Studying Synergies between SDEs and PDEs; Analysis of Kolmogorov and Feynman-Kac Results

Abstract

In this report, the synergy between stochastic differential equations (SDEs) and partial differential equations (PDEs) is studied. There are important results by Kolmogorov, Feynman and Kac that can be utilized to solve PDEs using SDEs and vice-versa. In this report, the results in the articles by Black-Scholes, and Harrison are analyzed and their importance critiqued. These synergies between PDEs and SDEs are poised to be utilized in the energy markets, demand-response programs and reserve planning for the grid. The report is presented by instantiating the analysis to European options; the general theory is applicable widely.