Journées MAS 2010

Algorithmes Stochastiques (pdf)

Session organisée par Peggy Cénac (Université de Bourgogne)

Un algorithme stochastique est un outil d'optimisation particulièrement utile lorsque l'on observe les données "en ligne". Il permet, entre autres choses, d'estimer les paramètres de modèles statistique avec des procédures généralement simples de mise à jour qui ne nécessitent pas la ré-estimation parfois très coûteuse en temps de calcul. Ces approches itératives possèdent de bonnes propriétés de convergence et atteignent même dans certains cas la vitesse optimale. Dans le vaste domaine d'applications, citons l'économie, la finance, la biologie, la physique mathématique, l'automatique, le traitement d'images jusqu'à la statistique non paramétrique. L'objet de cette session est de présenter quelques uns des derniers résultats théoriques ainsi que des applications récentes dans ce riche domaine.

Exposé de 20 minutes Jérôme Lelong (ENSIMAG) Un cadre pour les méthodes de Monte Carlo adaptatives transparents

Adaptive Monte Carlo methods are powerful variance reduction techniques. In this work, we propose a mathematical setting which greatly relaxes the assumptions needed by for the adaptive importance sampling techniques presented by Arouna in 2003. We establish the convergence and asymptotic normality of the adaptive Monte Carlo estimator under local assumptions which are easily verifiable in practice. We present one way of approximating the optimal importance sampling parameter using a randomly truncated stochastic algorithm. Finally, we apply this technique to the valuation of financial derivatives and our numerical experiments show a significant variance reduction.

Exposé de 20 minutes Véronique Maume-Deschamps (ISFA, Université de Lyon 1) en collaboration avec Peggy Cénac et Clémentine Prieur Some multivariate risk indicators, estimation and application to optimal reserve allocation transparents

We consider a vectorial risk process X_i=(X_i^1,...,X_i^d). X_i^k corresponds to the gains of the k-th business line during the i-th period : X_i^k = G_i^k-L_i^k where G_i^k denotes the incomes and L_i^k denotes the losses. We are interested in the cumulative gain : Y_i^k = \sum_{p=1}^i X_p^k. For a given period i, the variables (X_i^k)_{k=1,...,d} may be dependent. Some temporal dependence (with respect to index i) may also be taken into account. Given an initial capital u we assume that it is allocated to each line of business. Let u_i denotes the initial capital of the i-th line of business, u_1+...+ u_d =u. We aim to optimize the capital allocation with respect to some risk indicator. We introduce new risks indicators, considering that the main risk drivers for the overall company have been identified and the global solvency capital requirement has been computed. These new indicators reveal the marginal solvency capitals for each line of business. A way to avoid as far as possible that some lines of business become insolvent too often could be to minimize these risk indicators, under a fixed total capital constraint. These might be achieved if some capital fungibility between lines of business or between entities is possible. One possible way to define optimality of the global reserve allocation is to minimize the expected sum of the penalties that each line of business would have to pay due to its temporary potential insolvency. A multivariate risk indicator that takes into account the dependence structure may be :
B(u_1,...,u_d) = \sum_{k=1}^d E(\sum_{p=1}^n\Id_{\{R_p^k<0 \}} \Id_{\{\sum_{k=1}^d R_p^k>0 \}}),
with R_p^k = u_k+Y_p^k. This risk indicator gives an indication on some average time to ruin, it has been introduced in Loisel 2004 and is not, in general, convex. We shall consider, given a differentiable and convex function g_k:\R~\rightarrow~\R satisfying g_k(x)\geq 0 for x\leq 0, k=1,...,d:
I(u_1,...,u_d) = \sum_{k=1}^d E(\sum_{p=1}^n g_k(R_p^k) \Id_{\{R_p^k<0 \}} \Id_{\{\sum_{k=1}^d R_p^k>0 \}}).
The function g_k represents the cost that the k-th business branch has to pay if it becomes insolvable. The problem is to find a minimum of I under the constraint v_1+...+v_d = u. Formally, we are looking for u^\star such that
I(u^\star) = \inf_{v_1+...+v_d = u} I(v).
Unless for some very specific models, we are not able to compute explicitly u^\star which realize the minimum. Thus, we propose to solve this minimization problem by using stochastic algorithms. We provide a proof of almost sure convergence of our algorithm as well as an estimation of the probability error. Then we perform some simulations.

Exposé de 20 minutes Pierre-André Zitt (IMB Dijon) Algorithmes stochastiques et diffusions : l'étude par inégalités fonctionnelles transparents

Les algorithmes classiques que sont le recuit simulé ou l'algorithme de Robbins-Monro ont des équivalents en temps continu, qui sont des processus de diffusion non-homogènes. Pour étudier la convergence (et donc l'efficacité) de ces versions continues, on peut faire appel à des inégalités fonctionnelles (inégalité de Poincaré, de Sobolev logarithmique ou leurs généralisations). Nous verrons quelques exemples de ces approches, en particulier pour montrer la convergence du recuit simulé dans un paysage de potentiel à croissance lente.