Conditional distributions, exchangeable particle systems, and stochastic partial differential equations

Abstract

Stochastic partial dierential equations whose solutions are probability-measurevalued processes are considered. Measure-valued processes of this type arise naturally as de Finetti measures of innite exchangeable systems of particles and as the solutions for ltering problems. In both these cases, the solution is the conditional distribution of the solution of a stochastic dierential equation. The main result states that, under mild nondegeneracy conditions on the coecients of the stochastic dierential equation, the conditional distribution of its solution charges any open set. Under stronger conditions we show that it is absolutely continuous with respect to Lebesgue measure and its density is positive almost everywhere. As applications we show the existence of a solution of a system of interacting diusions and study the properties of the solution of the nonlinear ltering equation within a framework that allows for the signal noise and the observation noise to be correlated. The work was motivated by a model of asset price determination in which the price is given as a quantile of the valuations of innitely many individual investors. MSC 2010 subject classications: 60H15, 60G09, 60G35, 60J25