Abstract
Let be X(t) = x − μt + σBt − Nt 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).
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Acknowledgments
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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Abundo, M., Furia, S. Joint Distribution of First-Passage Time and First-Passage Area of Certain Lévy Processes. Methodol Comput Appl Probab 21, 1283–1302 (2019). https://doi.org/10.1007/s11009-018-9677-5
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DOI: https://doi.org/10.1007/s11009-018-9677-5