Distribution of the first ladder height of a stationary
risk process perturbed by
α
-stable Lévy motion
Hanspeter Schmidli
∗Laboratory of Actuarial Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
Received 1 September 1998; received in revised form 1 August 2000; accepted 19 September 2000
Abstract
We consider a risk model described by an ergodic stationary marked point process. The model is perturbed by a Lévy process with no downward jumps. The (modified) ladder height is defined as the first epoch where an event of the marked point process leads to a new maximum. Properties of the process until the first ladder height are studied and results of Dufresne and Gerber [Insurance: Math. Econ. 10 (1991) 51], Furrer [Scand. Actuarial J. (1998) 59], Asmussen and Schmidt [Stochastic Process. Appl. 58 (1995) 105] and Asmussen et al. [ASTIN Bull. 25 (1995) 49] are generalized. © 2001 Elsevier Science B.V. All rights reserved.
MSC:M01; M13
Subj. Class:C60
Keywords:Perturbed risk model; Lévy process; Ladder heights; Marked point process; Markov modulated risk model
1. Introduction
Consider a stationary and ergodic marked point process (smpp) M = (σi, Ui, Mi) on a probability space
(Ω,F, P )with event times· · ·< σ−1< σ0≤0< σ1< σ2<· · · and marks(Ui, Mi)∈(0,∞)×E. HereEis
a Polish space with Borelσ-algebraEandMi is interpreted as an environmental variable. Let
Nt =
( P∞
i=1I0<σi≤t ift >0,
−P∞
i=0It <σ−i≤0 ift ≤0.
We callMa compound Poisson model ifE= {0}, i.e. there are no environmental marks, and. . . , σ−1−σ−2, σ0−
σ−1,−σ0, σ1, σ2−σ1, . . . are i.i.d. exponentially distributed random variables and(Ui)is an i.i.d. sequence of
positive random variables independent of(σi).
In this paper, we consider the process
St = Nt
X
i=1
Ui−t+ηBt(α), (1)
∗Tel.:+45-35-320788; fax:+45-35-320772.
E-mail address:[email protected] (H. Schmidli).
whereη >0 is some constant and(Bt(α))a spectrally positiveα-stable Lévy motion independent ofM. This is a process with independent and stationary increments, such that the increments follow a stable law (see Section 2). Spectrally positive means that there are no downward jumps. Ifη = 0, we call(St)an unperturbed risk model.
Because we will need thatE[|St|]<∞we assume 1< α ≤2. In the special caseα=2 the process(Bt(2))is a
Brownian motion withE[(Bt(2))2]=2t. It will be important for our approach that(St)has no downward jumps.
The reader should note that we prove Theorems 1 and 2 for any perturbation that is a Lévy process(Bt)with no
downward jumps andE[Bt]= 0. But the distributionH of the maximum of(ηBt −t :t ≥ 0)will then not be
given by (4).
The case α = 2 is widely discussed in the literature (see, for instance, Gerber, 1970; Dufresne and Gerber, 1991; Veraverbeke, 1993; Furrer and Schmidli, 1994; Schmidli, 1995). Furrer (1997, 1998) considers the case where 1 < α < 2 and Mis a compound Poisson process. In Schlegel (1998), a more general perturbation is considered.
Letλ=E[N1] be the intensity of the claim arrivals andµ=λ−1E[PNi=11 Ui] be the mean value of a typical claim.
We assume the net profit conditionρ=λµ≤1. This implies that limt→∞St = −∞. Letmt =sup{Ss : 0≤s < t},
τ+=inf{σk :k >0, Sσk > mσk},Lc=mτ+,Ld=Sτ+−Lc,Z+=Lc−S(τ+)−andM+=MN (τ+).τ+is then the
first time where a jump of the unperturbed model leads to a new maximum of the process(St). We will callτ+the
first (modified) ladder epoch. Note that inf{t >0 :St >0} =0 almost surely, and therefore a ladder epoch in the
classical sense cannot be defined.Lcis then the part of the ladder height due to the perturbation,Ldthe part due to the jump of the unperturbed model. We furthermore denote byU+=Ld+Z+the height of the jump leading to a
new ladder height. This also gives an interpretation ofZ+.M+denotes the environmental state at the ladder epoch.
Note thatLd,Z+,U+andM+are not defined ifτ+= ∞, whereasLcis well defined. Ifτ+<∞a second ladder
epochτ+(2) ∈(τ+,∞] can be defined. We denote the number of finite ladder epochs byKand use the superscript (k)to denote the random variables corresponding to thekth ladder epoch.
Ifα =2 andMis a compound Poisson model, Dufresne and Gerber (1991) showed thatK has a geometric distribution with parameterρ, andLcandLdare independent with densities
fLc(ℓc)=η
−2e−ℓc/η2, f
Ld(ℓd)=µ −1P[U
i > ℓd]. (2)
This shows thatLdhas the same distribution as in the unperturbed case (see, for instance, Rolski et al., 1999), and Lchas the same distribution as the maximum of(ηBt(2)−t :t ≥0). LetGdenote the distribution function ofLd
and letHα denote the distribution function of the maximum of(ηBt(α)−t:t≥0). Then it follows that in the case
α=2 andMis a compound Poisson model
P
"
sup
t≥0 St ≤u
#
=(1−ρ) ∞
X
n=0
ρn(G∗n∗Hα∗(n+1))(u). (3)
Furrer (1998) proved that (3) also holds for 1< α <2 as long asMis a compound Poisson process. His approach
did, however, not show whether or notHα andGstill can be interpreted as the distribution functions ofLc and Ld. We will prove that in this paper. In fact, we will prove that the random variables considered at the first ladder
epoch have this property wheneverMis an smpp. Our approach will moreover indicate that the compound Poisson model is the only one where the ladder heights can be split in this way. The stationarity is the property that makes our approach work. IfMis not a compound Poisson model, then the process will not be anymore in its stationary state at the first ladder epochτ+and thereforeLcandLdat the next ladder epoch will have a different form and be
dependent on the previous ladder heights ifE6= {0}.
In Section 4 we will apply the results to prove that ruin in the stationary perturbed risk model in a Markovian environment is more likely than in the perturbed classical risk model with the same claim arrival intensityλand the same typical claim size distributionF0(x)=λ−1E[PN1
2. Preliminaries
Stable distributions are the only distributions that can be obtained as weak limits of normalized sums of i.i.d. random variables. The logarithm of its characteristic functionF (r)ˆ =E[eirX] must then be of the form
logF (r)ˆ =
(
−σα|r|α(1−iβsign(r)tan(π α/2))+iµr ifα6=1,
−σ|r|(1+iβ2/πsign(r)ln|r|)+iµr ifα=1,
where 0 < α ≤ 2,|β| ≤ 1,σ ≥ 0 andm∈ R. This is the normal distribution (with variance 2σ2) ifα =2, in which caseβ does not have any influence. The parameterµis a location parameter,σ a scale parameter andβ a shape parameter.
Definition 1. A cadlag process(Bt(α))is called a (standard)α-stable Lévy motion if
1. B0(α)=0.
2. (Bt(α))has independent increments.
3. For 0≤s < t,Bt(α)−Bs(α) has a stable distribution with parametersα∈(0,2],σ =(t−s)1/α,β ∈[−1,1]
andµ=0.
Forα = 2, we obtain the Brownian motion withE[(B1(2))2] = 2. From the theory of Lévy processes (see, for instance, Furrer, 1997), it follows that forβ =1 the process has no jumps downwards. We call anα-stable Lévy motion with β = 1 spectrally positive. In this case for 1 < α ≤ 2 the distribution Hα of the maximum of
(ηBtα−t:t≥0)is given by
1−Hα(x)=
∞
X
n=0
(axα−1)n
Γ (1+(α−1)n), (4)
where
a= cos(απ/2) ηα ,
see Furrer (1998). For our applications, we consider(Bt(α))as a process onR, i.e.(−B−(α)t−:t ≥0)is an independent copy of(Bt(α):t ≥0).
In recent work, Asmussen and Schmidt (1995) showed the result corresponding to Theorem 1 for unperturbed risk processes. LetP0denote the Palm probability measure and let(σi0, Ui0, Mi0)denote the marked point process starting at a typical claim epoch. For the definition of Palm probabilities, see for instance Rolski et al. (1999), Baccelli and Brémaud (1987), Franken et al. (1982) or König and Schmidt (1992).
Proposition 1. IfMis ergodic,η=0 (i.e.Lc=0a.s.)andρ≤1,then P[M+∈F, Ld≥y, Z+≥z, τ+<∞]=λ
Z ∞
z+y
P0[U ≥x, M ∈F] dx
for everyz, y≥0, F ∈E.
The special caseF =Eshows that in the unperturbed case the distribution of the first ascending ladder height is independent of the law of the smpp. Because the perturbed caseη >0 is very similar to the unperturbed case one expects such a result to hold for any ergodic smpp.
The main tool used by Asmussen and Schmidt (1995) is the following lemma, which also can be found in König and Schmidt (1992) or Rolski et al. (1999). Denote byΘt the shift operator, i.e. the shifted processM◦Θt has
Lemma 1. We have
whereφis an arbitrary non-negative measurable functional.
3. Main result
thatE[τ1]=1 (see, for instance, Furrer, 1997), andXτ1 =0. Moreover, because of the independent and stationary increments limn→∞τn/n=E[τ1]=1. Because(Xt)is a stationary ergodic process we have by the strong law of
The latter expression is the probability thatXt exceeds the levelb under the stationary initial measure for(Xt).
Inverting the time it follows as in Asmussen and Petersen (1989) that
lim
t→∞P[Xt ≥b]=P[supt≥0ηB
(α)
t −t ≥b]=1−Hα(b).
We can now prove the main result of this paper.
Note that
This condition is fulfilled in the interval [0,−σ−10 ]. Thereafter, the condition is not fulfilled until the first epoch
t ≥ −σ−10 whereS˜t = ˜S(−σ−1)−and so on. If we cut out all intervals in which (9) is not fulfilled then the pieces left follow, by the strong Markov property of the Lévy motion and the fact that(Bt(α))only admits positive jumps, the same law as(ηBt(α)−t ). Thus
leave the process as it is until the next jump caused by the unperturbed process,σn−σk say. Then we cut out the
can be obtained directly from(St)by cutting out the pieces of(St)betweenσn and the last time before the jump
where(St)was at the levelSσn. We had to go via the construction above because the last time beforeσnwhere(St)
was at a certain level is not a stopping time. It follows readily that(B¯t(α):t ≥0)and(In:n≥1)are independent.
Indeed, replacing(B¯t(α):t ≥0)by an independent copy and going backward in the construction (not changing the pieces finally cut out) yields a perturbed risk model following the same law as(St). We have
P[τ+> n, sup 0≤t <σn
St ≥ℓc]=P
"
In=1, sup
0≤t≤An
¯
Bt,n(α)−t ≥ℓc
#
.
Thus, lettingn→ ∞,
P[τ+= ∞, Lc≥ℓc]=P[τ+= ∞]P
"
sup
0≤t <∞ ¯
Bt(α)−t ≥ℓc
#
proving (7).
Remark. Note that in the proof of (7) we did not use the stationarity ofM. Thus,
P[τ+= ∞, Lc≥ℓc]=P[τ+= ∞](1−Hα(ℓc)) (10)
for any risk model perturbed by a Lévy process with no downward jumps and zero mean value with limt→∞St =
−∞.
The above theorem shows that (2) remains valid for any smpp ifα=2. We also have generalized Proposition 1 to the perturbed case.
Let us now consider the compound Poisson case. The lack of memory property of the exponential distribution assures that the smpp is in the stationary state at timeτ+.
Corollary 1. LetMbe a compound Poisson model. Then K,(L(n)c )and(L(n)d )are independent, the number of
ladder heights K has a geometric distribution with parameterρ,Lchas distribution functionHαandLdis absolutely
continuous with densityµ−1P[Ui > x].
This corollary leads to the Pollaczek–Khinchin type formula (3) obtained by Dufresne and Gerber (1991), and Furrer (1998). Let us consider the joint distribution of(Lc, Ld, U+, Z+, M+).
Theorem 2. Assume thatMis ergodic and thatρ ≤ 1.Then the joint(defective)distribution of(Lc, Ld, U+, Z+, M+)can be described as follows:
1. P[τ+<∞]=ρ.
2. Lcis independent of the random variables(Ld, U+, Z+, M+)and of the event{τ+<∞},and has the distribution
functionHα(i.e.Lchas the same distribution as the maximum of anα-stable Lévy motion with drift−1).
3. The conditional distribution of(U+, M+)givenτ+<∞is obtained from the Palm distributionP0of(U0, M0)
by the change of measure given by the likelihood ratioµ−1U0,i.e.
E[g(U+, M+)|τ+<∞]=
1
µE
0[U0g(U0, M0)]
for every non-negative measurable function g.
4. The conditional distribution of(Z+, Ld)given U+,M+,τ+ < ∞ is that of(U+V , U+(1−V ))where V is
uniformly distributed on(0,1)and independent ofU+, M+, τ+<∞.
4. The perturbed risk process in a Markovian environment
Let(Mt)be an irreducible time homogeneous Markov chain in continuous time with state space{1,2, . . . , p},
intensity matrixΛ=(Λij)and stationary initial distributionπ=(π1, . . . , πp). Letλibe the claim arrival intensity
andFi be the claim size distribution ifMt =i. This risk process was considered by Janssen (1980), Janssen and
Reinhard (1985), Asmussen (1989) in the unperturbed case, and by Schmidli (1995) in the perturbed case with
α=2. The intensity of the smpp is then given by
λ=
p
X
i=1 πiλi,
and the claim size distribution of a typical claim becomes
F0(x)= 1 λ
p
X
i=1
πiλiFi(x).
Without loss of generality we can assume that
λ1≤λ2≤ · · · ≤λp. (11)
We denote byψ(u)=P[sup{St :t >0}> u] the ruin probability of the (perturbed) risk process in a Markovian
environment.
In a recent paper, Asmussen et al. (1995) compared the unperturbed risk process in a Markovian environment with the unperturbed standard compound Poisson risk process with claim arrival intensityλand claim size distribution
F0. They showed under the assumptions (11),
1−F1(x)≤1−F2(x)≤ · · · ≤1−Fp(x) for allx ≥0, (12)
and
X
n≥l
Λjn≤
X
n≥l
Λkn for allj, k, lwithj ≤k,andl≤jorl > k (13)
thatψ(u)≥ψ∗(u)whereψ∗(u)is the ruin probability of the standard compound Poisson risk process. The main tool in their proof was Proposition 1. It is therefore not surprising that the corresponding result also holds for the perturbed risk process. For the rest of this section we denote by(St∗)the (perturbed) standard compound Poisson risk process with claim arrival intensityλand claim size distributionF0, and byψ∗(u)its ruin probability.
Proposition 2. Assume that the conditions(11)–(13)hold. Thenψ∗(u)≤ψ(u).
Proof. It follows as in Lemma 2.1 of Asmussen et al. (1995) thatψi(u)≤ψj(u)for any 1≤ i≤ j ≤pwhere
ψi(u)=P[sup{St :t >0}> u|M0=i] denotes the ruin probability if the process starts in statei. Let furthermore G∗(x)=P[Lc+Ld≤x] andGi(x)=P[Lc+Ld≤x|M+=i]. Note that, by Theorem 2,
ψ∗(u)=(1−ρ)(1−Hα(u))+ρ(1−G∗(u))+ρ
Z u
0
ψ∗(u−x)dG∗(x),
ψ(u)=(1−ρ)(1−Hα(u))+ρ(1−G∗(u))+ρ p
X
i=1
P[Mτ+ =i|τ+<∞]
Z u
0
ψi(u−x)dGi(x).
Denote byτ+(i)the epoch of theith ladder height. As in Asmussen et al. (1995) the following proposition can be proved.
Proposition 3. Assume that conditions(11)–(13)hold. Then
ρk ≤P[τ+(1)<∞, . . . , τ+(k) <∞]
for anyk∈N.
Acknowledgements
The author thanks Hansjörg Furrer for a fruitful discussion on the topic.
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