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

Mario Abundo ()
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Mario Abundo: Università Tor Vergata

Methodology and Computing in Applied Probability, 2010, vol. 12, issue 3, 473-490

Abstract: 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.

Keywords: First-passage time; Diffusion; Poisson process; Primary 60J60; Secondary 60H05, 60H10 (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/s11009-008-9115-1

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