Open Access
February, 1978 On the Gap Between Deterministic and Stochastic Ordinary Differential Equations
Hector J. Sussmann
Ann. Probab. 6(1): 19-41 (February, 1978). DOI: 10.1214/aop/1176995608


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.


Download Citation

Hector J. Sussmann. "On the Gap Between Deterministic and Stochastic Ordinary Differential Equations." Ann. Probab. 6 (1) 19 - 41, February, 1978.


Published: February, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0391.60056
MathSciNet: MR461664
Digital Object Identifier: 10.1214/aop/1176995608

Primary: 60H10
Secondary: 34F05

Keywords: sample paths , Stochastic differential equations

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 1 • February, 1978
Back to Top