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)].


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
Full Text: DOI


[1] BAIER, U. and FREIDLIN, M. I. (1977). Theorems on large deviations and stability under random perturbations. Dokl. Akad. Nauk USSR 235 253-256.
[2] BERNARD, P. and RACHAD, A. (2000). Moy ennisation d’un oscillateur stochastique quasiconservatif. C. R. Acad. Sci. Paris 331 1029-1032. · Zbl 0976.70019
[3] BOGOLUBOV, N. and MITROPOLSKII, A. (1961). Asy mptotics Methods in Non-linear Mechanics. Gordon and Breach, New York.
[4] DEMBO, A. and ZEITOUNI, O. (1998). Large Deviations Techniques and Their Applications, 2nd ed. Jones and Bartlett, Boston. · Zbl 0896.60013
[5] DEUSCHEL, J. D. and STROOCK, D. W. (1989). Large Deviations. Academic Press, New York. · Zbl 0675.60086
[6] DOWN, D., MEy N, P. and TWEEDIE, R. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23 1671-1691. · Zbl 0852.60075
[7] FENG, J. and KURTZ, T. (2000). Large deviations for stochastic processes. Preprint. Available at http://www.math.wisc.edu/kurtz/feng. URL: · Zbl 1113.60002
[8] FREIDLIN, M. I. (1978). The averaging principle and theorem on large deviations. Uspekhi Mat. Nauk 33 107-160. · Zbl 0416.60029
[9] FREIDLIN, M. I. and WENTZELL, W. D. (1998). Random Perturbations of Dy namical Sy stems, 2nd ed. Springer, New York.
[10] GUILLIN, A. (2001). Moderate deviations of inhomogeneous functionals of Markov processes and application to averaging. Stochastic Process. Appl. 92 287-313. · Zbl 1047.60022
[11] KHASMINSKII, R. Z. (1968). On the principle of averaging for the Itô stochastic differential equations. Ky bernetika (Czechoslovakia) 4 260-279. · Zbl 0231.60045
[12] KHASMINSKII, R. Z. (1980). Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Rockville, MD. · Zbl 0855.93090
[13] KIEFER, Y. (2000). Averaging and climate models. In Stochastic Climate Models. Birkhäuser, Boston.
[14] KLEBANER, F. C. and LIPTSER, R. S. (1999). Moderate deviations for randomly perturbed dy namical sy stems. Stochastic Process. Appl. 80 157-176. · Zbl 0961.60042
[15] LIPTSER, R. S. (1994). The Bogolubov averaging principle for semimartingales. Proc. Steklov Inst. Math. 4 000-000.
[16] LIPTSER, R. S. (1996). Large deviations for two scaled diffusions. Probab. Theory Related Fields 106 71-104. · Zbl 0855.60030
[17] LIPTSER, R. S. and PUHALSKII, A. A. (1992). Limit theorems on large deviations for semimartingales. Stochastics Stochastics Rep. 38 201-249. · Zbl 0749.60027
[18] LIPTSER, R. S. and SPOKOINY, V. (1999). Moderate deviations ty pe evaluation for integral functional of diffusion processes. Electron. J. Probab. 4 1-25. · Zbl 0932.60030
[19] LIPTSER, R. S. and STOy ANOV, J. (1990). Stochastic version of the averaging principle for diffusion ty pe processes. Stochastics Stochastics Rep. 32 145-163. · Zbl 0729.60047
[20] NUMMELIN, E. (1984). General Irreducible Markov Chains and Non-negative Operators. Cambridge Univ. Press. · Zbl 0551.60066
[21] PARDOUX, E. and VERETENNIKOV, A. Y. (2000). On Poisson equation and diffusion approximation, 2. · Zbl 1054.60064
[22] PARDOUX, E. and VERETENNIKOV, A. Y. (2001). On Poisson equation and diffusion approximation, 1. Ann. Probab. 29 1061-1085. · Zbl 1029.60053
[23] PUHALSKII, A. A. (1990). Large deviations of semimartingales via convergence of the predictable characteristics. Stochastics Stochastics Rep. 49 27-85. · Zbl 0827.60017
[24] RACHAD, A. (1999). Principes de moy ennisation pour des oscillateurs stochastiques non linéaires. Thèse, Univ. Blaise Pascal.
[25] REVUZ, D. and YOR, M. (1994). Continuous Martingales and Brownian Motion, 2nd ed. Springer, Berlin. · Zbl 0804.60001
[26] SANDERS, J. A. and VERHULST, F. (1985). Averaging Method in Non-linear Dy namical Sy stems. Springer, Berlin.
[27] VERETENNIKOV, A. Y. (1999a). On large deviations for stochastic differential equations with a small diffusion and averaging. Theory Probab. Appl. 43 335-337. · Zbl 0953.60042
[28] VERETENNIKOV, A. Y. (1999b). On large deviations in the averaging principle for stochastic differential equations with complete dependence. Theory Probab. Appl. 43 664-666. · Zbl 0958.60030
[29] VERETENNIKOV, A. Y. (1999c). On large deviations in the averaging principle for stochastic differential equations with complete dependence. Ann. Probab. 27 284-296. · Zbl 0939.60012
[30] WU, L. (2001). Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian sy stems. Stochastic Process. Appl. 91 205-238. · Zbl 1047.60059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.