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How the sojourn time distributions of Brownian motion are affected by different forms of conditioning

Author

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  • Beghin, L.
  • Nikitin, Y.
  • Orsingher, E.

Abstract

In this paper we study the distribution of the sojourn time [Gamma]t=meas{s 0}, where B(t), t>0 is a Brownian motion (with or without drift), under different conditions at an intermediate time u[less-than-or-equals, slant]t (and possibly with an additional condition at time t). We obtain different forms of the arc-sine law, which display a "bell-shaped" structure (instead of the usual "U-shaped" classical density) when (B(u)=0) is assumed. When the conditions (B(u)=0,B(t)[greater, less]0) are taken into account, an asymmetrical bell-shaped density is obtained.

Suggested Citation

  • Beghin, L. & Nikitin, Y. & Orsingher, E., 2003. "How the sojourn time distributions of Brownian motion are affected by different forms of conditioning," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 291-302, December.
    • Handle: RePEc:eee:stapro:v:65:y:2003:i:4:p:291-302
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    References listed on IDEAS

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    1. Nikitin, Yakov & Orsingher, Enzo, 2000. "The intermediate arc-sine law," Statistics & Probability Letters, Elsevier, vol. 49(2), pages 119-125, August.
    2. Vadim Linetsky, 1999. "Step Options," Mathematical Finance, Wiley Blackwell, vol. 9(1), pages 55-96, January.
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    Cited by:

    1. Iafrate, F. & Orsingher, E., 2021. "On the sojourn time of a generalized Brownian meander," Statistics & Probability Letters, Elsevier, vol. 168(C).
    2. Seuret, Stéphane & Yang, Xiaochuan, 2019. "On sojourn of Brownian motion inside moving boundaries," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 978-994.

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