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Zbl 1037.60073
Kou, S. G.; Wang, Hui
First passage times of a jump diffusion process.
(English)
[J] Adv. Appl. Probab. 35, No. 2, 504-531 (2003). ISSN 0001-8678

The jump diffusion process $$X_t=\sigma W_t+\mu t+\sum_{i=1}^{N_t} Y_i, \quad X_0=0,$$ is considered where $W$ is a standard Brownian motion, $N$ is a Poisson process with rate $\lambda$, $\sigma$ and $\mu$ are positive constants. The i.i.d. r.v.s $(Y_i)_{i\geq 1}$ have a double exponential distribution given by the density $$f_Y(y)=p\cdot\eta_1 e^{-\eta_1 y}\bold{1}_{\{y\geq 0\}}+ q\cdot\eta_2 e^{-\vert \eta_2\vert y}\bold{1}_{\{y< 0\}},$$ $p,q\geq 0$, $p+q=1$, $\eta_1,\eta_2>0$. The authors derive the closed-form formulae for the Laplace transform of the first passage time $\tau_b=\inf\{t\geq 0: X_t\geq b\}$, $b>0$, as well as for $\bold{E}[e^{-\alpha \tau_b}\bold{1}_{\{X_{\tau_b}-b>y\}}]$ and $\bold{E}[e^{-\alpha \tau_b}\bold{1}_{\{X_{\tau_b}-b=y\}}]$, $y\geq 0$. Connections with renewal-type equations are discussed. The Laplace transform of the joint law of $X_t$ and $\max_{0\leq s\leq t}X_s$ is obtained in terms of special functions. The Gaver-Stehfest algorithm for the numerical inversion of Laplace transforms is tested.
[Ilya Pavlyukevitch (Berlin)]
MSC 2000:
*60J75 Jump processes
44A10 Laplace transform
60J27 Markov chains with continuous parameter
60G51 Processes with independent increments

Keywords: jump diffusion process; Lévy process; independent increments; Laplace transform; renewal theory; Gaver-Stehfest algorithm; Poisson process; first passage time; running maxima; double exponential distribution

Cited in: Zbl 1233.91286

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