Abstract. We introduce a variation of the proof for weak approximations that is suitable for studying the densities of stochastic processes which are evaluations of the flow generated by a stochastic differential equation on a random variable that may be anticipating. Our main assumption is that the process and the initial random variable have to be smooth in the Malliavin sense. Furthermore, if the inverse of the Malliavin covariance matrix associated with the process under consideration is sufficiently integrable, then approximations for densities and distributions can also be achieved. We apply these ideas to the case of stochastic differential equations with boundary conditions and the composition of two diffusions. 1.

Introduction Stochastic integrals with anticipating integrands have been studied for several years. As a result, usual stochastic calculus on Ito's integrals has been extended to some class of integrals where the integrated term is not adapted to the driving Brownian motion (see in particular [11]). Therefore it has been possible to deal with some stochastic differential equations containing parameters which anticipate the Brownian motion. The most investigated problem is the study of the solution of a stochastic differential equation of the form : 2 J. Garnier dX(t) = f(t; X(t))dt + d X k=1 oe k (t; X(t)) ffi dw k t ; X 2 IR d ; (1) with a boundary condition, which means that we impose that the solution<F59