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Transactions of the American Mathematical Society

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



Local behaviour of solutions of stochastic integral equations

Author: William J. Anderson
Journal: Trans. Amer. Math. Soc. 164 (1972), 309-321
MSC: Primary 60H20
MathSciNet review: 0297031
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Abstract: Let X denote the solution process of the stochastic equation $ dX(t) = a(X(t))dt + \sigma (X(t))dW(t)$. In this paper, conditions on $ a( \cdot )$ and $ \sigma ( \cdot )$ are given under which the sample paths of X are differentiate at $ t = 0$ with probability one. Variations of these results are obtained leading to a new uniqueness criterion for solutions of stochastic equations. If $ \sigma ( \cdot )$ is Hölder continuous with exponent greater than $ \tfrac{1}{2}$ and $ a( \cdot )$ satisfies a Lipschitz condition, it is shown that in the one-dimensional case the above equation has only one continuous solution.

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Keywords: Stochastic integral equations, sample path behaviour, differentiability of solution, uniqueness of solution
Article copyright: © Copyright 1972 American Mathematical Society