Local Behaviour of Solutions of Stochastic Integral Equations

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.