On the Gap Between Deterministic and Stochastic Ordinary Differential Equations

We consider stochastic differential equations $dx = f(x) dt + g(x) dw$, where x is a vector in n-dimensional space, and w is an arbitrary process with continuous sample paths. We show that the stochastic equation can be solved by simply solving, for each sample path of the process w, the corresponding nonstochastic ordinary differential equation. The precise requirements on the vector fields f and g are: (i) that g be continuously differentiable and (ii) that the entries of f and the partial derivatives of the entries of g be locally Lipschitzian. For the particular case of a Wiener process w, the solutions obtained this way turn out to be the solutions in the sense of Stratonovich.