Methodology and Computing in Applied Probability

, Volume 21, Issue 4, pp 1283–1302 | Cite as

Joint Distribution of First-Passage Time and First-Passage Area of Certain Lévy Processes

  • Mario AbundoEmail author
  • Sara Furia


Let be X(t) = xμt + σBtNt a Lévy process starting from x > 0, where μ ≥ 0, σ ≥ 0, Bt is a standard BM, and Nt is a homogeneous Poisson process with intensity 𝜃 > 0, starting from zero. We study the joint distribution of the first-passage time below zero, τ(x), and the first-passage area, A(x), swept out by X till the time τ(x). In particular, we establish differential-difference equations with outer conditions for the Laplace transforms of τ(x) and A(x), and for their joint moments. In a special case (μ = σ = 0), we show an algorithm to find recursively the moments E[τ(x)mA(x)n], for any integers m and n; moreover, we obtain the expected value of the time average of X till the time τ(x).


First-passage time First-passage area Jump-diffusion Lévy process 

Mathematics Subject Classification (2010)

60J60 60H05 60H10 


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We would like to express particular thanks to the anonymous reviewer for his/her useful comments, leading to improved presentation.

This research was funded by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità “Tor Vergata”RomeItaly

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