A generalized formula of Ito and some other properties of stochastic flows

Summary A stochastic differential equation with smooth coefficients is considered, which defines a continuous flow _t , (, ^.) of C ⁺⁸ mappings of R ^d in R ^d . If z _t is a continuous semi-martingale, _t ,(,z_t)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, _t (, ^.) is an onto diffeomorphism. If z _t is a continuous semi-martingale, _t ^–1 ,(,z _t ) is proved to be a semi-martingale, whose Ito decomposition is explicitly found.The support of the University of British Columbia is gratefully acknowledged.