Convergence by Viscosity Methods in Multiscale Financial Models with Stochastic Volatility

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 Hamilton-Jacobi-Bellman 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.