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Intégrales progressive, rétrograde et symétrique de processus non adaptés. (Forward, backward and symmetric integrals of non adapted processes). (French) Zbl 0723.60058

Through a convolution procedure, we introduce three different stochastic integrals: they generalize the usual Itô forward and backward stochastic integrals and the usual Stratonovich integral. The paper is organized in two parts. The first one contains techniques of usual stochastic calculus: we state integration by parts formulas and we are interested in a stochastic differential equation with anticipative initial condition. In the second part, the integrator process is just the Brownian motion: the three integrals above will be shown to be equal to the Skorokhod integral of the integrand plus a trace term involving the “Malliavin derivative”.

MSC:

60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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