Summary
A stochastic differential equation with smooth coefficients is considered, which defines a continuous flow φ t , (ω, .) of C +8 mappings of R d in R d. If z t is a continuous semi-martingale, φ t ,(ω,zt)s> is proved to be a semi-martingale, for which an Ito type formula is established. It is shown that a.s., for any t,φ t (ω, .) is an onto diffeomorphism. If z t is a continuous semi-martingale, φ −1t ,(ω,z t ) is proved to be a semi-martingale, whose Ito decomposition is explicitly found.
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Bismut, J.M. A generalized formula of Ito and some other properties of stochastic flows. Z. Wahrscheinlichkeitstheorie verw Gebiete 55, 331–350 (1981). https://doi.org/10.1007/BF00532124
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DOI: https://doi.org/10.1007/BF00532124