IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v13y2025i10p1629-d1657001.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/13/10/1629/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/13/10/1629/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Mario Abundo, 2013. "On the First-Passage Area of a One-Dimensional Jump-Diffusion Process," Methodology and Computing in Applied Probability, Springer, vol. 15(1), pages 85-103, March.
    2. Lefebvre, Mario, 2024. "Exact solution to a first-passage problem for an Ornstein–Uhlenbeck process with jumps and its integral," Statistics & Probability Letters, Elsevier, vol. 205(C).
    3. Jun Peng & Zaiming Liu, 2011. "First Passage Time Moments of Jump-Diffusions with Markovian Switching," International Journal of Stochastic Analysis, Hindawi, vol. 2011, pages 1-11, March.
    4. Mario Abundo, 2023. "The First-Passage Area of Wiener Process with Stochastic Resetting," Methodology and Computing in Applied Probability, Springer, vol. 25(4), pages 1-25, December.
    5. Mario Abundo, 2010. "On the First Hitting Time of a One-dimensional Diffusion and a Compound Poisson Process," Methodology and Computing in Applied Probability, Springer, vol. 12(3), pages 473-490, September.
    6. Zhou, Jiang & Wu, Lan, 2015. "Occupation times of refracted double exponential jump diffusion processes," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 218-227.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Zhang, Jian-Xun & Hu, Chang-Hua & He, Xiao & Si, Xiao-Sheng & Liu, Yang & Zhou, Dong-Hua, 2017. "Lifetime prognostics for deteriorating systems with time-varying random jumps," Reliability Engineering and System Safety, Elsevier, vol. 167(C), pages 338-350.
    2. Mario Lefebvre, 2021. "Moments of First-Passage Places for Jump-Diffusion Processes," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 245-253, February.
    3. Mario Abundo, 2023. "The First-Passage Area of Wiener Process with Stochastic Resetting," Methodology and Computing in Applied Probability, Springer, vol. 25(4), pages 1-25, December.
    4. Mario Abundo, 2013. "On the First-Passage Area of a One-Dimensional Jump-Diffusion Process," Methodology and Computing in Applied Probability, Springer, vol. 15(1), pages 85-103, March.
    5. Tung-Lung Wu, 2020. "Boundary Crossing Probabilities of Jump Diffusion Processes to Time-Dependent Boundaries," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 13-24, March.
    6. Song, Shiyu & Wang, Suxin & Wang, Yongjin, 2016. "On some properties of reflected skew Brownian motions and applications to dispersion in heterogeneous media," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 456(C), pages 90-105.
    7. Mario Abundo & Danilo Del Vescovo, 2017. "On the Joint Distribution of First-passage Time and First-passage Area of Drifted Brownian Motion," Methodology and Computing in Applied Probability, Springer, vol. 19(3), pages 985-996, September.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:13:y:2025:i:10:p:1629-:d:1657001. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.