Veretennikov, A. Yu. On approximate large deviations for 1D diffusion. (English) Zbl 1041.60028 Georgian Math. J. 10, No. 2, 381-399 (2003). Let \(X\) satisfy the SDE on the torus \(T^1\): \(X_t=x+\int_0^t\sigma(X_s)\,dB_s\,\,(\text{mod}\,\,1)\), \(t\geq 0\), where \(B\) is a standard Brownian motion. Let \(X^h\) be its standard Euler approximation, i.e.\(X_t^h=x+\int_0^t\sigma(X_{[s/h]h}^h)\,dB_s\). Define the corresponding semigroup operators on \(C(T^1)\): \(A^\beta\varphi(x)=E_x\varphi(X_1)\exp(\beta\int_0^1 f(X_s)\,ds)\) and \(A^{h,\beta}\varphi(x)=E_x\varphi(X^h_1)\exp(\beta\int_0^1 f(X^h_s)\,ds)\). If \(\sigma\in C^3_b\), \(\sigma^{-1}>0\) and \(f\in C^1_b\), then for any positive \(b\) there exists \(C\) such that for any \(| \beta| <b\), \(\| A^\beta-A^{h,\beta} \| _{C(T^1)}\leq \| A^\beta-A^{h,\beta} \| _{C(R^1)}\leq C\sqrt{h}\). As a corollary, estimates on the rate functions of \(X\) and \(X^h\) are given. Reviewer: Ilya Pavlyukevitch (Berlin) MSC: 60F10 Large deviations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 65C30 Numerical solutions to stochastic differential and integral equations Keywords:ItĂ´ equation; Euler scheme; large deviations PDFBibTeX XMLCite \textit{A. Yu. Veretennikov}, Georgian Math. J. 10, No. 2, 381--399 (2003; Zbl 1041.60028) Full Text: EuDML