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A theorem on large deviations for one class of diffusion processes. (English. Russian original) Zbl 0838.60024
Theory Probab. Appl. 39, No. 3, 437-447 (1994); translation from Teor. Veroyatn. Primen. 39, No. 3, 554-566 (1994).
The author proves large deviations as $$\varepsilon \to 0$$ for trajectories of $$R^d$$-valued diffusions which solve the stochastic differential equation $\xi^\varepsilon (t) = x + \varepsilon \int^t_0 \sigma^\varepsilon \bigl( s, \xi^\varepsilon (s)\bigr)dW(s)$ with the coefficient $$\sigma^\varepsilon (t,x)$$ that depends on a small parameter $$\varepsilon$$. The main assumption of the paper is that the limit $$\lim_{\varepsilon \to 0} \varepsilon^2 \ln E \exp (\varepsilon^{-2} \int^T_0 (\psi, d \xi^\varepsilon))$$ exists for all piecewise smooth functions $$\psi$$ and is given by a suitable bilinear expression in $$\psi$$.
MSC:
 60F10 Large deviations 60J60 Diffusion processes 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)