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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Stochastic equations with discontinuous drift

Author: Edward D. Conway
Journal: Trans. Amer. Math. Soc. 157 (1971), 235-245
MSC: Primary 60.75
MathSciNet review: 0275532
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Abstract: We study stochastic differential equations, $ dx = adt + \sigma d\beta $ where $ \beta $ denotes a Brownian motion. By relaxing the definition of solutions we are able to prove existence theorems assuming only that $ a$ is measurable, $ \sigma $ is continuous and that both grow linearly at infinity. Nondegeneracy is not assumed. The relaxed definition of solution is an extension of A. F. Filippov's definition in the deterministic case. When $ \sigma $ is constant we prove one-sided uniqueness and approximation theorems under the assumption that $ a$ satisfies a one-sided Lipschitz condition.

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Keywords: Stochastic integral of Itô, stochastic differential equations, stochastic integral equations, discontinuous coefficients
Article copyright: © Copyright 1971 American Mathematical Society