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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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