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On the First Hitting Time of a One-dimensional Diffusion and a Compound Poisson Process

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  • Mario Abundo

    (Università Tor Vergata)

Abstract

It is studied the first-passage time (FPT) of a time homogeneous one-dimensional diffusion, driven by the stochastic differential equation dX(t) = μ(X(t))dt + σ(X(t)) dB t , X(0) = x 0, through b + Y(t), where b > x 0 and Y(t) is a compound Poisson process with rate λ > 0 starting at 0, which is independent of the Brownian motion B t . In particular, the FPT density is investigated, generalizing a previous result, already known in the case when X(t) = μt + B t , for which the FPT density is the solution of a certain integral equation. A numerical method is shown to calculate approximately the FPT density; some examples and numerical results are also reported.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:12:y:2010:i:3:d:10.1007_s11009-008-9115-1
    DOI: 10.1007/s11009-008-9115-1
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    References listed on IDEAS

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    1. Abundo, Mario, 2002. "Some conditional crossing results of Brownian motion over a piecewise-linear boundary," Statistics & Probability Letters, Elsevier, vol. 58(2), pages 131-145, June.
    2. Liqun Wang & Klaus Pötzelberger, 2007. "Crossing Probabilities for Diffusion Processes with Piecewise Continuous Boundaries," Methodology and Computing in Applied Probability, Springer, vol. 9(1), pages 21-40, March.
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    Cited by:

    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. 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.
    3. 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.
    4. 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.
    5. 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.

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