First-Hitting Problems for Jump-Diffusion Processes with State-Dependent Uniform Jumps

{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.

Uncontrolled Keywords

• Brownian motion
• Poisson process
• first-passage time
• Kolmogorov backward equation
• integro-differential equation