  We propose an unconstrained stochastic approximation method of finding the
optimal measure change (in an a priori parametric family) for Monte Carlo
simulations. We consider different parametric families based on the Girsanov
theorem and the Esscher transform (or exponential-tilting). In a
multidimensional Gaussian framework, Arouna uses a projected Robbins-Monro
procedure to select the parameter minimizing the variance. In our approach, the
parameter (scalar or process) is selected by a classical Robbins-Monro
procedure without projection or truncation. To obtain this unconstrained
algorithm we intensively use the regularity of the density of the law without
assume smoothness of the payoff. We prove the convergence for a large class of
multidimensional distributions and diffusion processes. We illustrate the
effectiveness of our algorithm via pricing a Basket payoff under a
multidimensional NIG distribution, and pricing a barrier options in different
markets.
