Stochastic Differential Equations: Basic Theory

Abstract

The basic theoretical problems concerned with stochastic differential equations are, generally speaking, the same as those in the case of deterministic differential equations, namely: existence and uniqueness of a solution, analytical properties of the solutions, dependence of the solutions on parameters and initial values etc. Yet, the introduction of random elements into appropriate differential equations leads to new probabilistic problems and specific difficulties. For instance, the sense itself of a stochastic differential equation should be clearly defined since it can be different depending on the understanding of a stochastic process and its derivatives. For the analysis of stochastic differential equations, however, a crucial point is regularity of random functions occuring in a given equation.