## The Annals of Probability

### On the Gap Between Deterministic and Stochastic Ordinary Differential Equations

Hector J. Sussmann

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 6, Number 1 (1978), 19-41.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995608

Digital Object Identifier
doi:10.1214/aop/1176995608

Mathematical Reviews number (MathSciNet)
MR461664

Zentralblatt MATH identifier
0391.60056

JSTOR