Solution of Convection-Diffusion Partial Differential Equations Using Girsanov Theorem and Conditional Brownian Local Time

In this brief research note, we try to find solution of a large class of convection-diffusion backward or 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. Using Brownian Bridging technique, we find conditional local time of variance scaled Brownian motion. We use Girsanov theorem to add drift to this scaled Brownian motion and solve the PDE at points of interest by integrating conditional value of exponential martingale using conditional Brownian local time. In the Backward Kolmogorov equation, the final data is modeled as a sum of Dirac Deltas of finite width and total solution is a superposition of sum of individual solutions from Dirac Deltas