Schémas d'approximation multiniveaux pour la value-at-risk et l'expected shortfall

Evaluating the risk associated to a portfolio is important to regulators, market exchanges (or clearing houses) and investors. That is why significant research efforts have been spent to characterize financial risk measures according to various aspects: coherence, convexity, elicitability, identifiability and backtestability. The value-at-risk (VaR) and expected shortfall (ES) remain however the most widely used risk measures in finance. Given a portfolio of random future loss X and a confidence level c in (0,1), the VaR represents the quantile of X at level c (P(X > VaR) = 1-c) and the ES the average loss given that the latter exceeds the VaR, i.e. the superquantile at level c (ES = E[X|X > VaR]). Computing efficiently the VaR and ES of a given portfolio constitutes an active field of research. Rockafeller and Uryasev (2000) show that, under suitable assumptions, the VaR and ES can be jointly retrieved as solutions to a stochastic convex program, thus allowing to resort to stochastic approximation methods. In a realistic scenario, the loss of the portfolio is expressed in a nested way, i.e. X = E[f(Y, Z)|Y], where Y describes the risk factors influencing the portfolio up to some time horizon T > 0, Z the risk factors influencing it beyond T and f(Y, Z) the subsequent cash flows. The conditioning is done with respect to the information available at time T. When X cannot be simulated exactly, a natural solution consists in swapping X at each descent step with a nested Monte Carlo estimate thereof. This thesis endeavors to accelerate the stochastic approximation method to compute the VaR and ES in the nested loss case, where X = E[f(Y, Z)|Y] can only be simulated by Monte Carlo. The goalpost is to attain a quadratic complexity as is achieved by unbiased stochastic approximation in the case that X can be simulated exactly. We first revisit the nested stochastic approximation scheme to reach a complexity in O(n^3) to achieve an error in O(1/n). We then leverage the multilevel stochastic approximation paradigm, developed by Frikha (2016), to bring the VaR estimation cost down to O(n^2.5) and the ES estimation cost down to O(n^2 ln(n)^2). The nested and multilevel VaR and ES complexities above can only be reached under additional explicit, yet inaccessible, conditions on the initial stochastic approximation step size. Meeting such conditions requires an overhead finetuning phase, which only increases the total runtime of the nested and multilevel algorithms. To circumvent such conditions, we develop ergodic versions of these algorithms that leverage the Polyak-Ruppert averaging principle. To demonstrate the numerical stability increase of the averaged nested and multilevel algorithms, we provide central limit theorems that display such stability gains in their asymptotic covariance matrices. Eventually, to bridge the performance gap between multilevel stochastic approximation of the VaR in O(n^2.5) and its unbiased stochastic approximation (where the loss X is directly simulated) in O(n^2), we explore the effect of adaptively refining the Monte Carlo innovations digested by the multilevel scheme that fall too close to the discontinuities of the involved stochastic gradients but could potentially be on the wrong side. This allows to bring the nested VaR estimation down to O(n^2.5 ln(n)^0.5) and the multilevel VaR estimation down to O(n^2 ln(n)^2.5). All of the above results are exemplified through suitable numerical studies including realistic financial scenarios.