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

Mario Lefebvre () and Ibrahim Elmojtaba
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Mario Lefebvre: Department of Mathematics and Industrial Engineering, Polytechnique Montréal, Station Centre-Ville, P.O. Box 6079, Montréal, QC H3C 3A7, Canada
Ibrahim Elmojtaba: Department of Mathematics, College of Science, Sultan Qaboos University, Al-khod, P.O. Box 36, Muscat 123, Oman

Mathematics, 2025, vol. 13, issue 10, 1-14

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 ( − 2 x , 0 ) when x is positive and on the interval ( 0 , − 2 x ) 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.

Keywords: Brownian motion; Poisson process; first-passage time; Kolmogorov backward equation; integro-differential equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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