| In this thesis, we consider second order parabolic equations with coefficients that vary both in space and in time (non-autonomous). We derive closed-form approximations to the associated fundamental solution by extending the Dyson-Taylor commutator method recently developed by Costantinescu, Costanzino, Mazzucato and Nistor 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 can give an approximation of the solution for any fixed time and within any given tolerance. To extend our results to large time, we introduce the so called bootstrap scheme, and show that the total error is still under control in this scheme, but the short time limitation can be removed. For applications, we adapt our ideas to Kolmogorov backward equations that appear in various research fields, such as option pricing. We also numerically compare our results with many other methods in the literature and show that our Dyson-Taylor Commutator method is algorithmically more elementary, it works for more general PDEs, and it gives fairly accurate approximations that are good enough in practice. |
