Stochastic Differential Equations with Nonlocal Sample Dependence

Abstract

Stochastic ordinary differential equations are investigated for which the coefficients depend on nonlocal properties of the current random variable in the sample space such as the expected value or the second moment. The approach here covers a broad class of functional dependence of the right-hand side on the current random state and is not restricted to pathwise relations. Existence and uniqueness of solutions is obtained as a limiting process by freezing the coefficients over short time intervals and applying existence and uniqueness results and appropriate estimates for stochastic ordinary differential equations.

Keywords:

• Existence and uniqueness theorems
• It\^o stochastic differential equations
• Nonlocal dependence
• Strong solutions

Mathematics Subject Classification:

• Primary 60H35
• Secondary 60H10, 60H25

This work was partially supported by DFG grant LO 273/5-1.