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The aim of the paper is to establish a convergence theorem formulti-dimensional stochastic approximation when the "innovations" satisfy some"light" averaging properties in the presence of a pathwise Lyapunov function.These averaging assumptions allow us to unify apparently remote frameworkswhere the innovations are simulated (possibly deterministic like in Quasi-MonteCarlo simulation) or exogenous (like market data) with ergodic properties. Wepropose several fields of applications and illustrate our results on fiveexamples mainly motivated by Finance.

Stochastic Approximation with Averaging Innovation Applied to Finance