On some estimates in quasi sure limit theorem for SDE's (Q1275920)

Let \[ dX(t)= \sigma\bigl(X(t)\bigr) dw(t)+ b\bigl(X(t)\bigr)dt, \quad X(0)=X_0,\tag{1} \] be a stochastic differential equation, where \(\sigma:R^d \to R^d\times R^r\) and \(b:R^d\to R^d\) are smooth functions with bounded derivatives of all orders, \(w(t)\) is the \(r\)-dimensional standard Brownian motion. The author considers the Stroock-Varadhan \(w_n(t)\) and Malliavin \(w_\varepsilon (t)\) approximation of Brownian motion. It is shown that for every \(p,r'\) \[ \sup_n\bigl\| X_n(t)- X_n(s)\bigr \|_{p,r'}< C| t-s|^p \quad \text{and} \quad\sup_{\varepsilon}\bigl\| X_\varepsilon(t)-X_\varepsilon (s) \bigr\|_{2p,r'}<C| t-s|^p, \] where \(X_n(t)\) and \(X_\varepsilon(t)\) denote the unique solutions of (1) with \(w_n\) and \(w_\varepsilon\), respectively, and \(\|\cdot\|_{p,r'}\) is the usual Sobolev norm in Malliavin calculus.