The aim of the paper is to establish a convergence theorem for
multi-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 frameworks
where the innovations are simulated (possibly deterministic like in Quasi-Monte
Carlo simulation) or exogenous (like market data) with ergodic properties. We
propose several fields of applications and illustrate our results on five
examples mainly motivated by Finance
