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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.

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