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On the Joint Distribution of First-passage Time and First-passage Area of Drifted Brownian Motion

Author

Listed:
  • Mario Abundo

    (Università “Tor Vergata”, via della Ricerca Scientifica)

  • Danilo Del Vescovo

    (Università “Tor Vergata”, via della Ricerca Scientifica)

Abstract

For drifted Brownian motion X(t) = x-µ t + B t (µ > 0) starting from x > 0, we study the joint distribution of the first-passage time below zero ,t(x), and the first-passage area ,A(x), swept out by X till the time t(x). In particular, we establish differential equations with boundary conditions for the joint moments E[t(x) m A(x) n ], and we present an algorithm to find recursively them, for any m and n. Finally, the expected value of the time average of X till the time t(x) is obtained.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:19:y:2017:i:3:d:10.1007_s11009-017-9546-7
    DOI: 10.1007/s11009-017-9546-7
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    References listed on IDEAS

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    1. Mario Abundo, 2013. "On the First-Passage Area of a One-Dimensional Jump-Diffusion Process," Methodology and Computing in Applied Probability, Springer, vol. 15(1), pages 85-103, March.
    2. 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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    Cited by:

    1. 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.
    2. Abdolsaeed Toomaj & Antonio Di Crescenzo, 2020. "Connections between Weighted Generalized Cumulative Residual Entropy and Variance," Mathematics, MDPI, vol. 8(7), pages 1-27, July.

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    1. Mario Abundo, 2013. "On the First-Passage Area of a One-Dimensional Jump-Diffusion Process," Methodology and Computing in Applied Probability, Springer, vol. 15(1), pages 85-103, March.
    2. 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.

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