an:00796429
Zbl 0838.60024
Makhno, S. Ya.
A theorem on large deviations for one class of diffusion processes
EN
Theory Probab. Appl. 39, No. 3, 437-447 (1994); translation from Teor. Veroyatn. Primen. 39, No. 3, 554-566 (1994).
00025518
1994
j
60F10 60J60 60H10
small parameter expansions; large deviations; diffusions; stochastic differential equation
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 \(\psi\).
W.Bryc (Cincinnati)