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Averaging principle of SDE with small diffusion: Moderate deviations. (English) Zbl 1016.60031

The main subject of the paper is the formulation of moderate deviations for the averaging principle of a stochastic differential equation (SDE) with small diffusion, in the case when the fast random environment is an exponentially ergodic Markov process and does not depend on the Wiener process driving the SDE. The original SDE has the form \[ dX_t^{\varepsilon}=b(X_t^{\varepsilon},\xi_{t/\varepsilon})+\sqrt{\varepsilon}a(X_t^{\varepsilon},\xi_{t/\varepsilon})dW_t,\quad X_0^{\varepsilon}=x_0, \] and the large deviation estimation is obtained for the process \(\eta_t^{\varepsilon}=(X_t^{\varepsilon}-\overline x_t)/\sqrt \varepsilon h(\varepsilon)\) in the space of continuous trajectories, as \(\varepsilon\) decreases to \(0\), where \(h(\varepsilon)\) is some deviation scale, and \(d\bar{x}(t)=\bar{b}(\bar{x}(t))dt\), \(\bar{b}(x)=\mathbb{E}_{\pi}(b(x,\cdot))\). The author’s strategy is to show the exponential tightness and then the local moderate deviation principle, which relies on the new method involving a conditional Schilder’s theorem and a moderate deviation principle for inhomogeneous integral functionals of Markov processes, established by the author [Stochastic Processes Appl. 92, 287-313 (2001)].

MSC:

60F10 Large deviations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34C29 Averaging method for ordinary differential equations
60J25 Continuous-time Markov processes on general state spaces
60J60 Diffusion processes
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