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Dynamic construction of martingales of density functions

The density hypothesis on random times becomes now a standard in modeling of risks. One of the basic reasons to introduce the density hypothesis is the desire to have a computable credit risk model. However, recent work shows that merely an existence of a density function for the conditional law of the random times will not be enough for the purposes of some numerical implantation problems. It becomes necessary to have models with martingales of density functions evolving along with the development of the information flow, in particular, to have Markovian martingales of density functions determined by a stochastic differential equation. The quetion of constructing a martingale of density functions by a stochastic differential equation has been answered in one dimensional case. The aim of this note is to provide a solution in higher dimensional cases.