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First hitting time and place for pseudo-processes driven by the equation subject to a linear drift

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  • Lachal, Aimé

Abstract

Consider the high-order heat-type equation [not partial differential]u/[not partial differential]t=(-1)1+N/2[not partial differential]Nu/[not partial differential]xN for an even integer N>2, and introduce the related Markov pseudo-process (X(t))t[greater-or-equal, slanted]0. Let us define the drifted pseudo-process (Xb(t))t[greater-or-equal, slanted]0 by Xb(t)=X(t)+bt. In this paper, we study the following functionals related to (Xb(t))t[greater-or-equal, slanted]0: the maximum Mb(t) up to time t; the first hitting time of the half line (a,+[infinity]); and the hitting place at this time. We provide explicit expressions for the Laplace-Fourier transforms of the distributions of the vectors (Xb(t),Mb(t)) and , from which we deduce explicit expressions for the distribution of as well as for the escape pseudo-probability: . We also provide some boundary value problems satisfied by these distributions.

Suggested Citation

  • Lachal, Aimé, 2008. "First hitting time and place for pseudo-processes driven by the equation subject to a linear drift," Stochastic Processes and their Applications, Elsevier, vol. 118(1), pages 1-27, January.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:1:p:1-27
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    References listed on IDEAS

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    1. Beghin, L. & Orsingher, E. & Ragozina, T., 2001. "Joint distributions of the maximum and the process for higher-order diffusions," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 71-93, July.
    2. Beghin, Luisa & Hochberg, Kenneth J. & Orsingher, Enzo, 2000. "Conditional maximal distributions of processes related to higher-order heat-type equations," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 209-223, February.
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    Cited by:

    1. D'Ovidio, Mirko & Orsingher, Enzo, 2011. "Bessel processes and hyperbolic Brownian motions stopped at different random times," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 441-465, March.
    2. Aimé Lachal, 2012. "A Survey on the Pseudo-process Driven by the High-order Heat-type Equation $\boldsymbol{\partial/\partial t=\pm\partial^N\!/\partial x^N}$ Concerning the Hitting and Sojourn Times," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 549-566, September.
    3. Ibragimov, I.A. & Smorodina, N.V. & Faddeev, M.M., 2015. "Limit theorems for symmetric random walks and probabilistic approximation of the Cauchy problem solution for Schrödinger type evolution equations," Stochastic Processes and their Applications, Elsevier, vol. 125(12), pages 4455-4472.
    4. Lachal, Aimé, 2014. "First exit time from a bounded interval for pseudo-processes driven by the equation ∂/∂t=(−1)N−1∂2N/∂x2N," Stochastic Processes and their Applications, Elsevier, vol. 124(2), pages 1084-1111.

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