Gaussian component. Similarly to the conclusion of Theorem 2.3 forCramer–Lundberg processes, the effect of the Esscher transform is to exponentiallytilt the Levy measure so that the new Levy measure satisfies

Πγ(dx) =kαe−γx

x1+α dx, x> 0.

Note also thatψ ′γ(0+) = ψ ′(γ)> 0 and hence the process drifts to+∞.Just as above, we may derive, by inspection of (9.12), the following identity for

W(q)γ , theq-scale function of(X,Pγ):

W(q)γ (x) = e−γxxα−1

Eα ,α((q+ γα)xα), x≥ 0.

9.3 Analytic properties of scale functions

Despite the fact that, for most spectrally negative Levy process, we are unable toinvert the Laplace transform (9.10), we can nonetheless getan understanding of theshape of the general scale function. Here is a summary of someof the known facts,most of which can be derived from the Laplace transform (9.10).

Continuity at the origin

For all q≥ 0,

W(q)(0+) =

0 if σ > 0 or

∫(0,1)xΠ(dx) = ∞

c−1 if σ = 0 and∫(0,1) xΠ(dx)< ∞,

(9.13)

wherec= a+∫(−1,0)xΠ(dx).

Derivative at the origin

For all q≥ 0,

9.3 Analytic properties of scale functions 85

W(q)′+ (0+) =

2/σ2 if σ > 0∞ if σ = 0 andΠ((0,∞)) = ∞(q+Π((0,∞))/c2 if σ = 0 andΠ((0,∞))< ∞.

(9.14)

Behaviour at+∞ for q= 0

As x ↑ ∞ we have

W(x)∼

1/ψ ′(0+) if ψ ′(0+)> 0eΦ(0)x/ψ ′(Φ(0)) if ψ ′(0+)< 0.

(9.15)

WhenE(X1) = 0 a number of different asymptotic behaviours may occur. Forex-ample, ifφ(θ ) := ψ(θ )/θ satisfiesφ ′(0+)< ∞ thenW(x)∼ x/φ ′(0+) asx ↑ ∞.

Behaviour at+∞ for q> 0

As x ↑ ∞ we haveW(q)(x)∼ eΦ(q)x/ψ ′(Φ(q)) (9.16)

and thus there is asymptotic exponential growth.

Smoothness

It is known that ifX has paths of bounded variation, then, for allq≥ 0,W(q)|(0,∞) ∈C1(0,∞) if and only if Π has no atoms. In the case thatX has paths of unboundedvariation, it is known that, for allq≥ 0, W(q)|(0,∞) ∈C1(0,∞). Moreover ifσ > 0,thenC1(0,∞) may be replaced byC2(0,∞). Clearly this picture is incomplete.

Taking account of the fact that the Laplace transform ofW(q) is expressed interms ofψ(θ ), which, itself, can be considered as a type of analytical transform ofthe measureΠ , it is not surprising that there is an intimate connection between thesmoothness of the scale functions and the Levy measure. Whilst there are a numberof existing results connecting the two (see Sect. 9.5), a general result remains atlarge.

Concavity and convexity

If x 7→Π(x), x> 0, is a completely monotone function1, then, for allq> 0,W(q)′(x),x> 0, is convex. Note in particular, the latter implies that there exists ana∗≥ 0 such

1 A function f : (0,∞)→ [0,∞) is completely monotone if, for alln∈ N,

(−1)n dn f (x)dxn ≥ 0.


thatW(q) is concave on(0,a∗) and convex on(a∗,∞). In the case thatψ ′(0+) ≥ 0and q = 0, the same conclusion holds witha∗ = ∞, which is to say thatW is aconcave function. More generally, we have the following result.

Theorem 9.2.Suppose thatΠ is log-convex. Then for all q≥ 0, W(q) has a log-convex first derivative.

(Note that a completely monotone function is log-convex anda log-convex functionis also convex.)

9.4 Engineered scale functions

Within the class of spectrally negative Levy processes, there are a number of meth-ods for generating examples of scale functions withq = 0. Rather than trying toinvert (9.10) for a givenψ , the idea is to construct aψ corresponding to a givenW.We outline one method here, which is based around theWiener-Hopf factorisation.

For the purpose of this discussion, the Wiener-Hopf factorisation concerns theLaplace exponent ofX and takes the form:

ψ(θ ) = (θ −Φ(0))φ(θ ), θ ≥ 0, (9.17)

whereφ , is a so-calledBernstein function. Specifically, the termφ(θ ) must neces-sarily take the form

φ(θ ) = κ + δθ +∫

(0,∞)(1−e−θx)ϒ (dx), θ ≥ 0, (9.18)

whereκ ,δ ≥ 0 andϒ is a measure concentrated on(0,∞) which satisfies∫(0,∞(1∧

x)ϒ (dx) < ∞. Roughly speaking, this factorisation can be proved by recalling thatψ(Φ(0)) = 0 and then manually factoring out(θ −Φ(0)) from ψ by using integra-tion by parts to deal with the integral part of (9.7). It turnsout that

ϒ ((x,∞)) = eΦ(0)x∫ ∞

xe−Φ(0)uΠ(u)du for x> 0, (9.19)

δ = σ2/2 andκ = ψ ′(0+)∨0.It is remarkable thatφ is also the Laplace exponent of a subordinator2 which is

sent to the cemetery state+∞ after an independent and exponentially distributedrandom time with rateκ .3 If we write this processH = Ht : t ≥ 0 and letζ =inft > 0 : Ht =+∞, then, for allt ≥ 0,

2 A subordinator is a Levy process with non-decreasing paths.3 Recall our convention that an exponential random variable with rate 0 is defined to be infinite-valued with probability 1.

9.4 Engineered scale functions 87

φ(θ ) =−1t

logE(

e−θHt 1(t<ζ )

), θ ≥ 0.

What is even more remarkable is that the range ofHt : t < ζ agrees precisely withthe range of the process−Xt : t ≥ 0. Accordingly we callH the descending ladderheight process ofX.

In the special case thatΦ(0) = 0, that is to say, the processX does not drift to−∞, or equivalently thatψ ′(0+) ≥ 0, it can be shown that the scale functionWdescribes the renewal measure ofH. Indeed, recall that the renewal measure ofH isdefined by

U (dx) =∫ ∞

0dt ·P(Ht ∈ dx, t < ζ ), for x≥ 0. (9.20)

Calculating its Laplace transform we get the identity

∫ ∞

0e−θx

U (dx) =1

φ(θ )for θ > 0. (9.21)

Recall that we can integrate (9.10) by parts and get

∫

[0,∞)e−θxW(dx) =

θψ(θ )

=1

φ(θ ), θ ≥ 0.

Hence, it appears thatW agrees precisely with the renewal function,U , of the sub-ordinatorH that appears in the Wiener-Hopf factorisation.

It can be shown similarly that, whenΦ(0) > 0, the scale function is related tothe renewal measure ofH by the formula

W(x) = eΦ(0)x∫ x

0e−Φ(0)y

U (dy), x≥ 0. (9.22)

This relationship between scale functions and renewal measures of subordina-tors lies at the heart of the approach we shall describe in this section for engineeringscale functions. A key to the method is the fact that one can find in the literature sev-eral subordinators for which the renewal measure is known explicitly. Should thesesubordinators turn out to be the descending ladder height process of a spectrally neg-ative Levy process, then this would give an exact expression for its scale function.Said another way, we can build scale functions using the following approach.

Step 1.Choose a subordinator, sayH, with Laplace exponentφ , for which oneknows its renewal measure,U , or equivalently, in light of (9.21), one can explicitlyinvert the Laplace transform 1/φ(θ ).

Step 2.Choose a constantϕ ≥ 0 and verify whether the relation

ψ(θ ) := (θ −ϕ)φ(θ ), θ ≥ 0,

defines the Laplace exponent of a spectrally negative Levy process.


Step 3.Once steps 1 and 2 are verified, then the scale function of the spectrallynegative Levy process we have generated is given by (9.22).

Of course, for this method to be useful we should first providenecessary andsufficient conditions for the pair(H,ϕ) to belong to the Wiener-Hopf factorisationof a spectrally negative Levy process.

The following theorem shows how one may identify a spectrally negative LevyprocessX (called theparent process) for a given pair(H,ϕ). The proof follows bya straightforward manipulation of the Wiener-Hopf factorization (9.17).

Theorem 9.3.Suppose that H is a subordinator, killed at rateκ ≥ 0, with drift δ ≥ 0and Levy measureϒ which is absolutely continuous with non-increasing density.Suppose further thatϕ ≥ 0 is given such thatϕκ = 0. Then there exists a spectrallynegative Levy process X, henceforth referred to as the ‘parent process’, whose de-scending ladder height process is precisely the process H. The Levy triplet(a,σ ,Π)of the parent process is uniquely identified as follows. The Gaussian coefficient isgiven byσ =

√2δ . The Levy measure is given by

Π(x) = ϕϒ (x,∞)+dϒdx

(x).

Finally, a can be chosen such that

ψ(θ ) = (θ −ϕ)φ(θ ), (9.23)

for θ ≥ 0 whereφ(θ ) =− logE(e−θH1).Conversely, the killing rate, drift and Levy measure of the descending ladder

height process associated to a given spectrally negative Levy process X are alsogiven by the above formulae when one replacesϕ byΦ(0).

Let us conclude this section by presenting a concrete example of how thismethodology works in practice. Consider a spectrally negative Levy process whichis the parent process of a (killed) tempered stable subordinator. That is to say asubordinator with Laplace exponent given by

φ(θ ) = κ− cΓ (−α)((γ +θ )α− γα) ,

whereα ∈ (−1,1)\ 0, γ ≥ 0 andc> 0.The corresponding Levy measure is given by

Π(dx) = c(ϕ + γ)xα+1 e−γxdx+ c

(α +1)xα+2 e−γxdx, x> 0. (9.24)

This indicates that the jump part of the parent process is theresult of the independentsum of two spectrally negative tempered stable processes with stability parametersα andα +1. We also note from Theorem 9.3 thatσ = 0, indicating the absence ofa Gaussian component.

9.5 Comments 89

If 0 < α < 1, then the jump component is the sum of an infinite activity nega-tive tempered stable subordinator and an independent spectrally negative temperedstable process with infinite variation. If−1≤ α < 0, then the jump part of the par-ent process is the independent sum of a negative tempered stable subordinator withstability parameter 1+α and exponential parameterγ, and an independent negativecompound Poisson subordinator with jumps from a gamma distribution with −αdegrees of freedom and exponential parameterγ.

One easily deduces the following transformations as special examples of (9.12)for θ ,λ > 0, ∫ ∞

0e−θxxα−1

Eα ,α(λxα)dx=1

θ α −λ, (9.25)

and ∫ ∞

0e−θxλ−1x−α−1

E−α ,−α(λ−1x−α)dx=λ

λ −θ α −1, (9.26)

valid for α > 0 and α < 0, respectively. Together with the well-known rules forLaplace transforms concerning primitives and tilting, we may quickly deduce thefollowing expressions for the scale functions associated to the parent process withLaplace exponent given by (9.23) such thatκϕ = 0.

If 0 < α < 1, then, forx≥ 0,

W(x) =eϕx

−cΓ (−α)

∫ x

0e−(γ+ϕ)yyα−1

Eα ,α

(κ + cΓ (−α)γα

−cΓ (−α)yα)

dy.

If −1< α < 0, then, forx≥ 0,

W(x) =eϕx

κ + cΓ (−α)γα

+cΓ (−α)eϕx

(κ + cΓ (−α)γα)2

∫ x

0e−(γ+ϕ)yy−α−1

E−α ,−α

(cΓ (−α)y−α

κ + cΓ (−α)γα

)dy.

9.5 Comments

The case of a Cramer–Lundberg process with mixed exponential jumps can be gen-eralised by taking jumps whose distribution has a Laplace transform that is theratio of two polynomial functions of finite degree (also called a rational Laplacetransform). Another favourite class in the family of jump distributions with ratio-nal Laplace transform (which also contains the class of mixed exponential distri-butions) is the one of phase-type distributions. The tractability of the class of pro-cesses with jumps having rational transform can be routed back to early work ofBorovkov (1976) concerning the Wiener-Hopf factorisation. Many authors haveworked on these types of Cramer–Lundberg processes and it would be impos-sible to give a complete list here. We cite instead two of the most recent ref-erences which give a good overview in the context of Gerber-Shiu type prob-


lems. These are Asmussen and Albrecher (2010), Kuznetsov and Morales (2013)and Egami and Yamazaki (2012). The idea of engineering scalefunctions throughthe Wiener-Hopf factorisation comes from Hubalek and Kyprianou (2010) and Kyprianou and Rivero(2008). Analytical properties of scale functions have beendescribed in a variety ofdifferent papers. A recent summary of these and many more facts can be found inthe review on scale functions found in Cohen et al. (2012).

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gerber–shiu risk theory · gerber–shiu risk theory is not widely used. one might be more...

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