Exit identities for Lévy processes observed at Poisson arrival times

Transcription

1 Bernoulli 223, 216, DOI: 1.315/15-BEJ695 arxiv: v2 [math.pr] 17 Mar 216 Exit identities for Lévy processes observed at Poisson arrival times HANSJÖRG ALBRECHER1, JEVGENIJS IVANOVS 2 and XIAOWEN ZHOU 3 1 Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, CH-115 Lausanne, Switzerland and Swiss Finance Institute, Switzerland. hansjoerg.albrecher@unil.ch 2 Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, CH-115 Lausanne, Switzerland. jevgenijs.ivanovs@unil.ch 3 Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. W., Montreal, Quebec, Canada H3G 1M8. zhou@alcor.concordia.ca For a spectrally one-sided Lévy process, we extend various two-sided exit identities to the situation when the process is only observed at arrival epochs of an independent Poisson process. In addition, we consider exit problems of this type for processes reflected either from above or from below. The resulting Laplace transforms of the main quantities of interest are in terms of scale functions and turn out to be simple analogues of the classical formulas. Keywords: Cramér Lundberg risk model; dividends; exit problem; reflection; spectrally negative Levy process 1. Introduction Consider a spectrally-negative Lévy process X, that is, a Lévy process with only negative jumps, and which is not a.s. a non-increasing process. Let ψθ:=logee θx1, θ, be its Laplace exponent. Denote the law of X with X=x by P x and the corresponding expectation by E x. For a fixed a define the first passage times τ :=inf{t : Xt<}, τ+ a :=inf{t : Xt>a}. Furthermore, let T i be the arrival times of an independent Poisson process of rate >, and define the following stopping times: T :=min{t i: XT i <}, T + a :=min{t i: XT i >a}. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 216, Vol. 22, No. 3, This reprint differs from the original in pagination and typographic detail c 216 ISI/BS

2 2 H. Albrecher, J. Ivanovs and X. Zhou The latter times can be seen as first passage times when X is observed at Poisson arrival times by convention inf =min =. These quantities have useful interpretations in various fields of applied probability. For instance, if X serves as a model for the surplus process of an insurance portfolio over time, then P x τ < is the probability of ruin of the portfolio with initial capital x. Likewise, P x T < is the probability that ruin occurs and is detected, given that the process can only be monitored at discrete points in time modeled by an independent Poisson process. It is not hard to show that T τ a.s. and similarly T a + τ a + a.s. as the observation rate tends to, which may be used to retrieve the classical exit identities. Quantities related to T havebeen studied for a compound Poissonrisk model in [2], and in [18] a simple formula for P x T < wasestablished for generalspectrallynegative Lévy processes X. Ruin-related quantities under surplus-dependent observation rates were studied in [6] and for recent results on observation rates that change according to environmental conditions, we refer to [4]. Poissonian observation is also relevant in queueing contexts, see, for example, [1]. In practice one may interpret continuous and Poissonian observation as endogenous and exogenous monitoring of the process of interest, respectively. For example, in an insurance context τ may be understood as the time of ruin observed by the insurance company, whereas T a + may be considered as the first time when shareholders receive dividends they look at the company at discrete, here random, times. Hence, the shareholders receive dividends if the event {T a + <τ } occurs. Another example comes from reliability theory [2], where one considers a degradation process and assumes that τ is the time of failure and Ta :=min{t i: XT i <a} for some a> is the time at which the process is observed in its critical state necessitating replacement. Hence, the event {Ta <τ } signifies preventive replacement before failure. Finally, some identities involving both continuous and Poissonian observation lead to transforms of certain occupation times, see Remark 3.2. In this paper, we establish formulas for two-sided exit probabilities under Poissonian observation, as well as formulas for the joint transform of the exit time and the corresponding overshoot on the event of interest. In addition, we consider reflected processes and provide the joint transforms including the total amount of output dividends or input required capital to remain solvent up to the exit time. Note that the formulas also hold for spectrally-positive Lévy processes by simply exchanging the roles of the involved quantities. It turns out that the resulting formulas have a rather slim form in terms of first and second scale functions, and along the way it also proves useful to define a third scale function. The form of the expressions allows to interpret them as natural analogues of the respective counterparts under continuous observation. Finally, we note that discrete observation allows for a wide range of cases, and our list of exit identities is not exhaustive. We only consider the basic cases, that is, the ones where the corresponding events stay non-trivial if Poissonian observation is replaced by a continuous one, which, for example, excludes {Ta <τ } mentioned above. The remainder of this paper is organized as follows. Section 2 recalls some relevant exit identities under continuous observation. Section 3 contains the main results, and the proofs are given in Section 4. Finally, Section 5 gives an illustration of some identities for the case of Cramér Lundberg risk model with exponential claims.

3 Exit identities for Lévy processes observed at Poisson arrival times 3 Throughout this work, we use e u to denote an exponentially distributed r.v. with rate u>, which is independent of everything else. 2. Standard exit theory The most basic identity states that P τ + a < =e Φa, a, 1 where Φ is the right-most non-negative solution of ψθ=. Let us recall two fundamental functions which enter various exit identities. The first scale function Wx is a non-negative function, with Wx= for x<, continuous on [,, positive for positive x, and characterized by the transform e θx Wxdx=1/ψθ, θ>φ. It enters the basic two-sided exit identity for a> through P x τ a + <τ =Wx/Wa, x a, 2 see, for example, [17]. The so-called second scale function is defined by x Zx,θ:=e 1 ψθ θx e θy Wydy, x 3 and Zx,θ:=e θx for x<. Note that for θ =, Zx,θ reduces to Zx as defined in [17], Chapter 2. It is convenient to define Z as a function of two arguments, which allows to provide more general formulas. We refer to [15] for this definition and the following formulas in a more general setting of Markov additive processes. Note that for θ>φ we can rewrite Zx,θ in the form Zx,θ=ψθ It is known that for x a one has e θy Wx+ydy, x,θ>φ. 4 Moreover, for θ>φ we have E x e θxτ ;τ <τ+ a =Zx,θ WxZa,θ Wa. 5 lim Za,θ/Wa=ψθ/θ Φ 6 a

4 4 H. Albrecher, J. Ivanovs and X. Zhou and so we also find that E x e θxτ,τ < =Zx,θ Wx ψθ θ Φ. 7 Importantly, all the above results hold for an exponentially killed process X cf. [14]: For a killing rate q >, we write ψ q θ=ψθ q, then Φ q > is the positive solution to ψ q θ=, and W q x is defined by e θx W q xdx=1/ψ q θ, θ>φ q. With Z q x,θ defined through W q x and ψ q θ, formula 5 in the case of killing reads E x e qτ +θxτ ;τ <τ+ a =E x e θxτ ;τ <τ+ a,τ <e q =Z q x,θ W q x Z qa,θ W q a, and hence the information on the time of the exit is easily added. The same adaptations hold for the other exit identities above. For the sake of readability, we will often drop the index q in the sequel, if it does not cause confusion. In this case, Φ,W x,z x,θ should be interpreted as Φ +q,w +q x,z +q x,θ, respectively, that is, they correspond to the process killed at rate q and then additionally killed at rate. Finally, we will need the following identities which can readily be obtained from the known formulas for potential densities of X killed upon exiting a certain interval, see, for example, [11] or [17], Chapter 8.4: PXe dx =e Φ x /ψ Φ W xdx, 8 PXe dx,e <τ + a =e Φ a W a x W xdx, 9 P a Xe dx,e <τ =e Φ x W a W a xdx, 1 where a> and the killing rate q is implicit. 3. Results One of the first general results concerning T for q= and EX1> that was obtained in [18], where it was shown P x T = =ψ Φ Zx,Φ, x, 11 cf. 4. This leads to a strikingly simple identity for x=: PT = =ψ Φ /. A more general result was recently obtained in [4] for an arbitrary killing rate q : P x τ a + <T = Zx,Φ, x [,a]. 12 Za,Φ

5 Exit identities for Lévy processes observed at Poisson arrival times 5 It is easy to see that both these results also hold for x<. Note the resemblance of 2 and 12, and, moreover, Zx,Φ Za,Φ =P xτ a + <T P xτ a + <τ Wx = Wa as, 13 because in the limit the Poisson observation results in continuous observation of the process a.s. although it is hard to see the convergence of the ratio directly. We now present various exit identities for continuous and Poisson observations extending the standard exit theory. Theorem 3.1. For a,θ,x a and implicit killing rate q, we have E x e θxt ;T < = Zx,θ Zx,Φ ψθφ Φ ψθ θ Φ E x e θxt ;T <τ+ a = ψθ Zx,θ Zx,Φ Za,θ Za,Φ, 14, 15 E x e θxt+ a a ;T + a < = Φ Φ Φ +θ e Φa x, 16 E x e θxt+ a a ;T + a <τ = Φ +θ E x e θxτ ;τ <T+ a =Zx,θ Wx θ Φ Wx Za,Φ, 17 ψθ Za,θ, 18 Za,Φ where ratios for θ=φ and θ=φ should be interpreted in the limiting sense. Remark 3.1. When EX1 > and q = θ =, we have Φ = and Zx, = 1, so that 14 reduces to 11. Secondly, note the resemblance between 5 and 15, and that the first is retrieved from the second when cf. 13. Next, 16 is an extension of identity 1 which is retained for θ = and. This formula also implies that the overshoot XT + a a given {T + a < } and q = is exponentially distributed with rate Φ for all x a indeed it is not hard to establish the memoryless property of this overshoot. The identity 17 is a variation of 2 and 12, and identity 18 is the counterpart of 17 for the process X reproducing 5 for and 7 for. Remark 3.2. There is a close link between some of our results and transforms of certain occupation times: E x e θxτ ;τ <T+ a =E x e θxτ ;τ <,NA= =E x e θxτ τ 1 {Xt>a} dt ;τ <, wherea={t [,τ : Xt>a}and N is anindependent Poissonrandommeasurewith intensity dt. A similar identity also holds for P x τ a + <T. Hence, 12 and 18 for

6 6 H. Albrecher, J. Ivanovs and X. Zhou θ= can alternatively be obtained from [19], Corollaries1 and 2, and taking appropriate limits followed by somewhat tedious simplifications. The next result considers two-sided exit for exclusively Poissonian observation of the process. Theorem 3.2. For a,θ,x a and implicit killing rate q, we have E x e θxt ;T <T+ a = Zx,θ Zx,Φ Za,Φ,θ, 19 ψθ Za,Φ,Φ E x e θxt+ a a ;T + a <T = Φ +θ where we define a third scale function as Zx,Φ Za,Φ,Φ, 2 Zx,α,β:= ψαzx,β ψβzx,α, α,β. 21 α β Again there is a striking similarity between 15 and 19, as well as between 17 and 2. Note that for α=β the definition 21 results in Zx,α,α=ψ αzx,α ψαz x,α, where the differentiation of Z is with respect to the second argument. We now present results for reflected processes. Write E x for the law of X reflected at from below and E a x for the law of X reflected at a from above, and let R be the regulator at the corresponding barrier. That is Xt,Rt under E x and under E a x is given by Xt+ Xt +, Xt + and Xt Xt a +,Xt a + under E x, respectively, where Xt:=inf{Xs: s t}, Xt:=sup{Xs: s t}. Theorem 3.3. For a >, θ, ϑ, x a and implicit killing rate q, we have the following identities for the reflected processes: E xe ϑrt+ a θxt+ a a ;T + a < = ϑ Φ Zx,ϑ Φ +θψϑza,φ Za,ϑ = Φ +θ Zx, ϑ Za,ϑ,Φ, 22 E a x e ϑrt +θxt ;T < 23 = Zx,θ+Zx,Φ Waψθ θ+ϑza,θ ψθ Z, a,φ +ϑza,φ where the derivative of Z is taken with respect to the first argument.

7 Exit identities for Lévy processes observed at Poisson arrival times 7 Note that T a + and T can be infinite due to the implicit killing rate q. This result for = is to be compared with E xe ϑrτ+ a ;τ + a < = Zx,ϑ Za,ϑ, 24 E a x e ϑrτ +θxτ ;τ < 25 =Zx,θ+Wx Waψθ θ+ϑza,θ W +a+ϑwa, where W + denotes the right derivative of W see, e.g., [17], Theorem 8.1, for ϑ = and [15] for the general case. Furthermore, letting a in 23 and using 6 we obtain 14, which provides a nice check ϑ indeed cancels out. Similarly, 22 leads to 16 if we put a= +x and let x note that 16 depends only on the difference. Finally, we note that yet another exit identity for a reflected process with Poissonian observations can be found in [5], Corollary 6.1. In particular, letting ρ y :=inf{t : Rt>y} be the first passage time of R, it holds that P a x ρ y <T = Zx,Φ Za,Φ exp Z a,φ Za,Φ y, 26 whereq isimplicitand x [,a].ifx issomesurplusprocess,itisnaturaltointerpret Rt as the dividend payments up to time t according to a horizontal dividend barrier strategy. Identity 26 then allows to obtain the expected discounted dividends until ruin: E a x e qt 1 {t<t }drt=ea x e qρy 1 {ρy<t }dy = E a x ;ρ e qρy y <T dy 27 = Z qx,φ +q exp Z q a,φ +q Z q a,φ +q Z q a,φ +q y dy = Z qx,φ +q Z qa,φ +q, where x [,a]. In the case of continuous observations, that is, =, this expression reduces to W q x/w q+a, see, for example, [21], Proposition 2. Also, in the absence of discounting q=, one obtains from 23 for θ= that E a ae ϑrt =Z a,φ /Z a,φ +ϑza,φ, that is, the distribution of total dividend payments until ruin observed at Poissonian times is exponentially distributed with parameter Z a,φ /Za,Φ, if the initial surplus level is at the barrier. The exponential parameter reduces to W + a/wa for, cf. [21], Section 5.

8 8 H. Albrecher, J. Ivanovs and X. Zhou Remark 3.3. We emphasize again that each of the formulas in Theorems can also be written for q> in an explicit form, see also 27. For instance, 15 can be read as E x e qt +θxt ;T <τ+ a = 4. Proofs ψ q θ Z q a,θ Z q x,θ Z q x,φ +q Z q a,φ +q Some of the proofs below will rely on the following intriguing identity first observed in [19], Equation 6: p q which as a consequence yields W p a xw q xdx=w p a W q a, 28 W q a xw q xdx= W qa. q The following result generalizes the second part of [19], Equation 6: Lemma 4.1. For θ,α,p,q, it holds that p q Proof. First, we show that p q W p a xz q x,θdx=z p a,θ Z q a,θ. W p a x x =e θa e θx W p xdx e θx y W q ydydx e θx W q xdx by taking transforms of both sides. The left-hand side gives, for sufficiently large s, x e p q sa W p a x e θx y p q W q ydydx da= s θψ p sψ q s, and for the right-hand side we have e sa e θa e θx W p xdx da= 1 s θψ p s and similarly for the second term. Then 29 follows by noting that 1 ψ p s 1 ψ q s = p q ψ p sψ q s.. 29

9 Exit identities for Lévy processes observed at Poisson arrival times 9 Finally, using 29 we get p q =p q W p a xz q x,θdx =e θa p q ψ q θ =Z p a,θ Z q a,θ W p a xe θx dx ψ q θe θa e θx W p xdx e θx W p xdx+e θa ψ q θ e θx W q xdx e θx W q xdx finishing the proof Proof of Theorem 3.1 We split the proof into several parts. Proof of Equation 15. Denoting fx,θ,a:=e x e θxt ;T <τ+ a one can write, using the strong Markov property, fx,θ,a= P x Xτ dz,τ <τ+ a P zτ + <e f;θ,a+e z e θxe,e <τ +. Recall that P z τ + <e =e Φ z,z and also E z e θxe,e <τ + =E ze θxe e Φ z Ee θxe = ψθ eθz e Φ z 3 for θ small enough such that ψθ<. The result can then be analytically continued to any θ. Thus, using 5 we arrive at fx,θ,a= Zx,Φ Za,Φ Wx f,θ,a Wa ψθ + Zx,θ Za,θ Wx Wa Note that due to Z,θ=1 we get for x= Za,Φ W Wa f,θ,a= ψθ ψθ. Za,Φ W Wa Za,θW Wa This equation is trivial when W=, that is, when X has sample paths of unbounded variation, see, for example, [17], Equation 8.26, but otherwise we have f,θ,a= 1 Za,θ 32 ψθ Za,Φ. 31

10 1 H. Albrecher, J. Ivanovs and X. Zhou and the result follows combining 31 and 32. It is only left to show that 32 holds also when X has sample paths of unbounded variation, that is, when W=. For x,a, we have f,θ,a=pτ + x <e fx,θ,a+ax+bx, where Ax:=Ee θxe ;e <τ + x,xe <, 33 Bx:= It is well known that for θ x x Pe <τ + x,xe dyfy,θ,a. e θy Wydy/Wx as x, 34 which can be seen by interpreting the ratio of scale functions. In a similar way one can show, using 28, that W x/wx 1 as x. Using 9, observe that Pe <τ + x,xe =e Φ x x as x. Next, using 3 observe that Bx=oWx and W ydy=owx Ax :=Ee θxe ;e <τ + x EeθXe ;e <τ + x,xe = ψθ 1 eθ Φ x +owx. Plugging 33 into 31 and rearranging it we obtain [ fx,θ,a 1 Zx,Φ Za,Φ Wx ] e Φ x Wa = ψθ Zx,Φ e θ Φx Zx,θ+ Wx Wa Za,θ Za,Φ e θ Φx +owx. 35 Divide 35 by Wx and take the limit as x, using the representation 3 and then also 34, to obtain 1 f,θ,aza,φ Wa = 1 ψθ Wa Za,θ Za,Φ, which immediately yields 32.

11 Exit identities for Lévy processes observed at Poisson arrival times 11 Proof of Equation 17. We only need to consider x [,a]. Putting we write fx,θ:=e x e θxt+ a a ;T + a <τ fx,θ=p x τ + a <τ Wx fa,θ= fa,θ. 36 Wa Using 1 and conditioning on the first Poisson observation time, we get fa,θ= a e θx a W ae Φ x dx+ Using 36, we obtain fa,θ Wa W a = W awa e Φa. Φ +θ Wxe Φ x dx+ W ae Φ x W a xfx,θdx. W xwa xdx With the help of 28, the expression in the brackets reduces to W a 1 Wxe Φx dx =W ae Φa Za,Φ, which shows that fa,θ= Wa Φ +θ Za,Φ, completing the proof in view of 36. Proof of Equations 14 and 16. Identity 14 for θ > Φ follows immediately from 15 and 6; by analytic continuation it is also true for any θ. Similarly, 16 follows from 17 by plugging in x+u and a+u instead of x and u, respectively, letting u and using 6 together with lim Wx+u/Wa+u=P xτ + u a < =e Φa x. Proof of Equation 18. Considerfx:=E x e θxτ ;τ <T+ a, whichcanbe written as fx=p x τ a + <τ fa+e xe θxτ ;τ <τ+ a 37 = Wx Wa fa+zx,θ WxZa,θ Wa and also fa= P a Xe dx,e <τ fx+e ae θxτ ;τ <e,

12 12 H. Albrecher, J. Ivanovs and X. Zhou where the last term is Z a,θ W a ψθ θ Φ according to 7. Hence, we can determine fa by plugging in 37 and computing the integrals of Wx and Zx,θ with respect to 1. In particular, using 28 we find Next we compute e Φ x Zx,θdx = P a Xe dx,e <τ Wx =W a W ae Φ x W a xwxdx e Φ x Wxdx W a Wa =Wa Za,Φ e Φ a W a. = 1 e θ Φa 1 ψθ θ Φ θ Φ e Φa Za,θ 1+ψθ = 1 θ Φ e θ Φ a which together with Lemma 4.1 implies that T := = W a θ Φ P a Xe dy,e <τ Zy,θ e Φ a Za,θ 1+ψθ e θy Wydy e Φy Wydy, e Φy Wydy e Φy Wydy Z a,θ+za,θ. This finally yields fa= 1 Za,Φ e Φ a W a fa+t 1 Za,Φ e Φ a W a Za, θ Wa Wa +Z a,θ W a ψθ θ Φ, which reduces to Za,Φ e Φ a W a fa Wa = W a θ Φ e Φ a Za,θ ψθe Φ a Za,Φ +Za,Φ e Φ a W a Wa Za,θ,

13 Exit identities for Lévy processes observed at Poisson arrival times 13 and hence Now the result follows from 37. fa=za,θ Wa ψθ Za,θ. θ Φ Za,Φ 4.2. Proof of Theorem 3.2 Using 3 and changing the order of integration, we can show that α β e αx Zx,βdx=1+e αa Za,α 1ψβ/ψα e αa Za,β, and hence Z has an alternative representation Za,α,β=e αaψα ψβ α β Plugging in α=φ and β =θ, we obtain ψα e αa x Zx,βdx. e Φ x Zx,θdx= ψθ θ Φ e Φ a Za,Φ,θ. 38 Proof of Equation 19. Defining fx,θ:=e x e θxt ;T <T+ a for x a, we write fx,θ=e x e θxt ;T <τ+ a +P x τ + a <T fa,θ. Plugging in the corresponding identities, we first get for x= that f,θ= 1 Za,θ + fa,θ ψθ Za,Φ Za,Φ, and some simplifications yield fx,θ= ψθ Zx,θ Zx,Φ +f,θzx,φ. 39 Using 8 and conditioning on the first observation epoch, we get f,θ= ψ Φ where the latter term evaluates to f,θ 1 ψ Φ e Φx fx,θdx+ e θx PXe dx, ψθ + e Φx Zx,Φ dx ψ Φ θ Φ. Plugging in 39, we get

14 14 H. Albrecher, J. Ivanovs and X. Zhou = ψ Φ ψθ Using 38 this reduces to e Φ x Zx,θ Zx,Φ dx+ f,θe Φ a Za,Φ,Φ /ψ Φ ψθ + ψ Φ θ Φ. =e Φ a ψ Φ ψθ Za,Φ,Φ Za,Φ,θ, which readily leads to f,θ= 1 Za,Φ,θ ψθ Za,Φ,Φ and then the result follows from 39. Proof of Equation 2. We write for x a that E x e θxt+ a a =E x e θxt+ a a ;T + a <T Using 16, we obtain + P x XT dy,t <T+ a E y e θxt+ a a. E x e θxt+ a a ;T a + <T = Φ Φ Φ +θ e Φa e Φx E x e ΦXT ;T <T+ a. With 21, we see that Za,Φ,Φ= Φ Φ eφa and then it follows from 19 that E x e ΦXT ;T <T+ a =e Φx Zx,Φ Φ Φ eφa / Za,Φ,Φ, which completes the proof Proof of Theorem 3.3 Proof of Equation 22. Let fx:=e xe ϑrt+ a θxt+ a a ;T + a <, then fx=e x e ϑrτ+ a ;τ + a < fa= Zx,ϑ Za,ϑ fa according to 24, and also fa=e a e θxt+ a a ;T + a <τ +E ae ϑxτ ;τ <T+ a f,

15 Exit identities for Lévy processes observed at Poisson arrival times 15 which is equal to Za,ϑf. Plugging in 17 and 18 we solve for f: f= ϑ Φ Φ +θψϑza,φ Za,ϑ and the result follows. Proof of Equation 23. Let fx=e a x e ϑrt +θxt ;T <, then fx=e x e θxt ;T <τ+ a +P xτ + a <T fa, 4 which using 15 and 12 yields f= 1 Za,θ + fa ψθ Za,Φ Za,Φ. 41 Also fa= Using 3, we arrive at E a ae ϑrτ ;Xτ dz,τ < eφ z f+e z e θxe ;e <τ +. fa=e a a e ϑrτ +Φ Xτ f + ψθ Ea a e ϑrτ +θxτ E a a e ϑrτ +Φ Xτ. 42 Substituting 25 and 41 into 42, we obtain after some simplifications fa Φ +ϑ Wa Za,Φ = Wa ψθ Za,θ +Φ θza,θ, ψθ Za,Φ which yields the result after plugging fa into 4 and yet another round of simplifications. The above proofs mostly rely on the strong Markov property and various identities from fluctuation theory. We note that often there are several possibilities to approach a problem, but some of them may require significantly more effort to obtain a simple formula resembling the classical case. One could, for instance, consider using exit theory of random walks for a purely Poissonian observation. This approach builds upon some general formulas, see, for example, [12], Theorem 4, ignoring the crucial assumption of one-sided jumps. Consequently, one loses structure, making it hard to rewrite these formulas in terms of scale functions. Martingale techniques, as used, for example, in [9] to

16 16 H. Albrecher, J. Ivanovs and X. Zhou obtain some classical exit identities, also do not seem to be immediately appropriate for our setting. Moreover, one needs to guess the right martingale and for this one typically needs to know already the resulting expression. Finally, independent exponential interobservation times may suggest using Wiener Hopf factorization, exploited, for example, in [16] to design simulation algorithms. Indeed, this factorization is one way to prove 8 1, which are building blocks of our results. 5. Cramér Lundberg risk model with exponential claims As mentioned in Section 1, one application area for identities of the above type is the ruin analysis for an insurance portfolio with surplus value Xt=x+ct St at time t, where x is the initial capital and c > is a constant premium intensity. The classical Cramér Lundberg risk model in this context assumes St to be a compound Poisson process, where independent and identically distributed claims arrive according to a homogeneous Poisson process with rate ν see, e.g., [7]. Assume now that claims are Expη distributed. Then ψ q θ=cθ ν1 Ee θeη q=cθ νθ θ+η q. If either q > or c ν/η the usual safety loading condition is c ν/η>, then the scale function has the form W q x=u q e Φqx v q e Rqx, where u q,v q > and R q,φ q not simultaneously. Moreover, R q and Φ q are the two roots of ψ q θ=, see Figure 1 note that R is the classical Lundberg adjustment coefficient, and u q θ Φ q v q θ+r q = 1 ψ q θ yielding u q =1/ψ qφ q =Φ q,v q = 1/ψ q R q =R q. So we also obtain Z q x,θ =e θx 1 ψ q θ uq uq e Φqx =ψ q θ v qe Rqx θ Φ q θ+r q Φ q θ eφq θx 1 = ψ qθφ q θ Φ q e Φqx e Rqx +e Rqx. v q R q θ e Rq θx 1

17 Exit identities for Lévy processes observed at Poisson arrival times 17 Figure 1. The function ψθ and the inverses R q and Φ q. Consider 14, which for the present model immediately simplifies to E x e qt +θxt ;T < =e Rqx ψ qθφ q Φ +q /Φ q θ ψ q θ =e Rqx 1 ψ qθ Φ +q θ. ψ +q θ Φ q θ Since ψ +q θ θ+η c =θ+r +q θ Φ +q, we get E x e qt +θxt ;T < =e Rqx 1 θ+r q =e RqxR +q R q θ+r +q R +q +θ, which agrees with the result of [2], Example 4.2 take an exponential penalty function w 2 y = e θy for the overshoot. Note also that R +q η as, because θ = η is the asymptote of ψ q θ. Hence, we also retain the classical formula for the Laplace transform of the discounted ruin deficit under continuous observation E x e qτ +θxτ ;τ < =e Rqxη R q η+θ, cf. [13], Equation 5.42, which can alternatively be obtained using a direct argument exchange the meaning of claims and interarrivals. Next, identity 15 simplifies to E x e qt +θxt ;T <τ+ a = e Rqa+Φqx e Φqa Rqx ψ q θφ q /θ Φ q Φ q /Φ +q Φ q ψ +q θ Φ q /Φ +q Φ q e Φqa e Rqa +e Rqa = R +q R q R +q +θ e Φqa+Rqa x e Φqx e Φqa+Rqa 1+Φ +q Φ q /Φ. q

18 18 H. Albrecher, J. Ivanovs and X. Zhou It is not hard to see that Φ +q Φ q / 1/c as, and hence we have E x e qτ +θxτ ;τ <τ+ a = η R q η+θ = η R q η+θ e Φqa+Rqa x e Φqx e Φqa+Rqa 1+ψ q Φ q/c e Φqa+Rqa x e Φqx e Φqa+Rqa +R q η/φ q +η, where the last equality follows from the observation that ψqθ θ Φ q = c θ+rq θ+η and hence ψ qφ q =c Φq+Rq Φ. q+η Finally, 27 provides a formula for the expected continuously discounted dividends until ruin: E a x e qt Φ q/φ +q Φ q e Φqx e Rqx +e Rqx 1 {t<t }drt = Φ q /Φ +q Φ q Φ q e Φqa +R q e Rqa R q e Rqa because Φ +q Φ q Φ q = eφqx +e Rqx Φ +q Φ q /Φ q 1 Φ q e Φqa R q e Rqa Φ +q Φ q /Φ q 1 43 = R +q +Φ q e Φqx R +q R q e Rqx R +q +Φ q Φ q e Φqa +R q R +q R q e Rqa, = Φq+Rq Φ q+r +q. This expression is similar to [1], Equation 24, but not identical, because there the dividends are paid at Poissonian times only see also [8] for a spectrally positive model setup. Identity 43 is, however, the analogue of [3], Equation 2, where it was derived for a diffusion process Xt. Acknowledgements We would like to thank two anonymous referees for helpful remarks concerning the presentation of this paper. H. Albrecher and J. Ivanovs are supported by the Swiss National Science Foundation Project , and X. Zhou is supported by an NSERC grant. References [1] Albrecher, H., Cheung, E.C.K. and Thonhauser, S Randomized observation periods for the compound Poisson risk model dividends. Astin Bull MR [2] Albrecher, H., Cheung, E.C.K. and Thonhauser, S Randomized observation periods for the compound Poisson risk model: The discounted penalty function. Scand. Actuar. J MR317613

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