On the approximate solutions of the Stratonovitch equation. (English) Zbl 0901.60028

Summary: The authors present a new method for proving the convergence of the classical approximations of the Stratonovich equation \[ X_t = x_0 +\int_0^t \sigma(X_s)\circ dW_s + \int_0^t\beta(X_s) ds. \] The Liouville space \(\mathcal J_{\alpha,p}\) is used, where it turns out that the calculations are simpler even than with uniform convergence. The main point is the isomorphism \(\mathcal J_{\alpha,p}(L^p)\approx L^p(\mathcal J_{\alpha,p})\) which is a sharpening of the Kolmogorov lemma. With the classical regularity conditions on \(\sigma\) and \(\beta\), the convergence of approximate solutions in each fractional Liouville-valued Sobolev space \(W^{r,p}(\mathcal J_{\alpha,p})\) for suitable values of \(\alpha\) and \(p\) is proved. The \(p\)-admissibility of the space \(W^{r,p}(\mathcal J_{\alpha,p})\) allows to obtain easily convergence in the space \(\mathcal L^1(\Omega,c_{r,p},\mathcal J_{\alpha,p})\) which is the natural space of \(\mathcal J_{\alpha,p}\)-valued quasi-continuous functions on the Wiener space \(\Omega\). As an application, an improvement of the classical support theorem for the capacity \(c_{r,p}\) is obtained.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G17 Sample path properties
60H07 Stochastic calculus of variations and the Malliavin calculus