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January, 1987 A Class of Markov Processes which Admit Local Times
Vlad Bally, Lucretiu Stoica
Ann. Probab. 15(1): 241-262 (January, 1987). DOI: 10.1214/aop/1176992266

Abstract

A class of standard processes which admit local times at each point is considered. The following regularity properties are assumed: $T_x \rightarrow T_a = 0$ (as $x \rightarrow a$) in $P^a$-probability and $P^a(T_b < \infty) > 0$ for all pairs of points $a, b (T_x = \inf\{t > 0: X_t = x\})$. The class under consideration turns out to be very large. It is already known that a wide class of processes with independent increments fulfill our hypothesis. We also observe that the class is left invariant by the usual transformations: time change, subprocess and $u$-process ($h$-path) transformations. The first important result of the paper is that every continuous additive functional may be represented as a mixture (integral) of local times. This theorem is used to prove two further results. The first one asserts that every process in the class has a dual process which remains in the class. Particularly Hunt's hypothesis (F) is satisfied. The second one generalises the occupation time and downcrossing approximating models. Such approximation theorems are proved for a C.A.F. whose representing measure is given.

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Vlad Bally. Lucretiu Stoica. "A Class of Markov Processes which Admit Local Times." Ann. Probab. 15 (1) 241 - 262, January, 1987. https://doi.org/10.1214/aop/1176992266

Information

Published: January, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0615.60069
MathSciNet: MR877600
Digital Object Identifier: 10.1214/aop/1176992266

Subjects:
Primary: 60J55
Secondary: 60J45

Keywords: Additive functionals , approximation by occupation times and downcrossings , Duality , Local times , representing measures , Standard processes

Rights: Copyright © 1987 Institute of Mathematical Statistics

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Vol.15 • No. 1 • January, 1987
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