Stochastic Differential Equations

In this chapter we introduce stochastic differential equations (SDEs) and discuss existence and uniqueness questions. The geometric and linear equations are studied in some detail and their most important properties are derived. We then discuss the connection between SDEs and partial differential equations (PDEs). In particular we prove the Feynman–Kač representation theorem which provides the solution to a parabolic PDE in terms of an expected value connected to a certain SDE. We also discuss and derive the Kolmogorov forward and backward equations.