We obtain an expansion of the implicit weak discretization error for the target of stochastic approximation algorithms introduced and studied in [Frikha2013]. This allows us to extend and develop the Richardson-Romberg extrapolation method for Monte
Carlo linear estimator (introduced in [Talay & Tubaro 1990] and deeply studied in [Pag{`e}s 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.
Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are two risk measures which are widely used in the practice of risk management. This paper deals with the problem of computing both VaR and CVaR using stochastic approximation (with decreasing
steps): we propose a first Robbins-Monro procedure based on Rockaffelar-Uryasevs identity for the CVaR. The convergence rate of this algorithm to its target satisfies a Gaussian Central Limit Theorem. As a second step, in order to speed up the initial procedure, we propose a recursive importance sampling (I.S.) procedure which induces a significant variance reduction of both VaR and CVaR procedures. This idea, which goes back to the seminal paper of B. Arouna, follows a new approach introduced by V. Lemaire and G. Pag`es. Finally, we consider a deterministic moving risk level to speed up the initialization phase of the algorithm. We prove that the convergence rate of the resulting procedure is ruled by a Central Limit Theorem with minimal variance and its efficiency is illustrated by considering several typical energy portfolios.
