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Estimations de grandes déviations pour des systèmes où apparaissent un bruit gaussien et un bruit non gaussien. (Large deviations estimations for systems with Gaussian and non-Gaussian noise). (French) Zbl 0597.60025
The abstract reads: ”We obtain large deviations estimates for stochastic differential systems perturbed by both a rapid process satisfying some result of large deviations, and an independent white noise.”
The following system is considered: \[ (S_{\xi,\sigma})dX_ t^{\epsilon}=b(X_ t^{\epsilon},\xi_{t/\epsilon})+\sqrt{\epsilon \sigma}(X_ t\quad^{\epsilon})dW_ t,\quad t\in [0,T],\quad X_ 0^{\epsilon}=x\in {\mathbb{R}}^ d, \] where b and \(\sigma\) are Lipschitz bounded functions, W is a d-dimensional Brownian process and \(\xi\) is a stochastic process independent of W. The main result is as follows: For a Borel set A of \(C_ x([0,T],{\mathbb{R}}^ d)\) \[ -\inf_{\phi \in A^ 0}S(\phi)\leq \lim \inf_{\epsilon \to 0}\log E(X^{\epsilon}(x)\in A)o \limsup_{\epsilon \to 0}\log E(X^{\epsilon}(x)\in A)\leq -\inf_{\phi \in \bar A}S(\phi), \] where S is a functional, \(A^ 0\) and \(\bar A\) are the interior and the closure of A, respectively.
Reviewer: D.Szynal
60F10 Large deviations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: Numdam EuDML