First-Hitting Problems for Jump-Diffusion Processes with State-Dependent Uniform Jumps
Mario Lefebvre, Ibrahim Elmojtaba
Source abstract
Let {X(t),t≥0} be a one-dimensional jump-diffusion process whose continuous part is either a Wiener, Ornstein–Uhlenbeck, or generalized Bessel process. The process starts at X(0)=x∈[−d,d]. Let τ(x) be the first time that X(t)=0 or |X(t)|=d. The jumps follow a uniform distribution on the interval (−2x,0) when x is positive and on the interval (0,−2x) when x is negative. We are interested in the moment-generating function of τ(x), its mean, and the probability that X[τ(x)]=0. We must solve integro-differential equations, subject to the appropriate boundary conditions. Analytical and numerical results are presented.
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