Stochastic Differential Equations and Stochastic Flows

Abstract

In this chapter, we show that solutions of a continuous symmetric stochastic differential equation (SDE) on a Euclidean space define a continuous stochastic flow of diffeomorphisms and that solutions of an SDE with diffeomorphic jumps define a right continuous stochastic flow of diffeomorphisms. Sections 3.1 and 3.2 are introductions. Definitions of these SDEs and stochastic flows will be given and the geometric property of solutions will be explained as well as basic facts. Rigorous proof of these facts will be given in Sects. 3.3, 3.4, 3.5, 3.6, 3.7, 3.8 and 3.9. In Sect. 3.3, we study another Itô SDE with parameter, called the master equation. Applying results of Sect. 3.3, we show in Sect. 3.4 that solutions of the original SDE define a stochastic flow of C∞-maps. For the proof of the diffeomorphic property, we need further arguments. In Sect. 3.5 we consider backward SDE and backward stochastic flow of C∞-maps. Further, the forward–backward calculus for stochastic flow will be discussed in Sects. 3.5, 3.6 and 3.8. These facts will be applied in Sects. 3.7 and 3.9 for proving the diffeomorphic property of solutions.