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.