Asymptotic expansions of evolution equations with fast volatility - Howison, S; Reisinger, C; Sircar, R; Wang, Z (Society for Industrial and Applied Mathematics, 2025)
We study Kolmogorov forward equations (KFE) 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, CUP, 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 homogenised one-dimensional SPDEs.