The Annals of Probability

On the Gap Between Deterministic and Stochastic Ordinary Differential Equations

Hector J. Sussmann

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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

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

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]

Stochastic differential equations sample paths


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.

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