Local behaviour of solutions of stochastic integral equations

Anderson, William J.

doi:10.1090/S0002-9947-1972-0297031-9

Local behaviour of solutions of stochastic integral equations
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by William J. Anderson PDF

Trans. Amer. Math. Soc. 164 (1972), 309-321 Request permission

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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