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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

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.

Citation

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

Information

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

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

Subjects:
Primary: 60H10
Secondary: 34F05

Rights: Copyright © 1978 Institute of Mathematical Statistics

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Vol.6 • No. 1 • February, 1978
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