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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

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  • Aimé Lachal

    (Institut Camille Jordan, UMR CNRS 5208, Université de Lyon
    Institut National des Sciences Appliquées de Lyon, Pôle de Mathématiques Bât. L. de Vinci)

Abstract

Fix an integer N > 2 and let X = (X(t)) t ≥ 0 be the pseudo-process driven by the high-order heat-type equation $\partial/\partial t=\pm\partial^N\!/\partial x^N$ . The denomination “pseudo-process” means that X is related to a signed measure (which is not a probability measure) with total mass equal to one. In this survey, we present several explicit results and discuss some problems concerning the pseudo-distributions of various functionals of the pseudo-process X: the first or last overshooting times of a single barrier {a} or a double barrier {a, b} by X; the sojourn times of X in the intervals [a, + ∞ ) and [a, b] up to a fixed time; the maximum or minimum of X up to a fixed time.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:14:y:2012:i:3:d:10.1007_s11009-011-9245-8
    DOI: 10.1007/s11009-011-9245-8
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    References listed on IDEAS

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    1. 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.
    2. Hochberg, Kenneth J. & Orsingher, Enzo, 1994. "The arc-sine law and its analogs for processes governed by signed and complex measures," Stochastic Processes and their Applications, Elsevier, vol. 52(2), pages 273-292, August.
    3. 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.
    4. 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.
    5. Beghin, L. & Orsingher, E., 2005. "The distribution of the local time for "pseudoprocesses" and its connection with fractional diffusion equations," Stochastic Processes and their Applications, Elsevier, vol. 115(6), pages 1017-1040, June.
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    Cited by:

    1. Ansari, Alireza & Askari, Hassan, 2014. "On fractional calculus of A2n+1(x) function," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 487-497.
    2. 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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