A multi-step Richardson-Romberg extrapolation method for stochastic approximation

We obtain an expansion of the implicit weak discretization error for the target of stochastic approximation algorithms introduced and studied in Frikha (2013). This allows us to extend and develop the Richardson-Romberg extrapolation method for Monte Carlo linear estimator (introduced in Talay and Tubaro (1990) and deeply studied in Pagès (2007)) to the framework of stochastic optimization by means of stochastic approximation algorithm. We notably apply the method to the estimation of the quantile of diffusion processes. Numerical results confirm the theoretical analysis and show a significant reduction in the initial computational cost.