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Temporal symmetry of some classes of stochastic processesLéonard, Christian, Roelly, Sylvie, Zambrini, Jean-Claude January 2013 (has links)
In this article we analyse the structure of Markov processes and reciprocal processes to underline their time symmetrical properties, and to compare them. Our originality consists in adopting a unifying approach of reciprocal processes, independently of special frameworks in which the theory was developped till now (diffusions, or pure jump processes). This leads to some new results, too.
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Reciprocal class of jump processesConforti, Giovanni, Dai Pra, Paolo, Roelly, Sylvie January 2014 (has links)
Processes having the same bridges as a given reference Markov process
constitute its reciprocal class. In this paper we study the reciprocal class
of compound Poisson processes whose jumps belong to a finite set A in R^d.
We propose a characterization of the reciprocal class as the unique set of
probability measures on which a family of time and space transformations
induces the same density, expressed in terms of the reciprocal invariants.
The geometry of A plays a crucial role in the design of the transformations,
and we use tools from discrete geometry to obtain an optimal characterization.
We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state.
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Duality formula for the bridges of a Brownian diffusion : application to gradient driftsRoelly, Sylvie, Thieullen, Michèle January 2005 (has links)
In this paper, we consider families of time Markov fields (or reciprocal classes) which have the same bridges as a Brownian diffusion. We characterize each class as the set of solutions of an integration by parts formula on the space of continuous paths C[0; 1]; R-d) Our techniques provide a characterization of gradient diffusions by a duality formula and, in case of reversibility, a generalization of a result of Kolmogorov.
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