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Joint distributions of the maximum and the process for higher-order diffusions

Author

Listed:
  • Beghin, L.
  • Orsingher, E.
  • Ragozina, T.

Abstract

For processes X(t),t>0 governed by signed measures whose density is the fundamental solution of third and fourth-order heat-type equations (higher-order diffusions) the explicit form of the joint distribution of (max0[less-than-or-equals, slant]s[less-than-or-equals, slant]t X(s),X(t)) is derived. The expressions presented include all results obtained so far and, for the third-order case, prove to be genuine probability distributions. The case of more general fourth-order equations is also investigated and the distribution of the maximum is derived.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:94:y:2001:i:1:p:71-93
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    References listed on IDEAS

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    1. 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.
    2. Jumarie, Guy, 1999. "Complex-valued Wiener measure: An approach via random walk in the complex plane," Statistics & Probability Letters, Elsevier, vol. 42(1), pages 61-67, March.
    3. 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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    Citations

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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. 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.
    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.
    5. 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.

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