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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 \(\psi\).
60F10 Large deviations
60J60 Diffusion processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)