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

Lefebvre, Mario; Elmojtaba, Ibrahim

doi:10.3390/math13101629

Open AccessArticle

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

by

Mario Lefebvre

^1,*

and

Ibrahim Elmojtaba

²

¹

Department of Mathematics and Industrial Engineering, Polytechnique Montréal, Station Centre-Ville, P.O. Box 6079, Montréal, QC H3C 3A7, Canada

²

Department of Mathematics, College of Science, Sultan Qaboos University, Al-khod, P.O. Box 36, Muscat 123, Oman

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(10), 1629; https://doi.org/10.3390/math13101629

Submission received: 13 March 2025 / Revised: 24 April 2025 / Accepted: 13 May 2025 / Published: 16 May 2025

(This article belongs to the Section D1: Probability and Statistics)

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Abstract

Let

{X (t), t \geq 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 \in [- 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

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MDPI and ACS Style

Lefebvre, M.; Elmojtaba, I. First-Hitting Problems for Jump-Diffusion Processes with State-Dependent Uniform Jumps. Mathematics 2025, 13, 1629. https://doi.org/10.3390/math13101629

AMA Style

Lefebvre M, Elmojtaba I. First-Hitting Problems for Jump-Diffusion Processes with State-Dependent Uniform Jumps. Mathematics. 2025; 13(10):1629. https://doi.org/10.3390/math13101629

Chicago/Turabian Style

Lefebvre, Mario, and Ibrahim Elmojtaba. 2025. "First-Hitting Problems for Jump-Diffusion Processes with State-Dependent Uniform Jumps" Mathematics 13, no. 10: 1629. https://doi.org/10.3390/math13101629

APA Style

Lefebvre, M., & Elmojtaba, I. (2025). First-Hitting Problems for Jump-Diffusion Processes with State-Dependent Uniform Jumps. Mathematics, 13(10), 1629. https://doi.org/10.3390/math13101629

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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

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