Stochastic flows for SDEs with non-Lipschitz coefficient. (Q1415377)

A stochastic differential equation \[ dX_t=\sum_{n=1}^\infty\sigma_n(X_t)dW_t^n+b(X_t)dt,\quad X_0=x\in{\mathbb R}, \] is considered, where \(W^n\) are Brownian motions, \(n=1,2,\dots\), and none of the \(\sigma_n\)'s or \(b\) are Lipschitz. Conditions on coefficients are given which imply that the solution is a.s.\ continuous in \(x\) and \(t\) for small \(t\). Under a stronger assumption, the joint continuity of the solution in all \(x\) and \(t\) is proved (this is the main result of the paper). Hence the homeomorphic property of the mapping \(x\mapsto X(x,t,\omega)\) is deduced as well.