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On approximate large deviations for 1D diffusion. (English) Zbl 1041.60028
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.
##### 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:
Itô equation; Euler scheme; large deviations
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