Section: New Results

A stochastic parametric representation for the density of a Markov process

Participant : Vlad Bally.

Classical results in the PDE theory (due to A. Friedmann) assert that, under uniform ellipticity conditions, the law of a diffusion process has a continuous density (the approach of A. Friedmann is analytical and concerns PDE's instead of the corresponding diffusion process). The method developed by A. Friedmann becomes well known as the "parametric method". In collaboration with A. Kohatzu Higa [49] we gave a probabilistic approach which represents the probabilistic counterpart of the parametric method. We obtained a probabilistic representation for the density of the law of the solution of a SDE and more generaly, for a class of Markov processes including solutions of jump type SDE's. This representation may be considered as a perfect simulation scheme and so represents a starting point for Monte Carlo simulation. However the random variable which appears in the stochastic representation has infinite variance, so direct simulation gives unstable results (as some preliminary tests have proved). In order to obtain an efficient simulation scheme some more work on the reduction of variance has to be done.