Analytic Solution of Convection-Diffusion Partial Differential Equations Using Girsanov Theorem, Feynman-Kac Formula and Brownian Bridge

In this brief research note, we try to find solution of a large class of convection-diffusion forward Kolmogorov equations of the type that typically appear in theory of derivatives pricing and stochastic volatility modeling. Our technique is based on a change of coordinates that makes the diffusion part of associated SDE linear. We use Girsanov theorem to add drift to this scaled Brownian motion and Feynman-Kac Formula is used to take into account killing term in the PDE. We solve the PDE at points of interest by analytically calculating conditional value of Girsanov exponential and Feynman Exponential using Brownian Bridging