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Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes



PDF/Soon/ps0709.pdf.url ESAIM: PS ESAIM: Probability and Statistics Janvier 2008, Vol. 12, p. 58–93 DOI: 10.1051/ps:2007034 ASYMPTOTIC BEHAVIOR OF THE HITTING TIME, OVERSHOOT AND UNDERSHOOT FOR SOME LE´VY PROCESSES Bernard Roynette1, Pierre Vallois1 and Agne`s Volpi1, 2 Abstract. Let (Xt, t ≥ 0) be a Le´vy process started at 0, with Le´vy measure ν. We consider the first passage time Tx of (Xt, t ≥ 0) to level x > 0, and Kx := XTx − x the overshoot and Lx := x−XTx− the undershoot. We first prove that the Laplace transform of the random triple (Tx,Kx, Lx) satisfies some kind of integral equation. Second, assuming that ν admits exponential moments, we show that (T˜x, Kx, Lx) converges in distribution as x→∞, where T˜x denotes a suitable renormalization of Tx. Mathematics Subject Classification. 60E10, 60F05, 60G17, 60G40, 60G51, 60J65, 60J75, 60J80, 60K05. Received February 16, 2007. Introduction 1. Let (Xt, t ≥ 0) be a Le´vy process, which is right continuous with left limits and starts at 0. Let Xt = σBt − c0t + Jt t ≥ 0 , (0.1) be the canonical decomposition, when c0 ∈ R, σ > 0 and (Bt , t≥ 0) is a one-dimensional Brownian motion started at 0. (Jt , t≥ 0) is a pure jump Le´vy process which is independent from (Bt , t≥ 0). In addition J0 = 0. Recall (see for instance Th. 2.1, Chap. 2, [13]) that (Jt , t≥ 0) is the sum of a compound Poisson process and a square integrable martingale whose jumps are of magnitude less than 1. For simplicity, we may assume that σ = 1. 2. We are interested in the first hitting time of level x > 0 Tx := inf {t ≥ 0; Xt > x} . (0.2) Keywords and phrases. Le´vy processes, ruin problem, hitting time, overshoot, undershoot, asymptotic estimates, functional equation. 1 De´partement de mathe´matiques, Institut E´lie Cartan,Universite´ Henri Poincare´, BP 239, 54506 Vandœuvre-le`s-Nancy cedex, France; [roynette;vallois] 2 ESSTIN, 2 rue Jean Lamour, Parc Robert Bentz, 54500 Vandœuvre-le`s-Nancy, France; [email protected]

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