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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Mario Lefebvre & Ibrahim Elmojtaba, 2025. "First-Hitting Problems for Jump-Diffusion Processes with State-Dependent Uniform Jumps," Mathematics, MDPI, vol. 13(10), pages 1-14, May.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:10:p:1629-:d:1657001
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