This paper develops the stochastic calculus of variations for Hilbert space-valued solutions to stochastic evolution equations whose operators satisfy a coercivity condition. An application is made to the solutions of a class of stochastic pde's which includes the Zakai equation of nonlinear filtering. In particular, a Lie algebraic criterion is presented that implies that all finite-dimensional projections of the solution define random variables which admit a density. This criterion generalizes hypoellipticity-type conditions for existence and regularity of densities for finite-dimensional stochastic differential equations.
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