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Continuity Estimates for Ruin ProbabilitiesFarida Enikeeva; Vladimir Kalashnikov; Deimante Rusaityte

To cite this Article Enikeeva, Farida , Kalashnikov, Vladimir and Rusaityte, Deimante(2001) 'Continuity Estimates for RuinProbabilities', Scandinavian Actuarial Journal, 2001: 1, 18 — 39To link to this Article: DOI: 10.1080/034612301750077293URL: http://dx.doi.org/10.1080/034612301750077293

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Scand. Actuarial J. 2001; 1: 18–39 ORIGINAL ARTICLE

Continuity Estimates for Ruin Probabilities

FARIDA ENIKEEVA,1 VLADIMIR KALASHNIKOV2 and DEIMANTERUSAITYTE

Enikeeva F, Kalashnikov V, Rusaityte D. Continuity estimates for ruinprobabilities. Scand. Actuarial J. 2001; 1: 18–39.

A method of continuity analysis of ruin probabilities with respect tovariation of parameters governing risk processes is proposed. It is basedon the representation of the ruin probability as the stationary probabilityof a reversed process. We apply Kartashov’s technique designed forcontinuity analysis of stationary distributions of general Markov chainsin order to obtain desired continuity estimates. The method is illustratedby the Sparre Andersen and Markov modulated risk models. Key words:Ruin probability, re×ersed process, general Marko× chain, continuity esti -mate, stationary distribution, operator norm.

1. INTRODUCTIONThe probability of ruin is one of the basic characteristics of risk processes. It cannot,however, be found in an explicit form for many models of interest. Furthermore,parameters governing these models are often unknown and one can only give somebounds for their values. In such a situation continuity estimates become crucial.

In a general form, the continuity problem can be stated as follows. Let R (t, a) bea risk process governed by a parameter a. For example, if R(t, a) is the classical riskprocess (see Grandell [9] and Kalashnikov [11]), then one can take a¾(l, c, B), wherel is the intensity of the Poisson occurrence process, c is the gross premium rate, Bis the distribution function (d.f.) of claim sizes. If R(t, a) is more complicated riskprocess, then one can de� ne other appropriate parameters. Let us note that even fora relatively simple classical model the parameter takes values from a functional space.For a given process R(t, a), one can de� ne a probability of ruin

ca (x)¾P inftE 0

R (t ; a)B0 R(0, a)¾x .

If denotes the space of possible values of the parameter then one can view atthe ruin probability as at a mapping c : “C, where C is a functional space of allpossible functions ca (x), xE0. Let us equip and C with metrics m and n

respectively. Then we can speak about continuity of the mapping c. Let us call theruin probability c continuous at point a if

1 Supported by Russian Foundation of Fundamental Research (grant 98-01-00855).2 Supported by The Society of Actuaries Committee on Knowledge Extension Research (CKER),

Russian Foundation of Fundamental Research (grant 90-01-00855), and INTAS (grant 98-1625).

© 2001 Taylor & Francis. ISSN 0346-1238

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Scand. Actuarial J. 1 Continuity estimates for ruin probabilities 19

m(a, a Æ)“0 n(ca, caÆ)“0.

The metrics m and n should be convenient from computational point of view andre� ect the core of the problem. For example, if the probability of ruin decaysexponentially as in the Cramer case, then it is reasonable to take the distance in theform

n(ca, caÆ)¾Ä

0

e ox ca¼caÆ (dx) (1.1)

for an appropriate constant o\0. If the probability of ruin decays like a powerfunction (like it is in the presence of large claims with Pareto tails), then it is naturalto consider a metric

n(ca, caÆ)¾Ä

0

x g ca ¼caÆ (dx) (1.2)

for an appropriate constant g\0. The same is true for the choice of metric m. If we� nd an inequality

n(ca, caÆ)0f (m(a, a Æ)), (1.3)

where the non-negative function f is such that f (0)¾0 and f(x)“0 as x“0, thenthe inequality (1.3) is called a continuity estimate as it gives us the possibility tobound the desired quantity n(c (a), c(a Æ)) in terms of the distance m (a, a Æ).

The main purpose of this paper is to propose a general approach allowing us toobtain appropriate continuity estimates for ruin probabilities. We con� ne ourselvesto the estimates of the distance (1.1). One can argue that, in practice, it is moreinteresting to estimate ca(u)¼caÆ(u) for some (or, all) u. However, we can use theinequality

supu

e ou ca (u)¼caÆ(u) 0Ä

0

e ox ca ¼caÆ (dx) (1.4)

in order to get the desired estimate:

ca (u)¼caÆ(u) 0e¼oun(ca, caÆ)

for any uE0. Note that the inequality (1.4) cannot be improved in the class ofmonotonically decreasing functions ca and caÆ.

This approach is based on the following three steps, each step being well-knownin mathematical and actuarial literature. But it seems that their combination opensnew possibilities.

The � rst step consists of identi� cation of the ruin probability with a stationaryprobability for a speci� c random process which is called a re×ersed process (exactde� nitions will be given later on). Such identi� cation is well-known and it has beeninvestigated in recent works by Asmussen [1, 2] where the reader can also � ndfurther references. Let us denote this reversed process by V(t, a). The identi� cationmentioned above means that


F. Enikee×a et al. Scand. Actuarial J. 120

ca (x)¾ limt“ Ä

P(V(t, a)\x).

The second step consists of the embedding V (t, a) into a Markov process byequipping it with supplementary coordinates. So, instead of V, we consider anenlarged process

W(t, a)¾ (V(t, a), Z(t, a)),

which is Markov. Such an embedding is widely used in queueing theory (see Cox[6], Gnedenko & Kovalenko [8]), complex systems theory (see Buslenko et al. [5]),control theory (see Davis [7]), and risk theory (see Schmidli [15]) in order to employstandard Markovain methods for the analysis of the underlying process. In theseterms,

ca (x)¾pa (Gx),

where pa (·) is stationary distribution of the Markov process W (t, a) and

Gx¾{W¾ (V, Z): V\x}.

What is important, one can still regard the probability of ruin as the stationarydistribution of a random process, but now this process is Markov for which theanalysis of stationary distributions can be provided by standard methods.

The third step consists of the application of the continuity theory givingquantitative estimates of possible deviations of stationary distributions of the twoMarkov processes under the comparison governed by parameters a and a Æ respec-tively. More exactly, let n(pa, paÆ) be an appropriate distance between stationarydistributions of two Markov processes W(t, a) and W (t, a Æ). And let m (a, a Æ) be anappropriate distance between a and a Æ. Then the continuity estimate means that theinequality of the form

n(pa, paÆ)0f(m (a, a Æ)) (1.5)

holds for all a, a ÆÏ , or may be, for a and a Æ belonging to a subset of , wherefunction f(x) is de� ned for xE0, f(0)¾0, and f is continuous at x¾0. Thisinequality is equivalent to (1.3).

Various methods can be used to obtain inequality (1.5). Let us list severalrelevant works in this direction: Borovkov [4], Kalashnikov [10, 11], Kartashov [13].Further references can be found in these works. Here we shall use the resultsobtained by Kartashov [13]. They are stated for general Markov chains that is, theydeal with the process W(t, a) having a discrete parameter t¾0, 1, 2 , … That iswhy we consider only discrete time processes. But we start with usual continuoustime risk processes successively reducing the problem to the discrete case.

We do not intent to develop a general theory in this paper but just illustrate howthe combination of the three mentioned steps work in speci� c situations. We limitourselves to only two examples. The � rst is the Sparre Andersen risk model, andhere our result is actually a reformulation of a Kartashov’s result (see [13]). It serves



as a simple illustration of the approach where the second step (embedding intoMarkov process) is absent. The second example is a Markov modulated riskprocess where all steps are non-degenerated and the continuity estimates are new.The approach can be generalized to more complicated risk processes.

The paper is organized as follows. In Section 2 we construct a discrete timereversed process which allows us to de� ne the ruin probability in terms of itsstationary distribution.

All further constructions are illustrated by the two examples mentioned above.These examples have a double enumeration: Examples 3.1, 4.1, and 5.1 deal withSparre Andersen model whereas Examples 3.2, 4.2, and 5.2 with the Markovmodulated model.

Section 3 shows and illustrates how to embed the reversed process into a generalMarkov chain.

In Section 4 we introduce operators associated with general Markov chains andde� ne norms of these operators and measures. These norms are used in order toobtain continuity estimates.

In Section 5, we state Kartashov’s results concerning the continuity of stationarydistributions of general Markov chains. These results are then applied to getcontinuity estimates for ruin probabilities in the S. Andersen and Markov modu-lated models (Theorems 1 and 2 respectively).

2. RUIN PROBABILITIES AND REVERSED PROCESSESLet R (t) be a risk process, about which we assume that R(0)¾xE0 is the initialcapital, T 0¾{T i

0}, iE0, are successive occurrence times, T00¾0, and that paths of

R(t) are right-continuous. Then the probability of ruin can be de� ned as

c(x)¾P inft Ï T0

R (t)B0 R(0)¾x .

For some reasons, we may want to consider a larger set of random times T ³T 0.For example, if we consider a Markov modulated risk process, we might considernot only occurrence times but also the times of state changing of the modulatingprocess. Let {Ti} be the set of moments comprising T , that is

T ¾{Ti}iE 0, T0¾0,

and let

Rn¾R(Tn)

be the embedded risk process. As T ³T 0 and the ruin can only occur at epochesfrom T 0, we have

c(x)¾P infn E 0

RnB0 R0¾x . (2.1)



Let

jn »1¾Rn»1¼Rn. (2.2)

Then,

Rn¾x»j1»···»jn. (2.3)

Hence,

c(x)¾P infn E 1

(j1» ··· »jn)B ¼x .

Let

c(x, N)¾P inf10 n 0 N

(j1»···»jn)B ¼x

be the probability of ruin after N steps of the embedded process. For a � xed NE1,de� ne the process {V n

(N)} by the following recurrent equation

V n»1(N) ¾(V n

(N)»h n»1(N) )», 00n0N¼1, V0

(N)¾0, (2.4)

where random variables hn(N) will be de� ned later. Then

V n(N)¾max(0, hn

(N), hn(N)»hn¼1

(N) ,…, h n(N)»···»h1

(N)), 10n0N. (2.5)

Let us regard that both {jn} and {h n(N)} are de� ned on the same probability space

and thus, both {Rn} and {Vn(N)} are also de� ned on the same probability space.

Now, let us require that the sequence {h n(N)} satis� es the conditions

{Vn(N)0RN¼n} {V n

(N)»h n»1(N) 0RN¼n¼jN¼n}, 00n0N¼1, (2.6)

{Vn(N)\RN¼n} {V n

(N)»h n»1(N) \RN¼n¼jN¼n}, 00n0N¼1. (2.7)

It possible to put hn »1(N) ¾ ¼jN¼n, which works in these arguments, but, in the case

where jk depend on the level of the risk process, we need in more generalconditions (2.6) and (2.7). The reason for this is the fact that the governing randomvariables h n

(N) are supposed to be dependent of the values of the reversed processwhich do not coincide (in general) with the values of the risk process. Anappropriate construction of variables h will be given below, after Assumption 1.

The following lemma is mostly algebraic rather than probabilistic and its proofuses arguments that are similar to those from works [2, 3]. However, there is somedifference due to the fact that we consider the discrete time case and use theconditions (2.6) and (2.7).

LEMMA 1. Gi×en relations (2.6) and (2.7),

c(x)¾ limN“ Ä

P(V n(N)\x).

Proof. Let us � x N\0 and assume that the ruin occurs within [1, N ] that is,min10 i0 N RiB0. Let



8¾min{i: RiB0}.

Then

c(x, N)¾P(80N).

By (2.7),

{80N}¦{R8 BVN¼8(N) }¦{R8¼1BVN¼8 »1

(N) }¦ ···¦{x¾R0BVN(N)}.

It follows that

c(x, N)0P(VN(N)\x).

In the opposite case, there is no ruin within [1, N ] (perhaps, f¾Ä), and hence, allRnE0, 00n0N. But V0

(N)¾00RN. Therefore

{8\N}¦{V0(N)0RN}S{8\N}¦{V0

(N)»h1(N)0RN¼jN}S{8\N}

¾{V0(N)»h1

(N)0RN¼1}S{8\N}¾{(V0(N)»h1

(N))» 0RN¼1}S{8\N}

¾{V1(N)0RN¼1}S{8\N}¦ ···¦ ¾{Vn

(N)0R0¾x}S{8\N}

¦{V n(N)0R0¾x}

which yields that

1¼c (x, N)0P(VN(N)0x),

or

P(VN(N)\x)0c(x, N).

Therefore

c(x, N)¾P(VN(N)\x),

which completes the proof.

The statement of Lemma 1 is too general to be applied as the increments ji maydepend on the risk process. In order to make the construction tractable, let usimpose the following restrictions which will be used in the rest of the paper.

Let {sn}nE 1 be a stationary sequence which is regarded (without loss of general-ity) as a part of another stationary sequence {sn}¼Ä B nB Ä , taking values from anappropriate Polish space , and

jn »1¾ f(Rn, sn »1), nE0, (2.8)

where f : ½ “ . Therefore,

Rn»1¾Rn »f(Rn, sn»1). (2.9)

Let us call sequence {sn} governing.



ASSUMPTION 1.

g(r, s)¾ r»f (r, s) (2.10)

is right-continuous with respect to r and has the following monotonicity property

{rBr Æ} {g(r, s)Bg(r Æ, s)} sÏS. (2.11)

The condition (2.11) is natural. It infers that if an insurance company has a surplusr which is less than r Æ then, at the next step, its reserve g(r, s) will be still less thang(r Æ, s).

For a � xed sÏ denote

g¼1(R, s)¾ inf{r : g (r, s)ER}. (2.12)

It follows that any R from the set of ×alues of g(r, s) satisfy the equation

R¾g¼1(R, s) »f(g¼1(R, s), s)

or

R¼f (g¼1(R, s), s)¾g¼1(R, s). (2.13)

If, in particular, f(r, s) does not depend on r, then, evidently,

g¼1(R, s)¾R¼ f(s).

For a � xed N\0, let

hn(N)¾ ¼ f(g¼1(Vn¼1

(N) , sN¼n »1), sN¼n »1)

and

V n»1(N) ¾(V n

(N)»h n»1(N) )», V0

(N)¾0. (2.14)

Let us check that, under Assumption 1, the conditions (2.6) and (2.7) hold. By(2.13), for any nBN,

V n(N)»h n»1

(N) ¾V n(N)¼ f(g¼1(Vn

(N), sN¼n), sN¼n)¾g¼1(Vn(N), sN¼n).

If Vn(N)0RN¼n, then, by monotonicity of g¼1,

g¼1(V n(N), sN¼n)0g¼1(RN¼n, sN¼n)¾RN¼n¼1.

If Vn(N)\RN¼n, then

g¼1(V n(N), sN¼n)\g¼1(RN¼n, sN¼n)¾RN¼n¼1.

Thus, (2.6) and (2.7) hold true and

c(x)¾ limN“ Ä

P(VN(N)\x).



Let us return to de� nition (2.14) of the sequence {Vn(N)}. Since {sn}¼Ä B nB Ä is

stationary, we can de� ne another sequence {Vn}nE 0 by the relation

Vn»1¾(Vn »hn»1)», V0¾0, (2.15)

where

hn »1¾ ¼ f(g¼1(Vn, s¼n), s¼n). (2.16)

Evidently, the following equality in distribution holds

{Vn}00 n0 N¾d

{V n(N)}00 n0 N,

and therefore,

c(x)¾ limn “ Ä

P(Vn\x). (2.17)

For brevity, the process {Vn} will be referred to as a re×ersed process.

3. THE REVERSED PROCESS AND MARKOV CHAINS

In general, sequence {Vn} is random with a complex correlation structure. In orderto use a standard technique for its study, it is convenient to embed this sequenceinto a Markov chain (see Cox [6], Gnedenko & Kovalenko [8]).

ASSUMPTION 2. Assume that the sequence {Vn} de� ned by (2.15) can be embeddedinto a Marko× chain

Wn¾(Vn, tn), (3.1)

where {tn} is a sequence taking ×alues from a Polish space . Thus, Wn takes ×aluesfrom ¾ » ½ .

Denote by

P(w, B)¾P(Wn »1¾ (Vn»1, tn)ÏB¦ » ½ Wn¾w¾(×, t))

the transition probability of chain (3.1) and by

Pf(w)¾E( f(Wn»1) Wn¾w) (3.2)

its shift operator. Operator P is de� ned for all functions f : “ such that theright-hand side of (3.2) is � nite.

EXAMPLE 3.1. Consider a S. Andersen model R(t) (see Grandell [9] and Kalash-nikov [12]). Let R(0)¾x and Zn be successive i.i.d. claim sizes with the commondistribution B and un be successive i.i.d. inter-occurrence times with the commondistribution A, T n

0¾u1»···»un, nE1. Let c\0 be the gross premium rate. Onecan regard that this model is de� ned by the triple (A, c, B). Take T ¾T 0. Then

Rn»1¾Rn »jn»1, nE0,



where

jn¾cun¼Zn.

Since {jn} are i.i.d., Assumption 2 holds if one takes sn¾jn and f (R, s)¾s. As{jn} consists of i.i.d.r.v’s, it is a stationary sequence and the process {Vn} can bede� ned by the equation

Vn»1¾(Vn¼j¼n)», V0¾0.

But j¼n ¾d

jn. Therefore, one can consider the process

Vn»1¾(Vn »Zn¼cun)», (3.3)

where Zn¼cun plays the role of hn»1.Process (3.3) is Markov and therefore, we do not need supplementary variables

tn to embed it into a Markov chain. By (3.3), its shift operator has the form

Pf(×)¾Ef((×»Z¼cu) »),

where Z and u are independent r.v’s distributed as Zn and un respectively.

EXAMPLE 3.2. Let R(t) (R (0)¾x)) be the following Markov modulated riskprocess (see [2]). Assume that J0(t) is a continuous-time Markov process with � nitestate space ¾{1, 2,…, m}. The occurrence times from a Cox process of intensitylJ 0(t). If T n

0 is the nth occurrence time and J0(Tn0)¾ i, then the corresponding claim

size Zn has the d.f. Bi(u)¾P(Zn0x) and does not depend on other characteristicsof the process.

Denote by ai the intensity with which J0(t) leaves state i that is, holding time ofJ0(t) at state i has the exponential distribution 1¼e ai u. upon leaving state i, J0(t)jumps to state j" i with probability p0

ij, j" i p ij0¾1. Denote by ci the gross premium

rate at the state i. It is natural to view at the Markov modulated risk process asde� ned by the following parameter

a¾ ((li)iÏ , (ai)i Ï , (p ij0)i, j Ï , (ci)i Ï , (Bi(u))i Ï ). (3.4)

We now consider the set T ¾{Tn}n E 0³T 0 of all occurrence times and jumps ofJ0(t). For this, let us introduce another continuous-time Markov process J(t)having the same state space and the following parameters ai »li—the intensity ofthe exponential holding time at state i,

pij¾aip ij

0

ai »li

, j" i, (3.5)

pii¾li

ai »li

, (3.6)

—the probabilities to jump i“ j, i, jÏ .



Let I0n be successive values of J0(t) (just after jumps) and In—successive values of

J(t) (also just after jumps). Evidently, In0 is a homogeneous Markov chain with the

stationary probabilities p i0, iÏE, satisfying the equations

p j0¾

i Ï

p i0p ij

0, jÏ . (3.7)

Similarly, In is also a homogeneous Markov chain having stationary probabilities

pi¾bp i0 1»

li

ai

, iÏ , (3.8)

where b is a norming coef� cient,

b¾i Ï

p i0 1»

li

ai

¼1

. (3.9)

Assume that the chain In is in the steady state that is, its initial distribution is thestationary one. Let Rn¾R (Tn). Then

Rn»1¾Rn »jn»1,

where

jn »1¾cInun

(In )¼dIn, In»1Z n

(In ),

un(i )¾Tn»1¼Tn is the nth inter-occurrence time given In¾I (Tn)¾ i (therefore, it is

exponentially distributed with the parameter ai »li), dij is the Kronecker delta, andZ n

(i ) is the payoff made at time Tn »1 (if it is an occurrence epoch) given that In¾ i(that is, it is a r.v. having the d.f. Bi). Putting

hn »1(N) ¾ ¼jN¼n¾ ¼cIN ¼n¼1

uN¼n¼1(IN ¼n¼1)»dIN¼n¼1,IN¼n

ZN¼n¼1(IN ¼n¼1),

and using the stationarity of the sequence h i(N), we de� ne the following reversed

process

Vn»1¾(Vn »dI¼n¼1,I¼nZ¼n¼1

(I¼n¼1)¼cI¼n¼1u¼n¼1

(I¼n¼1)) », nE0. (3.10)

The pair

Wn¾(Vn, I¼n), nE0, (3.11)

Forms a Markov chain taking values in » ½ . Let us write the shift operator Pfor this chain. The transition probabilities of the chain {I¼n}n E 0 are equal to

qij¾P(I¼n¼1¾ j I¼n¾ i )¾pj

pi

pij, i, jÏ . (3.12)

Now

Pf(×, i )¾E( f(Vn »1, I¼n¼1) Vn¾×, I¼n¾ i ).

Using (3.10) and (3.12), we obtain

Pf(×, i )¾piiEf((×»Z (i )¼ciu(i)) », i ) »

1pi j" i

pjpijEf((×¼cju( j))», j ). (3.13)



4. OPERATORS ASSOCIATED WITH MARKOV CHAINS AND THEIRNORMS

Let Wn be a homogeneous Markov chain de� ned on the measurable space ( , W )with transition probability P(w, G), wÏ , G¦W , and stationary distribution p.Then one can associate the following two standard linear operators with thetransition probability P (w, G). The � rst one is the shift operator P which operateson functions f : “ 1 by rule

Pf(w)¾ f (y)P (w, dy), wÏ . (4.1)

The other operator (designated as P*) operates on � nite measures m de� ned on Wby the rule

P*m(G)¾ P(w, G)m(dw), GÏW , (4.2)

(see [14]).Actually, we have to compare probability distributions and transition probabili-

ties of Markov chains which yields the necessity to consider measures taking bothpositive and negative values. Because of this, let us consider m as an element of aspace C of � nite measures (not necessarily positive) on ( , W ) and denote by C »

a positive cone in C . The space C can be equipped with a norm · and therefore,can viewed as a Banach space. In this paper, we shall only consider the norm

m 8¾ 8(w) m (dw), (4.3)

where 8 is a positive measurable functions such that

0Bk¾ supwÏ

18 (w)

BÄ (4.4)

(cf. [13]).Similarly, let us consider the kernel P(w, G) in (4.1) and (4.2) as not necessarily

a transition probability but as an element of C given w and as a measurablefunction of w provided that GÏW is � xed. Then one can de� ne a norm of theoperator P*:

P* ¾sup( P*m : m 01). (4.5)

In particular, if the norm of m has the form (4.3), then

P* 8¾ supwÏ

18 (w)

8 (y)P (w, dy). (4.6)

Note that the norm · 8 satis� es the following properties (see Kartashov [13])

m1 0 m1»m2 , m1, m2ÏC », (4.7)

m1 m2, m1, m1ÏC » m1¼m2 ¾ m1»m2 , (4.8)



m ( ) 0k m , mÏC , (4.9)

useful for more general constructions which are out of the scope of this paper.Denoting by F the space of measurable functions on , one can equip it with

the norm

f 8¾ supwÏ

f(w)8 (w)

, fÏF , (4.10)

which induces the operator norm

P 8 ¾ sup{ Pf 8 : f 801}¾ supwÏ

P8(w)8(w)

. (4.11)

Note that

P 8 ¾ P* 8 (4.12)

and

f 8¾ sup f(w)m (dw) : m 801 .

In the remainder of the paper we limit ourselves to only Markov chains havingthe unique stationary probability and satisfying the following assumption.

ASSUMPTION 3. (Kartashov [13]) For a Marko× chain with the transition probabil-ity P and the unique stationary probability p, there exists a probability measureGÏC » and a non-negati×e function hÏF such that

(i) h(w)p(dw)\0, h(w)G(dw)\0;(ii) the kernel K(w, G)¾P (w, G)¼h(w)G(G) is non-negative;(iii) K 8 0rB1, where h is the shift operator associated with the kernel K.

EXAMPLE 4.1. Let us return to the S. Andersen model. A transition kernel of thereversed process Vn has the form

P(×, G)¾P(×»Zn¼cunÏG, ×»Zn¼cun\0) »P(×»Zn¼cun00)d0(G), (4.13)

where d0(G) is the probability measure concentrate at 0.Let us assume that there exists an o\0 such that

E exp(o(Zn¼cun))¾rB1. (4.14)

In particular, condition (4.14) yields the positivity of the safety loading which, inturn, guarantees that the reversed process {Vn} has the limiting distribution.

We shall consider the norm · 8, where 8(×)¾e o×. For this, let us split thetransition probability of Markov chain Vn (cf. [13]):



P(×; G)¾K (×, G) »h (×)d0(G),

where h(×)¾P(×»Zn¼cun00). By (4.13), K (×, G)E0. We will show that thenorm of K is less than or equal to r. Actually,

K 8 ¾sup× E 0

e¼o×Ä

0

e oyK (×; dy)

¾ sup× E 0

e¼o× E(e o(× »Zn¼cun ); Zn¼cun »×\0)

0E e o(Zn¼cun )ÅrB1. (4.15)

Inequality (4.15) will be used for continuity estimate in the following section.

EXAMPLE 4.2. Let us turn to the Markov modulated risk process. Denote

bi(r)¾E exp(rZ (i)) (4.16)

and assume that bi(r)BÄ for some r\0 and all iÏ . We have also, for r\0,

E exp(¼ru (i))¾ai »li

ai »li »rci

. (4.17)

Put

pij(r)¾piiE e r(Z(i )¼ci u

(i ))¾libi(r):(ai »li »cir), j¾ i

pijE e rci u(i )

¾aip ij0:(ai »li »cir), j" i,

where probabilities pij are de� ned in (3.5) and (3.6). denote by P(r) the matrix withelements pij(r). This matrix is positive and therefore, its spectral radius Spr P(r) isequal to maximal eigenvalue d(r) which is positive. Note that d(0)¾1. De� ne

o*Å sup{r: Spr P(r)B1}. (4.18)

The constant o* is associated with the Cramer condition, cf. Grandell [9, § 4.4].

ASSUMPTION 4. Throughout the remainder of the example, we assume that o*\0and that o (0B o0o*) satis� es the inequality

Spr P(o)B1.

In particular, if we deal with the ‘‘usual case’’ Spr P(o*)¾1, then o should be lessthan o*. But if Spr P(o*)B1, then it is possible (and desirable) to choose o¾ o*.

Let us introduce another matrix Q (r) with elements

qij(r)¾qiiE e r(Z(i )¼ci u

(i )), j¾1

qijE e¼rcju(i )

, j" i,



where probabilities qij are de� ned in (3.12). Matrix Q(r) is associated with reversedprocess. Evidently,

Q(r)¾T¼1Pt(r)T,

where T is a diagonal matrix with diagonal elements pi, T¼1 is the inverse of T, andPt(r) is the transposition of P(r). Therefore, the maximal eigenvalue of Q(r) isequal to d (r). Let us denote by g(r) the eigenvalue (column) of the matrix Q (r)corresponding to the eigenvalue d(r) and by gP(r) the corresponding eigenvector ofPt(r). Then

g(r)¾T¼1gP(r).

Denote by gi(r) components of the vector g(r); they all are positive by thePerron–Frobenius theory. For de� niteness, let us consider that g1(r)¾1. Evidently,g(r) is a continuous function of r.

Now, let us return to the reversed process Wn (see (3.10) and (3.11)) and denoteits transition probability by P((×, i ); (G, j )), where G is a Borel set belonging to[0, Ä). The shift operator P de� ning this transition probability is given in (3.13).

Let us split the transition probability P in the following way

P((×, i ); (G, j ))¾K((×, i ); (G, j )) »h(×, i )G(G, j ), (4.19)

where

h(×, i )¾h i0 min

iÏP(×»Z (i )¼ciu

(i)00),

h i0¾

1pi

minj Ï

pji,

G(G, j )¾d0(G)pj.

Now we prove that kernel K satis� es Assumption 3. Evidently, Assumptions 3(i, ii)hold. The rest of the example is devoted to the proof that Assumption 3(iii) is alsotrue.

Denote

8r(×, i )¾gi(r) e r×, rE0, ×E0, iÏ . (4.20)

LEMMA 2. Let Assumption 4 hold. Then there exists a constant ×*¾×*(o)E0 suchthat

sup× E ×*

maxi Ï

K8o (×, i )8o (×, i )

¾r1(o)B1. (4.21)

Proof. Since K0P, it suf� ces to prove that

sup× E ×*

maxi Ï

P8o(×, i )8o(×, i )

¾r1(o)B1. (4.22)

Using (3.13), we have



gi(o)Ä

0

e oyP ((×, i ); (dy, i ))¾qiigi(o)E exp(o (×»Z (i)¼ciu(i )) »)

0qiigi(o) (E exp(o(×»Z (i)¼ciu(i)))»1)

¾qii(o)gi(o) e o× »qiigi(o). (4.23)

Similarly, if j" i,

gj(o)Ä

0

e oyP ((×, i ); (dy, j ))0qijgj(E exp(o(×¼cju( j))) »1)

¾qij(o)gj(o) e o× »qijgj(o). (4.24)

Taking into account that

j Ï

qij(o)gj(o)¾d(o)gi(o), iÏ ,

and

j Ï

qij¾1,

we have

P8o (×, i )8o (×, i )

¾1

gi(o) e o×j Ï

gj(o)Ä

0

e oyP((×, i ); (dy, j ))0d(o) »D (o) e¼o×,

where

D(o)¾maxiÏ

gi(o)

miniÏ

gi(o).

Taking

×*¾ ¼1o

ln1¼d(o)2D (o)

,

we arrive at the inequality (4.22) with

r1(o)¾1»d(o)

2B1,

which completes the proof.

Remark 1. It follows from the proof of Lemma 2 that

r1[o1, o2]Å supr Ï [o 1, o 2]

r1(r)B1 (4.25)

for any 0B o10 o20 o.Furthermore, for all rÏ [o1, o2], there exists the same constant ×* satisfying Lemma 2.

LEMMA 3. Let Assumption 4 hold. Then there exists 0B oÀ 0 o* such that, for any×*E0,



sup× 0 ×*

maxi Ï

K8o À(×, i )

8o À(×, i )

¾r2(×*)B1. (4.26)

Proof. Similarly to (4.23) and (4.24), we have

P80(×, i )80(×, i )

¾jÏ

P((×, i ); ([0, Ä), j ))¾d(0)¾1,

where 80(×, i )¾gi(0)Å1, and (4.19) yields that

K80(×, i )80(×, i )

¾1¼h i0

j Ï

pj mini Ï

P(×»Z (i)¼ciu(i )00).

Denote

s(×)¾mini Ï

P(×»Z (i)¼ciu(i )00).

Evidently, s(×)\0 for any ×E0, and s(×)“0 as ×“Ä.It follows that (4.26) holds for oÀ ¾0 and the left hand side of (4.26) does not

exceed

r02(×*)¾1¼ s(×*) min

i,j Ï

pij

pi

.

The continuity of gi(r) with respect to r and Assumption 4 infer that relation (4.26)holds for a positive (suf� ciently small) oÀ and appropriate r2(×*)B1, whichcompletes the proof.

Now, let o\0 satisfy Assumption 4 and o*0 o satisfy Lemma 3. Put

8(×, i )¾gi(r(×)) exp(r(×)×), ×E0, iÏ , (4.27)

where

r(×)¾oÀ »(o¼oÀ)x×

1»x×, x\0. (4.28)

Evidently, r(0)¾oÀ and r (×)“ o as ×“Ä. Note also that the function r(×)× satis� esthe Lipschitz condition

r(×1)×1¼r (×2)×2 0 o ×1¼×2 . (4.29)

Furthermore,

G 8¾iÏ

pigi(oÀ)

(see (4.19)) and

k0k [oÀ, o ]¾maxi Ï

1g i*

,

where



g i*¾ infr Ï [o À, o]

gi(r).

LEMMA 4. Let Assumption 4 hold. If 8 is de� ned by (4.27) and (4.28), thenK 8¾rB1 for suf� ciently small x\0.

Proof. Let

f× (z, i )¾gi(r(×)) exp(r (×)z), iÏ , zE0, ×E0.

Note that

f× (×, i )¾8(×, i ).

It follows, from the continuity of gi(r), the Lipschitz condition (4.29), and Assump-tion 4, that, for any d\0, there exists x1\0 such that

Kf× (×, i )f× (×, i )

¼K8(×, i )8(×, i )

0d

for all x0x1, iÏ , and ×E0. Therefore,

K 8 ¼ supi Ï , × E 0

Kf× (×, i )f× (×, i )

0d.

But, by Lemma 2 and Remark 1, there exist ×*E0 and 0Br1B1 such that

supi Ï , × E ×*

Kf× (×, i )f× (×, i )

0r1.

By Lemma 3, there exist 0Br2(×*)B1 and x2\0 such that, for all xBx2,

supi Ï , × E ×*

Kf× (×, i )f× (×, i )

0r2(×*).

This yields that

K 8 0max(r1, r2(×*))»d

for all x0min (x1, x2). Since d can be chosen as small as necessary, the lemma isproved.

5. CONTINUITY ESTIMATES

Let us state the following result proved in Kartashov [13, Theorem 8].

LEMMA 5. Let Assumption 3 hold and P 8BÄ. Then each Marko× chain with thetransition probability P Æ corresponding to the shift operator PÆ and satisfying

DÅ PÆ¼P 8B1¼r

1» p 8krÅD0, (5.1)



has a unique in×ariant probability measure pÆ and

p Æ¼p 80D p 8

D0¼D, (5.2)

where the norm p 8 can be estimated as

p 8 0G 8

1¼r. (5.3)

Lemma 5 serves as a source for desired continuity estimates in our paper. We shalluse its assertion the following simpi� ed form.

COROLLARY 1. Under the assumptions and notation of Lemma 5,

p Æ¼p 80D G 8

(1¼r)(D0¼D), (5.4)

if

DB(1¼r)2

1»(k G 8¼1)r

ÅD0. (5.5)

EXAMPLE 5.1. Let us � nish considering the S. Andersen risk process underAssumption (4.14). The process is completely determined by the triple a¾(A, c, B)(see Example 3.1). As we have seen, the probability ca (×) of ruin coincides with thestationary probability for the Markov chain {Vn} (see (3.3)) to exceed the level ×.Let a Æ¾(A Æ, c Æ, B Æ) be the triple, de� ning another risk process, its ruin probabilitybeing caÆ(×). Let o be the constant satisfying (4.14) and 8 (×)¾e o× (like in Example4.1). Then the distance n(ca, caÆ) between corresponding ruin probabilities can beexpressed in terms of the stationary distributions p and p Æ as follows:

n(ca, caÆ)¾ e ox ca¼caÆ (dx)¾ p¼p Æ 8. (5.6)

Denote

m(a, a Æ)¾ P¼PÆ 8 (5.7)

and

Ac(y)¾P(cu0y)¾A(y:c),

A ÆcÆ(y)¾P(c Æu Æ0y)¾A Æ(y:c Æ).

Then, by de� nition,

m(a, a Æ)¾ sup× E 0

e¼o×Ä

0

e oy P Æ(×; dy)¼P(×; dy)



0sup× E 0

e¼o×Ä

0

e o(× »z) B Æ¼B (dz)

»Ä

0

e o(× »z)B (dz)Ä

0

A ÆcÆ ¼Ac (dy)

¾E e oZÄ

0

A ÆcÆ ¼Ac (dy) »Ä

0

e oz B Æ¼B (dz). (5.8)

THEOREM 1. If the S. Andersen model satis� es condition (4.14), m (a, a Æ) is de� nedby (5.7), and n(ca, caÆ) by (5.6), then, for m (a, a Æ)0(1¼r)2,

n(ca, caÆ)0m (a, a Æ)

(1¼r)((1¼r)2¼m(a, a Æ)), (5.9)

where r is taken from (4.14).

Proof. In Example 4.1, it was shown that k¾ G 8¾1. Plugging these expres-sions into Corollary 1, we arrive at (5.9) .

Remark 2. One can replace m (a, a Æ) in the continuity estimate (5.9) by its upperbound (5.8), or any other appropriate bound expressed in terms of the triples a anda Æ.

EXAMPLE 5.2. We derive now a continuity estimate for the Markov modulatedrisk process which is governed by parameter a de� ned in (3.4). Let all assumptionsof Lemma 4 hold and function 8(×, i ) be de� ned by (4.27) and (4.28).

First of all, using the expressions for G 8 and k listed in Example 4.2, andCorollary 5.4, we arrive at the following general inequality

p Æ¼p 80PÆ¼P 8

(1¼r) (D0¼ PÆ¼P 8) j Ï

pjgj(oÀ), (5.10)

where 0BrB1 is taken from Lemma 4,

D0¾(1¼r)2

1»(B»1)r, (5.11)

B¾ maxi, jÏ ,r Ï [oÀ, o]

gj(oÀ)gi(r)

. (5.12)

The estimate (5.10) is valid if PÆ¼P 80D0.Now, let us estimate quantities p Æ¼p 8 and PÆ¼P 8 in terms of the risk

process. We start with p Æ¼p 8. Evidently, the probability of ruin can be presentedin the form

c(x)¾i Ï

Ä

x

p(d×, i ).



Denote

n(ca, caÆ )¾Ä

0

e o× ca¼caÆ (d×). (5.13)

By (4.28),

o×0o¼ oÀ

x»r (×)×. (5.14)

Actually, (5.14) is equivalent to the inequality

o×x0 o¼oÀ »r (×)×x, (5.15)

And (4.28) can be written as

r(×)×x¾oÀ »o×x¼ r(×).

Therefore, (5.15) is equivalent to o¼r (×)E0 that is, of course, true.It follows that

n(ca, caÆ )0e (o¼oÀ):xÄ

0

e r(×)× ca ¼caÆ (d×)

0e (o¼oÀ):x

i Ï

Ä

0

e r(×)× p Æ(d×, i )¼p(d×, i )

0k [oÀ, o ] e (o¼oÀ):x p Æ¼p 8.

Now, let

m(a, a Æ)¾ PÆ¼P 8

¾ sup× E 0

maxi Ï

jÏ

Ä

0

gj(r(y)) e r(y)y

gi(r(×)) e r(×)×P Æ((×, i ); (dy, j ))¼P ((×, i ); (dy, j ))

0C sup× E 0

maxiÏ

j Ï

Ä

0

e r(y)y¼r(×)× P Æ((×, i ); (dy, j ))¼P ((×, i ); (dy, j )) ,

where

C¾ supr 1, r 2Ï [o À, o],i,j Ï

gj(r1)gi(r2)

.

Let us denote, for all iÏ ,

A1i(y)¾P(ciu(i)0y)¾Ai(y :ci),

A Æ1i(y)¾P(c Æiu(i )Æ 0y)¾Ai(y :c Æi).

Then, similarly to (5.8), by the Lipschitz condition (4.29),

sup× E 0

Ä

0

e r(y)y¼r(×)× P Æ((×, i ); (dy, i ))¼P((×, i ); (dy, i ))



0bi(o)Ä

0

A Æ1i¼A1i (dy) »Ä

0

e oz B Æi ¼Bi (dz),

where bi(o) is de� ned in (4.16). Similarly, for j" i,

sup× E 0

Ä

0

e r(y)y¼r(×)× P Æ((×, i ); (dy, j ))¼P((×, i ); (dy, j ))

0Ä

0

A Æ1j¼A1j (dy) »Ä

0

B Æj¼Bj (dz).

This follows that

m(a, a Æ)0maxi Ï

bi(o)Ä

0

A Æ1i¼A1i (dy) »Ä

0

e oz B Æi ¼Bi (dz)

»j" i

Ä

0

( A Æ1j¼A1j » B Æj ¼Bj )(dy) . (5.16)

THEOREM 2. If the Marko× modulated risk process satis� es Assumption 4 ando*\0 is taken from Lemma 3, then the following continuity estimate holds

n(ca, caÆ)0k [oÀ, o ] e (o¼oÀ):xm (a, a Æ)(1¼r)(D0¼m (a, a Æ)) j Ï

pjgj(oÀ), (5.17)

where D0 is gi×en in (5.11) and (5.12), and m (a, a Æ) is bounded in (5.16).

ACKNOWLEDGEMENTS

We thank Ragnar Norberg for the friendly atmosphere he created in the Laboratory of ActuarialMathematics, University of Copenhagen, where this research was done. We also appreciate valuableremarks made by the anonymous referee.

REFERENCES

[1] Asmussen, S. (1995). Stationary distributions via � rst passage times. In: Dshalalow, J. (Ed.)Ad×ances in queueing, CRC Press, Boca Raton, 79–102.

[2] Asmussen, S. & Kella, O. (1996). Rate modulation in dams and ruin problems. Journal of AppliedAd×ances in Applied Probability 33, 523–535.

[3] Asmussen, S. & Petersen, S.S. (1998). Ruin probabilities expressed in terms of storage processes.Ad×ances in Applied Probability 20, 913–916.

[4] Borovkov, A. Asymptotic methods in queueing theory. J. Wiley & Sons, Chichester, 1984.[5] Buslenko, N. P., Kalashnikov, V. V. & Kovalenko, I. N. (1973). Lectures on complex systems

theory, Sov. Radio, Moscow (in Russian).[6] Cox, D. (1955). The analysis of non-Markovian stochastic processes by the inclusion of supplemen-

tary variables. Proc. Cambridge Philosophical Society 50, 433–441.[7] Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of non-diffusion

stochastic models. Journal of the Royal Statistical Society, Ser. B 46, 353–388.[8] Gnedenko, B. V. & Kovalenko, I. N. (1968). An introduction to queueing theory. Israel Program for

Scienti� c Translations, Jerusalem.



[9] Grandell, J. (1991). Aspects of risk theory. Springer, New York.[10] Kalashnikov, V. (1978). Qualitati×e analysis of complex systems beha×iour by the test functions.

Nauka, Moscow (in Russian).[11] Kalashnikov, V. (1994). Mathematical methods in queueing theory. Kluwer Academic Publishers,

Dordrecht.[12] Kalashnikov, V. (1997). Geometric sums: bounds for rare e×ents with applications. Kluwer Academic

Publishers, Dordrecht, 1997.[13] Kaztashov, N. V. (1986). Inequalities in theorems of ergodicity and stability for Markov chains

with common phase space, I, II. Theory Probability and Its Applications 30, 247–259, 507–515.[14] Meyn, S. & Tweedie, R. (1993). Marko× chains and stochastic stability. Springer, New York.[15] Schmidli, H. (1992). A general insurance model. Diss. ETH, Nr. 9881.

Address for correspondence:Vladimir KalashnikovLaboratory of Actuarial MathematicsUniversity of CopenhagenDK-2100 CopenhagenDenmarkE-mail: [email protected]


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