Approximate Solutions to Second Order Parabolic Equations Ii: Time-dependent Coefficients

We consider second order parabolic equations with coefficients that vary both in space and in time (non-autonomous). We derive closedform approximations to the associated fundamental solution by extending the Dyson-Taylor commutator method that we recently established for autonomous equations. We establish error bounds in Sobolev spaces and show that by including enough terms, our approximation can be proven to be accurate to arbitrary high order in the short-time limit. We show how our method extends to give an approximation of the solution for any fixed time and within any given tolerance. Some applications to option pricing are presented. In particular, we perform several numerical tests, and specifically include results on Stochastic Volatility models.