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Jump-type Levy Processes
Jump-type Levy Processes
Ernst Eberlein
Handbook of Financial Time Series
Jump-type Levy Processes
Outline
Table of contentsProbabilistic Structure of Levy Processes
Levy processLevy-Ito decompositionJump part
Probabilistic Structure of Levy ProcessesThe distribution of a Levy processLevy-Khintchine formulaIntegrability propertiesProperties of the process
Financial ModelingClassical modelExponential Levy modelPricing of derivativesModels for interest rates
Levy Processes with Jumps
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
Levy process
Levy process
I X = (Xt)t≥0 is a process with stationary and independentincrements
I underlying it is a filtered probability space (Ω,F , (Ft)t≥0,P)
I filtration (Ft)t≥0 is complete and right continuous
I proces Xt is Ft-adapted
I Levy process has version with cadlag path (Theorem 30 inProtter (2004)), i.e. right-continuous with limits from the left
? every Levy process is a semimartingale
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
Levy process
One-dimensional Levy process
Xt = X0 + bt +√
cWt + Zt +∑s≤t
∆XsI|∆Xs |>1 (1)
I b, c ≥ 0 are real numbers
I (Wt)t≥0 is standard Brownian motion
I (Zt)t≥0 is purely discontinuous martingale
I (Wt)t≥0 and (Zt)t≥0 are independent
I ∆Xs := Xs − Xs− denotes the jump at time s
? in case where c = 0, the process is purely discontinuous
? (1) is the so-called canonical representation forsemimartingales
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
Levy-Ito decomposition
Levy-Ito decomposition
I for semimartingale Y = (Yt)t≥0:
Yt −∑s≤t
∆YsI|∆Xs |>1
is a special semimartingale (see I.4.21 and I.4.24 in Jacod,Shiryaev (1987)), which admits unique decomposition intolocal martingale M = (Mt)t≥0 and a predictable process withfinite variation V = (Vt)t≥0, which for Levy processes is the(deterministic) linear function of time bt
I any local martingale M with (M0 = 0) admits a uniquedecomposition (see I.4.18 in Jacod, Shiryaev (1987)):
M = Mc + Md =√
cW + Z
where the second equality holds for Levy processes
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
Jump part
Jump part
I cadlag paths over finite intervals => any path has only finitenumber of jumps with absolute jump size larger than ε > 0(i.e. sum of jumps bigger than ε is a finite sum for each path)
I the sum of small jumps:∑s≤t
∆XsI|∆Xs |≤1 (2)
does not converge in general (infinitely many small jums), butone can force this sum to converge by compensating it, i.e. bysubtracting the corresponding average increase of the process
I the average increase can be expressed by the intensity F (dx)with which the jumps arrive
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
Jump part
Compensation
limε→0
∑s≤t
∆XsIε≤|∆Xs |≤1 − t
∫xIε≤|x |≤1F (dx)
(3)
I this limit exists in the sense of convergence in probability
I the sum represents the (finitely many) jumps
I the integral is the average increase of the process
I one cannot separate the difference, because neither of the twoexpressions has a finite limit as ε→ 0
? one can express the (3) using the random measure of jumpsof the process X denoted by µX
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
Jump part
Random measure of jumps
µX (ω; dt, dx) =∑s≤0
I|∆Xs |6=0δ(s,∆Xs(ω))(dt, dx)
I if a path of the process X given by ω has a jump of size∆Xs(ω) = x at time s, then the random measure µX (ω; ., .)places unit mass δ(s,x) at the point (s, x) ∈ R+ × R
I consequently for a time interval [0, t] and a set A ⊂ R,µX (ω; [0, t]× A) counts jumps of size within A:
µX (ω; [0, t]× A) = |(s, x) ∈ [0, t]× A|∆Xs(ω) = x|
I average number of jumps expressed by an intensity measure:
E[µX (.; [0, t]× A)
]= tF (A)
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
Jump part
Another expression
I the sum of the big jumps:∫ t
0
∫R
xI|x |>1µX (ds, dt)
I (Zt), the martingale of compensated small jumps:∫ t
0
∫R
xI|x |≤1
(µX (ds, dt)− dsF (dx)
)(4)
? µX (ω; dt, dx) is a random measure, i.e. it depends on ω
? dsF (dx) is a product measure on R+ × R? again these mesures cannot be separated in general
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
The distribution of a Levy process
The distribution of a Levy process
I the distribution of a Levy process X = (Xt)t>0 is completelydetermined by any of its marginal distributions L(Xt)
I lets consider L(X1) and arbitrary natural number n:
X1 = X1/n + (X2/n − X1/n) + . . .+ (Xn/n − Xn−1/n)
I by stationarity and independence of the increments, L(X1) isthe n-fold convolution of L(X1/n):
L(X1) = L(X1/n) ∗ · · · ∗ L(X1/n)
I consequently L(X1) and analogously any L(Xt) are infinitelydivisible distributions
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
The distribution of a Levy process
Infinitely divisible distributions
I conversely any infinitely divisible distribution ν generates in anatural way a Levy process X = (Xt)t>0 which is uniquelydetermined by setting L(X1) = ν
I if for n > 0, νn is the probability measure such thatν = νn ∗ · · · ∗ νn then one gets immediately for rational timepoints t = k/n L(Xt) as a k-fold convolution of νn
I for irrational time points t, L(Xt) is determined by acontinuity argument (see Chapter 14 in Breiman (1968))
I the process to be constructed has independent increments =>it is sufficient to know the one-dimensional distribution
I if a specific infinitely divisible distribution is chatacterizedby a few parameters the same hold for the correspondingLevy process (crucial for estimation of parameters)
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
Levy-Khintchine formula
Levy-Khintchine formula
I the Fourier transform E [exp(iuX1)] of a Levy process given aninfinitely divisible distribution ν = L(X1) is:
exp
[iub − 1
2u2c +
∫R
(eiux − 1− iuxI|x |≤1
)F (dx)
](5)
I the b, c,F determine the L(X1), and thus the process X
I (b,c,F) is called the Levy-Khintchine triplet or insemimartingale terminology the triplet of local characteristics
I the truncation function h(x) = xI|x |≤1 used in (5) could bereplaced by other versions of truncation functions (e.g.smooth one: x (identity) near the origin, else goes to zero)
? changing h results in a different drift parameter b, whereasthe diffusion coeficient c ≥ 0 and the Levy measure F remain
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
Levy-Khintchine formula
RemarksI Levy measure F does not have mass on 0 and satisfies:∫
Rmin(1, x2)F (dx) <∞
I conversely any measure on R with these two properties plusb ∈ R and c ≥ 0 defines via (5) an infinitely divisibledistribution and thus a Levy process.
I Levy-Khintchine formula (5) with ψ(u) (characteristicexponent):
E [exp(iuX1)] = exp(ψ(u))
I again by independence and stationarity of the increments:
E [exp(iuXt)] = exp(tψ(u))
? important for computation of derivative value E [f (XT )],parameters of X estimated as the parameters of L(X1)
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
Integrability properties
Integrability properties of Levy measure F
I finiteness of of moments of the process depends only onfrequency of large jumps since it is related to integration by Fover |x | > 1
Proposition (1)
Let X = (Xt)t≥0 be a Levy process with Levy measure F .
1. Xt has finite p-th moment for p ∈ R+, i.e. E [|Xt |p] <∞, ifand only if
∫|x |>1 |x |
pF (dx) <∞.
2. Xt has finite p-th exponential moment for p ∈ R+, i.e.E [exp(pXt)] <∞, if and only if
∫|x |>1 exp(px)F (dx) <∞.
Proof: see Theorem 25.3 in Sato (1999)
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
Integrability properties
Finite expectation of L(X1)I =>
∫|x |>1 xF (dx) <∞ => we can add
−∫
iuxI|x |>1F (dx) to the (5) and get:
E [exp(iuX1)] = exp
[iub − 1
2u2c +
∫R
(eiux − 1− iux
)F (dx)
]I =>
∫ t0
∫R xI|x |>1dsF (dx) exists => to the (4) we can add:∫ t
0
∫R
xI|x |>1
(µX (ds, dt)− dsF (dx)
)?∫ t
0
∫R xI|x |>1µ
X (ds, dt), always exists for every pathI as a result we get (1) in simpler representation:
Xt = X0+b∗t+√
cWt+Zt+
∫ t
0
∫R
x(µX (ds, dt)− dsF (dx)
)(6)
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
Integrability properties
Finite expectation of L(X1) II
I the drift coefficient b∗ = E [X1] because the W and Z aremartingales
I Levy processes used in finance have finite first moments, thanwe get the (6) representation
I the existence of moments is determined by the frequency ofthe big jumps
I the fine structure of the path is related to the frequency ofthe small jumps
? process has finite activity if almost all path have only finitenumber of jumps along any time interval of finite length
? process has infinite activity if almost all path have infinitelymany jumps along any time interval of finite length
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
Properties of the process
Activity of the process
Proposition (2)
Let X = (Xt)t≥0 be a Levy process with Levy measure F .
1. X has finite activity if F (R) <∞.
2. X has infinite activity if F (R) =∞.
I by definituon a Levy measure satisfies∫
R I|x |>1F (dx) <∞=> the assumption F (R) <∞ (or F (R) =∞) is equivalenttu assumption of finitenes (or infinitenes) of
∫R I|x |≤1F (dx)
I the path of Brownian motion have infinite variation
I whether the purely discontinuous component (Zt) in (1) orintegral in (6) has path with infinite variation depends on thefrequency of the small jumps
Jump-type Levy Processes
Probabilistic Structure of Levy Processes
Properties of the process
Variation of the process
Proposition (3)
Let X = (Xt)t≥0 be a Levy process with Levy measure F .
1. Almost all path of X have finite variation if c = 0 and∫|x |≤1 |x |F (dx) <∞.
2. Almost all path of X have infinite variation if c 6= 0 or∫|x |≤1 |x |F (dx) =∞.
Proof: see Theorem 21.9 in Sato (1999)I the integrability of F (
∫|x≤1 |x |F (dx) <∞) guarantees also
that the sum of the small jumps (2) converges (a.s.)? in this case one can separate the measures µX and F in (6):
∫ t
0
∫R
x(µX (ds, dt)− dsF (dx)
)=
∫ t
0
∫R
xµX (ds, dt)−t
∫R
xF (dx)
Jump-type Levy Processes
Ito formula for jump process
Ito formula for jump process
f (Xt) = f (X0) +
∫ t
0f ′(Xs)dX c
s +1
2
∫ t
0f ′′(Xs)dX c
s dX cs +
+∑
0<s≤t
[f (Xs)− f (Xs−)]
X cs = bt +
√cWt
Jump-type Levy Processes
Financial Modeling
Classical model
Classical model
dSt = µStdt + σStdWt
I the geometric Brownian motion goes back to Samuelson(1965)
St = S0 exp(σWt + (µ− σ2/2)t)
I the exponent of this price process is Levy process
I given (1): b = µ− σ2/2,√
c = σ, Zt = 0 and no big jumps
I log returns with time step 1 normally distributed
N(µ− σ2/2, σ2)
Jump-type Levy Processes
Financial Modeling
Exponential Levy model
Exponential Levy model
I one can identify a parametric distribution ν by fitting anempirical distribution
I considering the Levy process X = (Xt)t≥0 such thatL(X1) = ν, the model
St = S0 exp(Xt) (7)
produces log returns exactly equal to ν (infinitely divisible)
I the stochastic differential equation describing the process
dSt = St−(dXt + (c/2)dt + e∆Xt − 1−∆Xt)
I the distribution of the log returns of this process is not knownin general
Jump-type Levy Processes
Financial Modeling
Exponential Levy model
Exponential Levy model II
I a Levy version of differential equation
dSt = St−dXt
has the solution the stochastic exponential
St = S0 exp(Xt − ct/2)∏s≤t
(1 + ∆Xs) exp(−∆Xs)
I one can directly see from (1 + ∆Xs) that this model canproduce negative prices as soon as the driving Levy process Xhas negative jumps larger than 1
I the Levy measures of distributions used in finance have strictlypositive densities on the whole negative half line (negativejumps of arbitrary size)
Jump-type Levy Processes
Financial Modeling
Pricing of derivatives
Pricing of derivatives
I price process given by (7) has to be a martingale
? pricing is done by taking expectations under risk neutral(martingale) measure
I for (St)t≥0 to be a martingale, expectation has to be finite
? candidates for the driving process are Levy processes X with afinite first exponential moment E [exp(Xt)] <∞
I Proposition 1 characterizes these processes in terms of theirLevy measure
? the necessary assumption of finiteness of the first exponentialmoment a priori excludes stable processes
Jump-type Levy Processes
Financial Modeling
Pricing of derivatives
Xt = X0+bt+√
cWt +Zt +∫ t
0
∫R x(µX (ds, dt)− dsF (dx)
)
I let X be given in the representation (6) then St = S0 exp(Xt)is a martingale if
b = −c
2−∫
R(ex − 1− x)F (dx) (8)
I this can be seen by applying Ito formula to St = S0 exp(Xt),where (8) guarantees that the drift component is 0
Jump-type Levy Processes
Financial Modeling
Models for interest rates
Forward rate approach
I the dynamics of the instantaneous forward rate with matutityT at time t
f (t,T ) = f (0,T ) +
∫ t
0α(s,T )ds −
∫ t
0σ(s,T )dXs
I the coefficients α(s,T ) and σ(s,T ) can be deterministic orrandom
I one gets zero-coupon bond prices in a form comparable to thestock price (7)
B(t,T ) = B(0,T ) exp
(∫ t
0(r(s)− A(s,T ))ds +
∫ t
0Σ(s,T )dXs
),
? where r(s) = f (s, s) is the short rate and A(s,T ) and Σ(s,T )are derived from α(s,T ) and σ(s,T ) by integration
Jump-type Levy Processes
Financial Modeling
Models for interest rates
Levy LIBOR market modelI the forward LIBOR rates L(t,Tj) for the time points Tj
(0 ≤ j ≤ N) are chosen as the basic ratesI as a result of bacward induction one gets for each j the rate
L(t,Tj) = L(0,Tj) exp
(∫ t
0(λ(s,Tj)dX
Tj+1s
),
? where λ(s,Tj) is a volatility structure, X Tj+1 = (XTj+1t )t≥0 is
the process derived from an initial Levy processX TN = (X TN
t )t≥0 and the L(t,Tj) is considered under PTj+1
(the forward martingale measure)
I closely related to the LIBOR model is the forward processmodel, where forward processes
F (t,Tj ,Tj+1) = B(t,Tj)/B(t,Tj+1)
are chosen as the basic quantities
Jump-type Levy Processes
Levy Processes with Jumps
Levy Jump Diffusion
Poisson process
I the simplest Levy measure is ε1 (a point mass in 1)
I by adding an intensity parameter λ > 0, one gets F = λε1
I this Levy measure generates a process X = (Xt)t≥0 withjumps of size 1 which occur with an average rate of λ in aunit time interval
I X is called a Poisson process with intensity λ
I the drift parameter b in Fourier transform is E [X1], which isλ, therefore it takes the form:
E [exp(iuXt)] = exp[λt(e iu − 1)]
I any variable Xt has a Poisson distribution with parameter λt
P[Xt = k] = exp(−λt)(λt)k
k!
Jump-type Levy Processes
Levy Processes with Jumps
Levy Jump Diffusion
Exponentially distributed waiting times
I one can show that the successive waiting times from one jumpto the next are independent exponentially distributed randomvariables with parameter λ
I starting with a sequence (τi )i≥1 of independent exponentially(λ) distributed r.v. and setting Tn =
∑ni=1 τi , the associated
counting process
Nt =∑n≥1
ITn≤t
is a Poisson process with intensity λ
Jump-type Levy Processes
Levy Processes with Jumps
Levy Jump Diffusion
Compound Poisson processI a natural extension of the Poisson process is a process where
the jump size is randomI let Y = (Yt)t≤0 be a sequence of iid. r. v., L(Y1) = ν
Xt =Nt∑i=1
Yi ,
? where (Nt)t≥0 is a Poisson process (λ > 0) independent of(Yi )i≥0, defines a compound Poisson process X = (Xt)t≥0
with intensity λ and jump size distribution νI Fourier transform is given by
E [exp(iuXt)] = exp
[λt
∫R
(e iux − 1)ν(dx)
]I the Levy measure is given by F (A) = λν(A) for measurable
sets A in R
Jump-type Levy Processes
Levy Processes with Jumps
Levy Jump Diffusion
Levy jump diffusion
I a Levy jump diffusion is a Levy process where the jumpcomponent is given by a compound Poisson process
Xt = bt +√
cWt +Nt∑i=1
Yi ,
? where b ∈ R, c > 0, (Wt)t≥0 is a standard Brownian motion,(Nt)t≥0 is a Poisson process with intensity λ > 0 and (Yi )i≥0
is a sequence of iid. r.v. independent of (Nt)t≥0
I one can use e.g. normally or double-exponentially distributedjump sizes Yi
I any other distribution could be considered, but the question isif one can control explicitly the quantities one is interested in(for example, L(Xt))
Jump-type Levy Processes
Levy Processes with Jumps
Hyperbolic Levy porcesses
Hyperbolic Levy processes
I hyperbolic distributions which generate hyperbolic Levyprocesses X = (Xt)t≥0 - also called hyperbolic Levy motions-constitute a four-parameter class of distributions withLebesgue density
dH(x) =
√α2 − β2
2α δ K1(δ√α2 − β2)
exp
(−α√δ2 + (x − µ)2 + β(x − µ)
),
? where Kj denotes the modified Bessel function of the thirdkind
I α determines the shape, β with 0 ≤ |β| < α the skewness,µ ∈ R the location and δ > 0 is a scaling parameter
I taking the logarithm of dH , one gets a hyperbola
Jump-type Levy Processes
Levy Processes with Jumps
Hyperbolic Levy porcesses
The Fourier transform of a hyperbolic distribution
φH(u) = exp(iuµ)
(α2 − β2
α2 − (β + iu)2
)1/2K1(δ
√α2 − (β + iu)2)
K1(δ√α2 − β2)
I moments of all order exists and
E [X1] = µ+βδ√α2 − β2
K2(δ√α2 − β2)
K1(δ√α2 − β2)
I in case of hyperbolic distributions c = 0, which means thathyperbolic Levy motios are purely discontinuous processes
Jump-type Levy Processes
Levy Processes with Jumps
Generalized hyperbolic Levy processes
Generalized hyperbolic distributions
I hyperbolic distributions are a subclass of a more powerfullfive-parameter class, the generalized hyperbolic distributions(Barndorff-Nielsen (1978))
I the additional class parameter λ ∈ R has the value 1 forhyperbolic distributions
dGH(x) = a(λ, α, β, δ)(δ2 + (x − µ)2)(λ− 12
)/2·
·Kλ−1/2
(α√δ2 + (x − µ)2
)exp (β(x − µ)) ,
? where the normalizing constant is given by
a(λ, α, β, δ) =(α2 − β2)λ/2
√2παλ−1/2δλKλ(δ
√α2 − β2)
Jump-type Levy Processes
Levy Processes with Jumps
Generalized hyperbolic Levy processes
Generalized inverse Gaussian distributions
I generalized hyperbolic distribution can be represented as anormal mean-variance mixtures
dGH(x) =
∫ ∞0
dN(µ+βy)(x) · dGIG (x ;λ, δ,√α2 − β2)dy
? where the mixing distribution is generalized inverse Gaussianwith density
dGIG (x ;λ, δ, γ) =(γδ
)λ xλ−1
2Kλ(δγ)exp
(− 1
2x(δ2 + γ2x2)
)for x > 0
Jump-type Levy Processes
Levy Processes with Jumps
Generalized hyperbolic Levy processes
Generalized hyperbolic Levy processes
I the moment generating function MGH(u) for |u + β| < α:
MGH(u) = exp(µu)
(α2 − β2
α2 − (β + iu)2
)λ/2Kλ(δ
√α2 − (β + u)2)
Kλ(δ√α2 − β2)
I as a consequence, exponential moments are finite, which iscrucial fact for pricing of derivatives under martingalemeasures
I the Fourier transform φGH is obtained from the relationφGH(u) = MGH(iu)
I again c = 0, so generalized hyperbolic Levy motions arepurely discontinuous processes
I the Levy measure F has a closed-form density
Jump-type Levy Processes
Levy Processes with Jumps
Generalized hyperbolic Levy processes
Normal inverse Gaussian distributions
I setting λ = −1/2 we get the normal inverse Gaussiandistributions
dNIG (x) =αδK1(α
√δ2 + (x − µ)2)
π√δ2 + (x − µ)2
exp(β(x−µ)+δ√α2 − β2)
I their Fourier transform is simple becauseK−1/2(z) = K1/2(z) =
√π/(2z)e−z :
φNIG (u) = exp(iuµ) exp(δ√α2 − β2
)exp
(−δ√α2 − (β + iu)2
)I one can see that NIG are closed under convolution in
parameters δ and µ
Jump-type Levy Processes
Levy Processes with Jumps
α-Stable Levy processes
α-Stable Levy processes
I stable distributions constitute a four-parameter class ofdistributions with Fourier transform given by
φst(u) =
exp
(iuµ− |σu|α
(1− iβsign(u) tan πα
2
))ifα 6= 1,
exp(iuµ− |σu|
(1 + iβsign(u) 2
π ln |u|))
ifα = 1
I the parameter space is 0 < α ≤ 2, σ ≥ 0, −1 ≤ β ≤ 1 andµ ∈ R
I for α = 2 one gets the Gaussian distribution with mean µ andvariance 2σ2
I for α < 2 there is no Gaussian part, which means the paths ofan α-stable Levy motion are purely discontinuous
Jump-type Levy Processes
Levy Processes with Jumps
α-Stable Levy processes
Special cases of α-Stable distributions
I explicit densities are known in three cases only:
? Gaussian distribution, α = 2, β = 0
? Cauchy distribution, α = 1, β = 0
? Levy distribution, α = 1/2, β = 1
I usefulness in particular as a pricing model is limited for α 6= 2by the fact that finiteness of the first exponential moment innot satisfied
Jump-type Levy Processes
Levy Processes with Jumps
Meixner Levy processes
Meixner Levy processes
I the Fourier transform of Meixner distributions is given by
φM(u) =
(cos(β/2)
cosh((αu − iβ)/2)
)2δ
? for α > 0, |β| < π, δ > 0
I the corresponding Levy processes are purely discontinuouswith paths of infinite variation
I the density of Levy measure F is
gM(x) =δ
x
exp(βx/α)
sinh(πx/α)
Jump-type Levy Processes
Levy Processes with Jumps
CGMY and variance gamma Levy processes
CGMY and variance gamma Levy processes
I the class of CGMY distributions (infinitely divisible) extendsthe variance gamma model
I CGMY Levy processes have purely discontinuous paths andthe density of Levy mesure is given by
gCGMY (x) =
C exp(−G |x |)
|x |1+Y x < 0,
C exp(−Mx)x1+Y x > 0
? with parameter space C ,G ,M > 0 and Y ∈ (−∞, 2)
I the process has infinite activity iff Y ∈ [0, 2)
I the paths have infinite variation iff Y ∈ [1, 2)
I for Y = 0 one gets the three-parameter variance gammadistributions which are a subclass of the generalizedhyperbolic distributions
Jump-type Levy Processes
References
Barndorff-Nielsen, O.E. (1978): Hyperbolic distributions anddistributions on hyperbolae. Scandinavian Journal of Statistics5, 151-157.
Breiman, L (1968): Probability. Addison-Wesley, Reading.
Jacod, J., Shiryaev, A.N. (1987): Limit Theorems forStochastic Processes. Springer, New York.
Protter, P.E. (2004): Stochastic Integration and DifferentialEquations. (2nd ed.) Volume 21 of Applications ofMathematics. Springer, New York.
Samuelson, P. (1965): Rational theory of warrant pricing.Industrial Management Review 6:13-32.
Sato, K.-I. (1999): Levy Processes and Infinitely DivisibleDistributions. Cambridge University Press, Cambridge.