Passage times for a spectrally negative Lévy process with applications to risk theory

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1 Hong Kong Baptist University HKBU Institutional Repository HKBU Staff Publication 25 Passage times for a spectrally negative Lévy process with applications to risk theory Sung Nok Chiu Hong Kong Baptist University, snchiu@hkbu.edu.hk Chuancun Yin Hong Kong Baptist University, ccyin@qfnu.edu.cn This document is the authors' final version of the published article. Link to published article: Recommended Citation Chiu, Sung Nok, and Chuancun Yin. "Passage times for a spectrally negative Lévy process with applications to risk theory." Bernoulli 3.11 (25): This Journal Article is brought to you for free and open access by HKBU Institutional Repository. It has been accepted for inclusion in HKBU Staff Publication by an authorized administrator of HKBU Institutional Repository. For more information, please contact repository@hkbu.edu.hk.

2 Bernoulli 11(3), 25, Passage times for a spectrally negative Lévy process with applications to risk theory SUNG NOK CHIU 1 and CHUANCUN YIN 2,1 1 Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong. snchiu@math.hkbu.edu.hk 2 Department of Mathematics, Qufu Normal University, Shandong , P. R. China. ccyin@qfnu.edu.cn The distributions of the last passage time at a given level and the joint distributions of the last passage time, the first passage time and their difference for a general spectrally negative process are derived in the form of Laplace transforms. The results are applied to risk theory. Keywords: first passage time; last passage time; spectrally negative Lévy process; risk theory 1. Introduction Let X ¼ (Ù, F, F t, X (t), Ł t, P x ) denote the canonical realization of a spectrally negative Lévy process with P x fx () ¼ xg ¼1. Thus X is a Hunt process with stationary independent increments specified by where E x is the expectation with respect to P x, and ł(æ) ¼ aæ þ 1 2 ó 2 Æ 2 þ E e ÆX (t) ¼ e tł(æ), Æ >, (1:1) ð fe Æx 1 Æx1(x. )gí(dx), (1:2) with a 2 R, ó 2 >, and í is a non-negative measure supported on (, ) satisfying ð í(dx), 1 and ð x 2 í(dx), 1: The measure í and the function ł are called the Lévy measure and the Lévy exponent of X, respectively. Ð Such a Lévy process has bounded variation if and only if ó ¼ and jxjí(dx), 1. In this case the Lévy exponent can be re-expressed as ł(æ) ¼ bæ þ ð (e Æx 1)í(dx), # 25 ISI/BS

3 512 S.N. Chiu and C. Yin where b ¼ a Ð xí(dx) is known as the drift coefficient. If ó 2., X is said to have a Gaussian component. As usual, we also assume that X is not a subordinator. If Y is a spectrally positive Lévy process, then X :¼ Y is a spectrally negative Lévy process. Thus the results for Y can be obtained from the corresponding results for X. The following facts on ł can be found in Bertoin (1996): ł is strictly increasing and continuous on [z, 1), where z is the largest real zero of ł; ł() ¼ ; ł(æ)!1 as Æ!1; ł9(þ) ¼ E X (1); and ł9(æ). for Æ.. The right inverse of ł is denoted by ł. For x 2 R, denote by ô x ¼ infft > :X (t). xg and T x ¼ infft > :X (t), xg the first passage time above and below the level x, respectively; denote by l x ¼ supft > :X (t), xg and T9 x ¼ infft > T x : X (t). xg the last passage time below the level x and the first passage time above the level x after T x, respectively, with the usual convention that inf ¼1 and sup ¼. Since E X (1) < leads to lim inf t!1 X (t) ¼ almost surely (Zolotarev 1964), which implies l x ¼1 almost surely for any fixed x, we assume that E X (1)., which is equivalent to P flim t!1 X (t) ¼ þ1g ¼ 1 (Bingham 1975). For ease of presentation, let Q(x) ¼ 1 Q(x) ¼ P finf t> X (t), xg for x >, Q() :¼ lim x# Q(x) and E X (1) ¼ m. Lemma 1.1 (Bertoin, 1996, p. 189, Theorem 1). For the spectrally negative Lévy process X, ô x with x > is a subordinator with the Laplace exponent ł, that is, E e Æô x ¼ e xł (Æ), Æ > : (1:3) Consequently ł is differentiable. Lemma 1.2 (Zolotarev 1964). For the spectrally negative Lévy process X, Q(x) can be determined by Æ e Æx Q(x)dx ¼ 1 Æm ł(æ) : (1:4) It follows from (1.4) that Q() ¼ 1 if ó 2. or Ð jxjí(dx) ¼1 (see also Rogozin 1965; Prabhu 197), and that Q() ¼ ð ð xí(dx) xí(dx) a if ó ¼ and Ð jxjí(dx), 1. It is known that t(c) ¼ expfcx(t) ł(c)tg is a martingale under P for any c such that ł(c) is finite (see Avram et al. 24). Let P (c) denote the probability measure on F defined by dp (c) ¼ t(c), dp F t

4 Passage times for a spectrally negative Lévy process 513 for all < t, 1. We have for the stopping time T x (x < ), P (c) (T x, 1) ¼ E f Tx (c)1(t x, 1)g ¼E f Tx (c)g: (1:5) Under the measure P (c), X remains within the class of spectrally negative processes and the Laplace exponent of X is given by (see Avram et al. 24) ł c (Ł) ¼ ł(ł þ c) ł(c), Ł > minf c, g: (1:6) Definition 1.1 (Avram et al. 24; Bertoin 1997). Consider (X, P (c) ). For p >, the p-scale function W ( c p) :(, 1)! [, 1) is the unique function whose restriction to (, 1) is continuous and has Laplace transform e Łx W ( c p) (x)dx ¼fł c (Ł) pg, Ł. ł c ( p), and is defined to be identically zero for x <. Further, for every x >, the mapping p! W ( c p) (x) can be extended to p 2 C by W ( c p) (x) ¼ P 1 k¼ pk W (kþ1) c (x), where W k c denotes the kth convolution power of the function W (). Moreover, let Z ( c p) (x) ¼ 1 þ p ð x c W ( c p) (y)dy: Lemma 1.3 (Emery 1973). For Æ > and â >, the joint Laplace transform of T y and X (T y ) is given by E x (e ÆT yþâx(t y ) ) ¼ e âx fz ( p) â where p ¼ Æ ł(â). (x y) W ( p) â (x y) p=ł( p)g, x > y, (1:7) Lemma 1.4 (Doney 1991). (i) For any bounded variation spectrally negative Lévy process we have E (e ŁT9 ) ¼ 1 fb(ł )9(Ł)g for Ł.. (ii) For any spectrally negative Lévy process with paths of unbounded variation, T9 ¼ almost surely. The first passage time for Lévy processes has been well studied; see Doney (1991), Emery (1973), Prabhu (197), Rogers (199; 2) and Zolotarev (1964). This paper is more concerned with the last passage time and its joint distribution with the duration between the first and the last visit at x and the duration of the first period spent below the level x. More precisely, the main purpose of this paper is to develop results related to the random variables l x ô x, l x T x, l x and T9 x T x for spectrally negative Lévy processes. Explicit solutions of the Laplace transforms of the distributions are obtained. The outline of the paper is as follows. Section 2 gives some primary results for a Lévy process with bounded variation. Section 3 considers the general case, which of course includes Lévy processes with bounded variation as a special case. We present the bounded variation case separately to demonstrate a different methodology. Applications in risk theory are discussed in Section 4. â

5 514 S.N. Chiu and C. Yin 2. Special case This section considers the spectrally negative Lévy process with bounded variation, that is, the case of X with ó ¼ and Ð jxjí(dx), 1. Under these conditions, X is essentially a positive drift minus a subordinator (see Bingham 1975, Proposition 6). To avoid ambiguity, we use a different notation for the Lévy component in this case: ö(æ) ¼ bæ þ ð (e Æx 1)í(dx), where b ¼ a Ð xí(dx). We first give the Laplace transform of l x ô x, the duration between the first and the last visit at x, then deduce the Laplace transform of l x, and finally find the joint Laplace transform of T x, l x T x and T9 x T x. Theorem 2.1. For any y < x and Æ., we have E y e Æ(l x ô x ) ¼ m(ö )9(Æ): (2:1) Proof. We first consider y ¼. For x >, let S n (x) denote the time elapsed between the nth and the (n þ 1)th visit at x. Since x is both non-polar and non-regular, it is clear that l x ô x ¼ S 1 (x) þ...þ S N (x) on fø 2 Ù : S i (x)(ø), 1g, where N ¼ maxfk > 1:S k (x), 1g. The Markov property of the process implies that the random variables S i (x) are independent and identically distributed. Moreover, they are independent of N, which has a geometric distribution P (N ¼ k) ¼ Q()Q k (), k ¼, 1, 2,...: Note that P (l x ô x, 1) ¼ 1 and P fs 1 (x) > T x g¼1. Thus, for Æ., we have E e Æ(l x ô x ) ¼ E (e Æ(l x ô x ) jl x ô x, 1) ¼ X1 k¼ fe (e ÆS 1(x) js 1 (x), 1)g k P (N ¼ k) Q() ¼ 1 Q()E (e ÆS 1(x) js 1 (x), 1) : (2:2) The distribution of S 1 (x) does not depend on x, since the process X has stationary independent increments. It follows from Lemma 1.4 that which implies that E fe ÆS 1() 1(T, 1)g ¼E e ÆS 1() ¼ 1 fb(ö )9(Æ)g, E (e ÆS 1() js 1 (), 1) ¼ E (e ÆS 1() jt, 1) ¼ [1 fb(ö )9(Æ)g ]=Q(): Substituting this into (2.2) and noting that Q()b ¼ m, we obtain

6 Passage times for a spectrally negative Lévy process 515 E e Æ(l x ô x ) ¼ m(ö )9(Æ): (2:3) Since X has stationary independent increments, for x, y 2 R and y < x, and the result follows. E y e Æ(l x ô x ) ¼ E e Æ(l x y ô x y ), Theorem 2.2. For Æ. and x, y 2 R, 8 m(ö )9(Æ)e (x y)ö (Æ), y, x, (2:4) >< E y fe Æl x m(ö )9(Æ) m, y ¼ x, (2:5) 1(l x. )g ¼ b m(ö )9(Æ)P (ö (Æ)) >: (T x y, 1), y. x: (2:6) e (x y)ö (Æ) Proof. Since P fx (ô x ) ¼ xg ¼1, the strong Markov property of X at time ô x implies that, for x, s, t., P (l x ô x, t, ô x, s) ¼ E fp X (ôx )(l x, t)1(ô x, s)g ¼ P x (l x, t)p (ô x, s): (2:7) The process X has only negative jumps and P flim t!1 X (t) ¼ þ1g ¼ 1, thus P (ô x, 1) ¼ 1 for x.. Consequently, letting s!1 in (2.7) yields P (l x ô x, t) ¼ P x (l x, t), and hence (2.7) shows that l x ô x and ô x are independent with respect to P. As a result, by (2.3) and Lemma 1.1, we have, for x., Letting x # yields E e Æl x ¼ E e Æ(l x ô x ) E e Æô x ¼ m(ö )9(Æ)e xö (Æ) : E e Æl ¼ m(ö )9(Æ), and due to the stationarity and independence of increments, for any x 2 R, wehave E x e Æl x ¼ E e Æl ¼ m(ö )9(Æ), (2:8) which is equivalent to (2.5). For any y, x, P y (l x. ) ¼ P y (ô x, 1) ¼ 1 and P y (X (ô x ) ¼ x) ¼ 1. By the strong Markov property of X at time ô x, E y fe Æl x 1(l x. )g ¼E y [e Æô x E X (ôx )fe Æl x 1(l x > )g1(ô x, 1)] ¼ E e Æô x y E x e Æl x : (2:9) Equation (2.4) follows from (2.8), (2.9) and Lemma 1.1. Since X has stationary independent increments and is strong Markov, for any y. x, we have

7 516 S.N. Chiu and C. Yin E y fe Æl x 1(l x. )g ¼E fe Æl x y 1(l x y. )g ¼ E [e ÆT x y E X (Tx y )fe Æl x y 1(l x y > )g1(t x y, 1)] ¼ ð x y e Æt E z (e Æl x y )P ft x y 2 dt, X (T x y ) 2 dz, T x y, 1g, so that equation (2.6) follows from (2.4) and (1.5). h Theorem 2.3. Suppose that Æ, â, ä. and x, y 2 R. Ify. x, then E y fe ÆT x â(l x T x ) ä(t9 x T x ) 1(T x, 1)g ¼ m(ö )9(â)e (âþä) 1(Æ, xö â, ä, x, y), (2:1) where ( 1(Æ, â, ä, x, y) ¼ e ö (âþä) y Z ( p 1) ö (âþä) (y x) W ( p1) ö (âþä) (y x) p ) 1 ö ö (âþä) ( p, 1) in which p 1 ¼ Æ â ä. Proof. By the strong Markov property of X and (2.4), E y fe ÆT x â(l x T x ) ä(t9 x T x ) 1(T x, 1)g ¼ E y fe ÆT x E X (Tx )e âl x ät9 x 1(T x, 1)g ¼ E y fe ÆT x E X (Tx )fe (âþä)t9 x E x e âl x g1(t x, 1)g ¼ m(ö )9(â)E y fe ÆT x ö (âþä)fx X (T x )g 1(T x, 1)g ¼ m(ö )9(â)e xö (âþä) E y fe ÆT xþö (âþä)x (T x ) g, and the result follows from Lemma 1.3. h The following corollary generalizes the result in Doney (1991, equation (2.19)). Corollary 2.1. For any Æ, â., we have E fe ÆT â(l T ) ä(t9 T ) 1(T, 1)g ¼ m(ö Æ â ä )9(â) 1 bfö (Æ) ö : (â þ ä)g Proof. From the proof of Theorem 2.3 one finds

8 Passage times for a spectrally negative Lévy process 517 Since E fe ÆT â(l T ) ä(t9 T ) 1(T, 1)g ¼ m(ö )9(â)E fe ÆT þö (âþä)x(t ) g: E e ÆT þö (â)x (T ) ¼ 1 Æ â bfö (Æ) ö (â)g (Prabhu 197), the result follows. h 3. General case In the previous section, we computed the Laplace transforms of l x, l x ô x and the joint Laplace transform of T x, l x T x and T9 x T x for a spectrally negative Lévy process X with bounded variation. In this section, we consider the general Lévy process defined by (1.1) and (1.2), which includes the bounded variation case in Section 2. The argument used in Section 2, however, cannot be applied to the case where ó 6¼ or Ð jxjí(dx) ¼1, because then X has unbounded variation and hence S i (x) ¼ almost surely (see Lemma 1.4(ii)). Theorem 3.1. For Æ. and x, y 2 R, we have 8 m(ł )9(Æ)e (x y)ł (Æ), y, x, >< E y fe Æl x m(ł 1(l x. )g ¼ )9(Æ) Q(), y ¼ x, where Q() ¼ if ó 2. or jxjí(dx), 1. Ð m(ł )9(Æ)P (ł (Æ)) >: (T x y, 1), y. x, e (x y)ł (Æ) (3:1) (3:2) (3:3) Ð jxjí(dx) ¼1, and Q() ¼ m=b if ó 2 ¼ and Proof. Since fl, tg ¼fX (t)., inf s>t X (s). g, using the Markov property and right continuity, we have E e Æl ¼ Æ ¼ Æ e Æt P (l, t)dt e Æt Q(x)P fx (t) 2 dxgdt: Applying the identity tp fô x 2 dtgdx ¼ xp fx (t) 2 dxgdt (Bertoin 1996, p. 19, Corollary 3) to this leads to

9 518 S.N. Chiu and C. Yin E e Æl ¼ Æ x Q(x) te Æt P (ô x 2 dt) dx ¼ Æ ¼ Æ ¼ Æ(ł )9(Æ) ¼ m(ł )9(Æ), x Q(x) d dæ x Q(x) d (Æ) dæ fe xł g dx Q(x)e xł (Æ) dx e Æt P (ô x 2 dt) dx where (1.3) and (1.4) have been used in the third and last step, respectively. The process X has stationary independent increments, and thus, for any x 2 R, we have E x e Æl x ¼ E e Æl ¼ m(ł )9(Æ), (3:4) and (3.2) follows. Since (2.9) remains true for processes of unbounded variation, it, together with (1.3) and (3.4), yields (3.1). For any y. x, using (3.4) and an argument similar to the proof of (2.6), we obtain E y fe Æl x 1(l x. )g ¼ ð x y and (3.3) follows. m(ł )9(Æ) e Æt e (x y z)ł (Æ) P ft x y 2 dt, X (T x y ) 2 dz, T x y, 1g, h Remark 3.1. By using Lemma 1.2 or Kyprianou and Palmowski (25, Theorem 4), we have P (ł (Æ)) (T x y, 1) ¼ 1 ł9 ł (Æ)(þ)W () (y x), ł (Æ) and replacing ł by ö yields an expression for the probability P (ö (Æ)) (T x y, 1) in the special case presented in Section 2. Theorem 3.2. Suppose that Æ, â, ä. and x, y 2 R. Ify, x, then and if y. x, then where E y fe Æô x â(l x ô x ) 1(ô x, 1)g ¼m(ł )9(â)e (x y)ł (Æ), (3:5) E y fe ÆT x â(l x T x )þäx (T x ) 1(T x, 1)g ¼m(ł )9(â)e xł (â) 2(Æ, â, ä, x, y), (3:6)

10 Passage times for a spectrally negative Lévy process 519 2(Æ, â, ä, x, y) ¼ e k y Z ( p 2) k (y x) W ( p 2) k (y x) p 2 ł k ( p, 2) with p 2 ¼ Æ ł(ä þ ł (â)) and k ¼ ä þ ł (â). Proof. For any y, x 2 R such that y, x, P y (ô x, 1) ¼ 1 and P y (X (ô x ) ¼ x) ¼ 1. The strong Markov property of X at time ô x, the stationarity and independence of increments and equation (3.2) imply that, for Æ, â., E y fe Æô x â(l x ô x ) 1(ô x, 1)g ¼E y [e Æô x E X (ôx )fe âl x 1(l x > )g1(ô x, 1)] ¼ E y (e Æô x) E x fe âl x 1(l x > )g ¼ E (e Æô x y )E x (e âl x ): (3:7) Equation (3.5) follows from (1.3), (3.4) and (3.7). For any y, x 2 R such that y. x and Æ, â, ä., from the strong Markov property of X and (3.1), E y fe ÆT x â(l x T x )þäx(t x ) 1(T x, 1)g ¼ E y [e ÆT xþäx (T x ) E X (Tx )fe âl x 1(l x > )g1(t x, 1)] ¼ ð x e Ætþäz E z fe âl x 1(l x > )gp y ft x 2 dt, X (T x ) 2 dz, T x, 1g ¼ m(ł )9(â) and the result follows. ð x e Ætþäz e (x z)ł (â) P y ft x 2 dt, X (T x ) 2 dz, T x, 1g, h 4. Applications to risk theory Consider the surplus R(t) that is the classical risk process perturbed by diffusion: R(t) ¼ u þ ct þ ó B t XN t i¼1 Z i, t >, (4:1) where u > is the initial surplus, ó a non-negative constant, c the positive constant premium income rate, fb t, t > g the standard Brownian motion, fn t, t > g a Poisson process with intensity º., and fz k, k > 1g a sequence of non-negative independent and identically distributed claim amounts, such that fb t, t > g, fn t, t > g and fz k, k > 1g are independent. Denote by P and ì the distribution function and the mean, respectively, of the claim sizes Z k, with the condition that P() ¼. Assume that the safety loading c ºì

11 52 S.N. Chiu and C. Yin is positive, so that lim t!1 R(t) ¼1 almost surely if the process continues even when the surplus is negative. For simplicity, P is assumed to possess a density p. The diffusion term in (4.1) contributes an additional uncertainty of the premium income or the aggregate claims to the surplus. This model was first introduced and studied by Gerber (197). When ó ¼, (4.1) is called the classical risk process. We refer to Embrechts et al. (1997), Rolski et al. (1999) and Asmussen (2) for a complete presentation of risk theory. Denote the Laplace transform of p by ^p(æ) ¼ Ð 1 e Æx p(x)dx. ThenE e ÆfR(t) ug ¼ e tî(æ), where î(æ) ¼ cæ þ 1 2 ó 2 Æ 2 þ ºf ^p(æ) 1g. Obviously fr(t) u, t > g is a spectrally negative Lévy process with E fr(1) ug ¼c ºì. and initial value zero. Let T denote the time of ruin and Ø(u) ¼ P u (T, 1) the ultimate ruin probability. It is well known that Ø() is 1 if ó 6¼, and is ºì=c otherwise. If we use the notation in Section 1, we have T ¼ T and Ø(u) ¼ Q(u). One central topic of risk theory is to find the probability of ruin. Recently there has been growing interest in the distributions of some other random variables related to the surplus process. These random variables include the time when the surplus reaches some level for the first time (Gerber 199; Picard and Lefèvre 1994; Zhang and Wu 22), the time when the surplus crosses some level for the last time (Gerber 199), the duration of negative surplus (Egídio dos Reis 1993; Zhang and Wu 22) and the recovery time from negative surplus (Dickson and Egídio dos Reis 1997; Egídio dos Reis 2; Yang and Zhang 21). A scenario where these random variables make sense is as follows. Suppose that the portfolio under consideration is one of many belonging to a company so that it has other funds available to support negative surpluses for a while, in the hope that the portfolio will recover in the future. In such a situation perhaps a more interesting question is to find the distributions of the first and last passage times and of their difference for the surplus process at a given level. Gerber (199) considered this question for the classical risk model and obtained explicit results for their Laplace transforms and their moments. However, when we consider the time of last passage at level x and the duration between successive visits at x for the classical risk process perturbed by diffusion, the method of Gerber (199) is not applicable because the infinite oscillation of R(t) due to the Brownian motion B t yields an arbitrarily small duration. Nevertheless, the general results of spectrally negative Lévy processes in Sections 2 and 3 can be used directly. Using the same notation ô x, l x, T x and T9 x as before and applying the results in previous sections to the special Lévy process defined by equation (4.1), we obtain the following results. Theorem 4.1. For any y < x and Æ., we have E y e Æ(l x ô x ) ¼ c ºì c þ º ^p9(î (Æ)) : Theorem 4.2. For Æ. and x, y 2 R,

12 Passage times for a spectrally negative Lévy process >< E y fe Æl x 1(l x. )g ¼ c ºì c þ î (Æ)ó 2 þ º ^p9(î (Æ)) e (x y)î (Æ), y, x, º(c ºì) ^p9(î (Æ)) cfc þ º ^p9(î, y ¼ x, ó ¼, (Æ))g c ºì c þ î (Æ)ó 2 þ º ^p9(î, y ¼ x, ó 6¼, (Æ)) (c ºì)P (î (Æ)) (T x y, 1) >: c þ î (Æ)ó 2 þ º ^p9(î (Æ)) e (x y)î (Æ), y. x: Theorem 4.3. For any Æ, â, ä. and x, y 2 R, ify, x, then and if y. x, then E y fe Æô x â(l x ô x ) 1(ô x, 1)g ¼ (c ºì)e (x y)î (Æ) c þ î (â)ó 2 þ º ^p9(î (â)), E y fe ÆT x â(l x T x )þär(t x ) 1(T x, 1)g ¼ (c ºì)E e ÆT x yþfäþî (â)gr(t x y ) fc þ î (â)ó 2 þ º ^p9(î (â))ge xî (â), where E e ÆT x yþfäþî (â)gr(t x y ) can be computed by using Lemma 1.3. Theorem 4.4. Suppose that Æ, â, ä. and ó ¼. For any x, y 2 R, ify. x, then E y f ÆT x â(l x T x ) ä(t9 x T x ) 1(T x, 1)g c ºì ¼ c þ º ^p9(î e xî1 (âþä) E fe ÆT x yþî 1 (âþä)r(t x y) g, 1 (â)) where î 1 (Æ) ¼ cæ þ ºf ^p(æ) 1g and E e ÆT x yþî 1 (âþä)r(t x y) can be computed by using Lemma 1.3. Remark 4.1. Taking c ¼ 1, y ¼ and Æ ¼ s in the case of ó ¼ in Theorem 4.2, we get Gerber (199, p. 118, equation (43)). Acknowledgements We thank two anonymous referees for their detailed comments and valuable suggestions, which helped improve the presentation. This research was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKBU 248/2P) and the Natural Science Foundation of China (No ).

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