Abstract. We study singular perturbations of a class of stochastic control problems under assumptions motivated by models of financial markets with stochastic volatilities evolving on a fast time scale. We prove the convergence of the value function to the solution of a limit (effective) Cauchy problem for a parabolic equation of HJB type. We use methods of the theory of viscosity solutions and of the homogenization of fully nonlinear PDEs. We test the result on some financial examples, such as Merton portfolio optimization problem.

We consider a semigroup of operators in the Banach space C b (H) of uniformly continuous and bounded functions on a separable Hilbert space H. In particular, we deal with semigroups that are related to solution of stochastic PDEs in H and which are not, in general, strongly continuous. We prove an existence and uniqueness result for a measure valued equation involving this class of semigroups. Then we apply the result to a large class of second order differential operators in C b (H).

We consider a stochastic partial differential equation with a logarithmic nonlinearity with singularities at 1 and −1 and a constraint of conservation of the space average. The equation, driven by a trace-class space-time noise, contains a bi-Laplacian in the drift. We obtain existence of solution for equation with polynomial approximation of the nonlinearity. Tightness of this approximated sequence of solutions is proved, leading to a limit transition semi-group. We study the asymptotic properties of this semi-group, showing the existence and uniqueness of invariant measure, asymptotic strong Feller property and topological irreducibility.

We are concerned with a viscous Burgers equation forced by a perturbation of white noise type. We study the corresponding transition semigroup in a space of continuous functions weighted by a proper potential, and we show that the infinitesimal generator is the closure (with respect to a suitable topology) of the Kolmogorov operator associated to the stochastic equation. In the last part of the paper we use this result to solve the corresponding Fokker-Planck equation.

We consider stochastic semilinear partial differential equations with Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant measure and the existence of a solution for the corresponding Kolmogorov equation in the space L 2 (H; ν), where ν is the invariant measure. We also prove the closability of the derivative operator and an integration by parts formula. Finally, under boundness conditions on the nonlinear term, we prove a Poincaré inequality, a logarithmic Sobolev inequality and the ipercontractivity of the transition semigroup.