## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/aop/1176995608

**Mathematical Reviews number (MathSciNet)**

MR461664

**Zentralblatt MATH identifier**

0391.60056

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Secondary: 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]

**Keywords**

Stochastic differential equations sample paths

#### Citation

Sussmann, Hector J. On the Gap Between Deterministic and Stochastic Ordinary Differential Equations. Ann. Probab. 6 (1978), no. 1, 19--41. doi:10.1214/aop/1176995608. http://projecteuclid.org/euclid.aop/1176995608.