A nonlinear stochastic equation in a Hilbert space is considered, with constant but possibly degenerate diiusion term. Some smoothing properties for the associated transition semigroup are studied. In particular, strong Feller property and irreducibility are proved. The main tools are Malliavin calculus and Girsanov transformation.

In this paper we employ Malliavin calculus to derive a general stochastic maximum principle for stochastic partial differential equations with jumps under partial information. We apply this result to solve an optimal harvesting problem in the presence of partial information. Another application pertains to portfolio optimization under partial observation.

We develop a stochastic calculus for the fractional Brownian motion with Hurst parameter H > 2 using the techniques of the Malliavin calclulus. We establish estimates in Lp, maximal inequalities and a continuity criterion for the stochastic integral. Finally, we derive an Itô’s formula for integral processes.

We derive Edgeworth-type expansions for Skorohod and Itô integrals with respect to Brownian motion, based on cumulant operators defined by the Malliavin calculus. As a consequence we obtain Stein approximation bounds for stochastic integrals, which apply to SDE solutions and to multiple stochastic integrals.