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

The jump diffusion process

X t =σW t +μt+ i=1 N t Y i ,X 0 =0,

is considered where W is a standard Brownian motion, N is a Poisson process with rate λ, σ and μ are positive constants. The i.i.d. r.v.s (Y i ) i1 have a double exponential distribution given by the density

f Y (y)=p·η 1 e -η 1 y 1 {y0} +q·η 2 e -|η 2 |y 1 {y<0} ,

p,q0, p+q=1, η 1 ,η 2 >0. The authors derive the closed-form formulae for the Laplace transform of the first passage time τ b =inf{t0:X t b}, b>0, as well as for 𝐄[e -ατ b 1 {X τ b -b>y} ] and 𝐄[e -ατ b 1 {X τ b -b=y} ], y0. Connections with renewal-type equations are discussed. The Laplace transform of the joint law of X t and max 0st X s is obtained in terms of special functions. The Gaver-Stehfest algorithm for the numerical inversion of Laplace transforms is tested.


MSC:
60J75Jump processes
44A10Laplace transform
60J27Continuous-time Markov processes on discrete state spaces
60G51Processes with independent increments; Lévy processes