Large deviations analysis of reflected diffusions and constrained stochastic approximation algorithms in convex sets†

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Abstract
Let G be a bounded convex set. hen the solution to the Skorokhod problem for a given path 4 may be considered as a reflected version of 4 with respect to G. By use of estimates on the behavior of this deterministic transformation, large deviations type asymptotic estimates for reflected diffusions are shown to hold. Let ll, be the projection onto G and Gaussian sequence. Then from the diffusion results estimates on the asymptotic behavior of recursive algorithms of the type are shown to follow. Let 0 be a stable point of the algorithm, let D be a neighborhood of 0 with respect to G, let A = to be the piecewise linear interpolation of X",tarting at 0 and having interpolation interval E. Then estimates on P{xE(.) E A), the probability of escape from D before time 7; are obtained. This analysis yields an alternative to convergence results on the "asymptotic normality" of errors about 0, which are in any case not applicable if OESG. These and other estimates provided by the large deviations methods are often more useful in applications. Extensions are outlined for correlated noise, unbounded domains. and domains that are smooth with "convex corners".>

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