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

Author

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  • Mario Abundo

    (Università “Tor Vergata”)

  • Sara Furia

    (Università “Tor Vergata”)

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).

Suggested Citation

  • Mario Abundo & Sara Furia, 2019. "Joint Distribution of First-Passage Time and First-Passage Area of Certain Lévy Processes," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1283-1302, December.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:4:d:10.1007_s11009-018-9677-5
    DOI: 10.1007/s11009-018-9677-5
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    References listed on IDEAS

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    1. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
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    3. Mario Abundo & Danilo Del Vescovo, 2017. "On the Joint Distribution of First-passage Time and First-passage Area of Drifted Brownian Motion," Methodology and Computing in Applied Probability, Springer, vol. 19(3), pages 985-996, September.
    4. Philippe Jorion, 1988. "On Jump Processes in the Foreign Exchange and Stock Markets," The Review of Financial Studies, Society for Financial Studies, vol. 1(4), pages 427-445.
    5. Frank B. Knight, 2000. "The moments of the area under reflected Brownian bridge conditional on its local time at zero," International Journal of Stochastic Analysis, Hindawi, vol. 13, pages 1-26, January.
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