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

Aimé Lachal ()
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Aimé Lachal: Institut Camille Jordan, UMR CNRS 5208, Université de Lyon

Methodology and Computing in Applied Probability, 2012, vol. 14, issue 3, 549-566

Abstract: 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.

Keywords: Pseudo-process; Pseudo-distribution; First hitting or overshooting time; Sojourn time; Up-to-date maximum; Primary 60G20; Secondary 60J25 (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/s11009-011-9245-8

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