Continuity in a Pathwise sense with Respect to the Coefficients of Solutions of Stochastic Differential Equations

For SDE's of the form dX(t)=b(X(t))dt+sigma (X(t))dW(t)where b and sigma are Lipschitz continuous, it is shown that ifwe consider a fixed sigma in C^5, bounded and with boundedderivatives, the random field of solutions is pathwise locallyLipschitz continuous with respect to b when the space of driftcoefficients is the set of Lipschitz continuous functions of sublineargrowth endowed with the sup-norm. Furthermore it is shown that thisresult does not hold if we interchange the role of b and sigma.However for SDE's where the coefficient vector fields commutesuitably we show continuity with respect to the sup-norm on thecoefficients and a number of their derivatives.