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First passage times of a jump diffusion process. (English) Zbl 1037.60073

The jump diffusion process

${X}_{t}=\sigma {W}_{t}+\mu t+\sum _{i=1}^{{N}_{t}}{Y}_{i},\phantom{\rule{1.em}{0ex}}{X}_{0}=0,$

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 ${\left({Y}_{i}\right)}_{i\ge 1}$ have a double exponential distribution given by the density

${f}_{Y}\left(y\right)=p·{\eta }_{1}{e}^{-{\eta }_{1}y}{\mathbf{1}}_{\left\{y\ge 0\right\}}+q·{\eta }_{2}{e}^{-|{\eta }_{2}|y}{\mathbf{1}}_{\left\{y<0\right\}},$

$p,q\ge 0$, $p+q=1$, ${\eta }_{1},{\eta }_{2}>0$. The authors derive the closed-form formulae for the Laplace transform of the first passage time ${\tau }_{b}=inf\left\{t\ge 0:{X}_{t}\ge b\right\}$, $b>0$, as well as for $𝐄\left[{e}^{-\alpha {\tau }_{b}}{\mathbf{1}}_{\left\{{X}_{{\tau }_{b}}-b>y\right\}}\right]$ and $𝐄\left[{e}^{-\alpha {\tau }_{b}}{\mathbf{1}}_{\left\{{X}_{{\tau }_{b}}-b=y\right\}}\right]$, $y\ge 0$. Connections with renewal-type equations are discussed. The Laplace transform of the joint law of ${X}_{t}$ and ${max}_{0\le s\le t}{X}_{s}$ is obtained in terms of special functions. The Gaver-Stehfest algorithm for the numerical inversion of Laplace transforms is tested.

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