Une remarque sur la théorie des grandes déviations

Abstract

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Large deviation principle of Freidlin-Wentzell type for pinned diffusion processes

“…(ii) Because the pinned diffusion measure is a measure of finite energy integral, it satisfies a Freidlin-Wentzell type estimate. (iii) The general principle of Baldi-Sanz [3] then implies that the pinned diffusion measure satisfies a large deviation principle. However, it is unclear under what kind of assumptions on vector fields each step above holds true since there is no detailed explanation.…”

“…Note that, in their method, smoothness of the coefficients needs to be assumed. In our Proposition 6.1, however, we only assumed C 3 b , which is probably astonishing if we do not know rough path theory. In this sense, this is an improvement of Malliavin-Nualart's result.…”

“…The proof of the above theorem is a new proof of Ren's result. In fact, it is an improvement since we only assumed C 3 b , not C ∞ b , for the coefficients.…”

General Large Deviations and Functional Iterated Logarithm Law for Multivalued Stochastic Differential Equations

“…The following result establishes a Freidlin-Wentzell estimate for X ǫ . In the case of ordinary stochastic differential equations, it is proved in [3].…”

“…Theorem 2.3. Under (LD) 1 and (LD) 2 , {Z ǫ , ǫ ∈ (0, 1)} satisfies the Laplace principle with the rate function I(f ) given by (3). More precisely, for each real bounded continuous function g on S:…”