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

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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.

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  1. Dipartimento di Matematica, Università Tor Vergata, Via della Ricerca Scientifica, 00133, Roma, Italy

    Mario Abundo

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  1. Mario Abundo
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Correspondence to Mario Abundo.

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Abundo, M. On the First Hitting Time of a One-dimensional Diffusion and a Compound Poisson Process. Methodol Comput Appl Probab 12, 473–490 (2010). https://doi.org/10.1007/s11009-008-9115-1

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  • DOI: https://doi.org/10.1007/s11009-008-9115-1

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