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

Mario Abundo () and Sara Furia
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Mario Abundo: Università “Tor Vergata”
Sara Furia: Università “Tor Vergata”

Methodology and Computing in Applied Probability, 2019, vol. 21, issue 4, 1283-1302

Abstract: 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).

Keywords: First-passage time; First-passage area; Jump-diffusion; Lévy process; 60J60; 60H05; 60H10 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s11009-018-9677-5

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