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A generalized class of Lyons-Zheng processes. (English) Zbl 0988.60053

Stochastic processes that are the sum of a forward and a backward martingale have been first studied by T. J. Lyons and W. A. Zheng [in: Les processus stochastiques. Astérisque 157-158, 249-271 (1988; Zbl 0654.60059)] in connection with Dirichlet forms. The paper under review develops a general study of Lyons-Zheng processes, written as \(X = {1 \over 2} M^1 - {1 \over 2} M^2 + V\), where \(M^1\), resp. \(M^2\) is a local forward, resp. backward, martingale, \(V\) is a bounded variation process and \(M^1-M^2\) has zero quadratic variation. In particular a change of variable formula with respect to the Stratonovich (or symmetric) integral is obtained for \(f(X)\) where \(f\) is a \(C^1\) function. This implies a stability property for the class of Lyons-Zheng processes under transformations by \(C^1\) functions. Stochastic differential equations in Stratonovich sense are also considered, and an application to the representation of time reversed diffusions is given. Finally it is proved that Bessel processes of arbitrary dimension \(\delta>0\) (which are not semimartingales when \(\delta <1\)) are Lyons-Zheng processes.

MSC:

60H05 Stochastic integrals
60J60 Diffusion processes
60G48 Generalizations of martingales
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

Citations:

Zbl 0654.60059
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