Stochastic Differential Equations

Abstract

There exist different notions of a solution to a semilinear stochastic differential equation (SSDE). We define strong, weak (in the sense of duality), mild, and martingale solutions, and study the problem of existence and uniqueness. As in the deterministic case, for example in Pazy (Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York 1983), we first study solutions to a stochastic counterpart of the deterministic inhomogeneous Cauchy problem and we highlight the role played by the stochastic convolution. The SSDE’s we investigate are allowed to depend on the entire past of the solution which significantly broadens the field of applications. The existence result for mild solutions is first obtained for equations with Lipschitz coefficients. In the special case of equations depending only on the presence, we discuss the Markov property, dependence of the solution on the initial condition, including differentiability, and the Kolmogorov backward equation. We also study SSDE’s with continuous coefficients, and present an existence result for martingale solutions, but due to the failure of the Peano theorem, a compactness assumption is added for the associated semigroup, as in DaPrato and Zabczyk (Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge 1992). We also present an existence result for SSDE’s driven by a cylindrical Wiener process.

In case of SDE’s with continuous coefficients, we use Lipschitz-type approximation, as in the work of Gikhman and Skorokhod (The Theory of Stochastic Processes. Springer, Brelin, 1974), and prove the existence of weak solutions (in the stochastic sense). However, again due to the failure of the Peano theorem, the solution is in a larger Hilbert space, where the space containing the initial condition is compactly embedded. We call this technique the “method of compact embedding”, and use it later for finding solutions in the variational method and for models of spin systems.