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First passage times of a jump diffusion process. (English) Zbl 1037.60073
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 $f_Y(y)=p\cdot\eta_1 e^{-\eta_1 y}\mathbf{1}_{\{y\geq 0\}}+ q\cdot\eta_2 e^{-| \eta_2| y}\mathbf{1}_{\{y< 0\}},$ $$p,q\geq 0$$, $$p+q=1$$, $$\eta_1,\eta_2>0$$. 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 $$\mathbf{E}[e^{-\alpha \tau_b}\mathbf{1}_{\{X_{\tau_b}-b>y\}}]$$ and $$\mathbf{E}[e^{-\alpha \tau_b}\mathbf{1}_{\{X_{\tau_b}-b=y\}}]$$, $$y\geq 0$$. 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.

##### MSC:
 60J75 Jump processes (MSC2010) 44A10 Laplace transform 60J27 Continuous-time Markov processes on discrete state spaces 60G51 Processes with independent increments; Lévy processes
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