ASYMPTOTIC EXPANSIONS OF EVOLUTION EQUATIONS WITH FAST VOLATILITY

We study Kolmogorov forward equations (KFEs) and Zakai equations for diffusion processes with a fast mean-reverting stochastic volatility component. In the case of the KFE, a parabolic PDE in divergence form, we perform a matched asymptotic expansion up to first order in the small mean-reversion time. The solutions are expressed in terms of suitable PDEs with coefficients averaged over the ergodic distribution, in the spirit of extensive earlier work on the backward equation (see [J.-P. Fouque et al., Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge, University Press, 2011]). We then construct a sequence of approximations to the Zakai equation, a parabolic stochastic PDE (SPDE), and verify numerically for the first two terms weak convergence order half and order one, respectively, in the mean-reversion parameter. To this end, we give a novel numerical scheme for the original two-dimensional SPDE, which is robust in the small parameter regime, and compare derived functionals of marginals against those approximated by the solution of a sequence of homogenized one-dimensional SPDEs.