Weak approximations. A Malliavin calculus approach

We introduce a variation of the proof for weak approximations that is suita ble for studying the densities of stochastic processes which are evaluation s of the flow generated by a stochastic differential equation on a random v ariable that may be anticipating. Our main assumption is that the process a nd the initial random variable have to be smooth in the Malliavin sense. Fu rthermore, if the inverse of the Malliavin covariance matrix associated wit h the process under consideration is sufficiently integrable, then approxim ations for densities and distributions can also be achieved. We apply these ideas to the case of stochastic differential equations with boundary condi tions and the composition of two diffusions.