Stochastic Approximation with Averaging Innovation Applied to Finance

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

Optimal posting price of limit orders: learning by trading

Considering that a trader or a trading algorithm interacting with markets during continuous auctions can be modeled by an iterating procedure adjusting the price at which he posts orders at a given rhythm, this paper proposes a procedure minimizing his costs. We prove the a.s. convergence of the algorithm under assumptions on the cost function and give some practical criteria on model parameters to ensure that the conditions to use the algorithm are fulfilled (using notably the co-monotony principle). We illustrate our results with numerical experiments on both simulated data and using a financial market dataset

Optimal split of orders across liquidity pools: a stochastic algorithm approach

Evolutions of the trading landscape lead to the capability to exchange the same financial instrument on different venues. Because of liquidity issues, the trading firms split large orders across several trading destinations to optimize their execution. To solve this problem we devised two stochastic recursive learning procedures which adjust the proportions of the order to be sent to the different venues, one based on an optimization principle, the other on some reinforcement ideas. Both procedures are investigated from a theoretical point of view: we prove a.s. convergence of the optimization algorithm under some light ergodic (or "averaging") assumption on the input data process. No Markov property is needed. When the inputs are i.i.d. we show that the convergence rate is ruled by a Central Limit Theorem. Finally, the mutual performances of both algorithms are compared on simulated and real data with respect to an "oracle" strategy devised by an "insider" who knows a priori the executed quantities by every venues

Optimal split of orders across liquidity pools: a stochastic algorithm approach

Evolutions of the trading landscape lead to the capability to exchange the same financial instrument on different venues. Because of liquidity issues, the trading firms split large orders across several trading destinations to optimize their execution. To solve this problem we devised two stochastic recursive learning procedures which adjust the proportions of the order to be sent to the different venues, one based on an optimization principle, the other on some reinforcement ideas. Both procedures are investigated from a theoretical point of view: we prove a.s. convergence of the optimization algorithm under some light ergodic (or "averaging") assumption on the input data process. No Markov property is needed. When the inputs are i.i.d. we show that the convergence rate is ruled by a Central Limit Theorem. Finally, the mutual performances of both algorithms are compared on simulated and real data with respect to an "oracle" strategy devised by an "insider" who knows a priori the executed quantities by every venues.Asset allocation, Stochastic Lagrangian algorithm, reinforcement principle, monotone dynamic system

A New Perspective on Mary of Hungary’s Labours of Hercules Tapestries (Patrimonio Nacional, series 23)

peer reviewedThis article aims to shed new light on the twelve-piece set purchased in 1535 by the Governor of the former Netherlands, Mary of Hungary, from the Dermoyen workshop in Brussels. We shall see how this series’ success led the Flemish workshops to carry out numerous re-editions until the end of the century. The paper focuses on four main themes: the existing pieces and their iconography, the manufacture that woven them, the artists who provided the models to the weavers and finally the patron who chose the subject and gave these tapestries a precise function in her interior

Analyse d'Algorithmes Stochastiques Appliqués à la Finance

This thesis is about stochastic approximation analysis and application in Finance. In the first part, a convergence result for stochastic approximation where the innovations satisfy averaging assumptions with some rate is established. It is applied to different types of innovations and illustrated on examples mainly motivated by Finance. A result on "universal" rate of convergence is then presented when the innovations are uniformly distributed and compared to those obtained in the i.i.d. framework. The second part is devoted to applications. First an optimal allocation problem applied to dark pools is studied. The execution of the maximum of the desired quantity leads to the design of a constrained stochastic algorithm studied in the i.i.d. and averaging frameworks. The next chapter presents a constrained optimization stochastic algorithm with projection to find the optimal posting distance in a limit order book by minimizing the execution cost of a given quantity. Parameter implicitation and calibration in financial models using stochastic approximation are then studied and illustrated by examples of applications on Black-Scholes, Merton and pseudo-CEV models. The last chapter is about stochastic approximation application to randomized urn models used in clinical trials. Thanks to ODE and SDE methods, the consistency and asymptotic normality results of Bai and Hu are retrieved under less stringent assumptions on the generating matrices.Cette thèse porte sur l'analyse d'algorithmes stochastiques et leur application en Finance notamment et est composée de deux parties. Dans la première partie, nous présentons un résultat de convergence pour des algorithmes stochastiques où les innovations vérifient une hypothèse de moyennisation avec une certaine vitesse. Nous l'appliquons ensuite à différents types d'innovations (suites i.i.d., suites à discrépance faible, chaînes de Markov homogènes, fonctionnelles de processus \alpha-mélangeant) et nous l'illustrons à l'aide d'exemples motivés principalement par la Finance. Nous établissons ensuite un résultat de vitesse ''universelle'' de convergence dans le cadre d'innovations équiréparties dans [0,1]^q et nous confrontons nos résultats à ceux obtenus dans le cadre i.i.d.. La seconde partie est consacrée aux applications. Nous présentons d'abord un problème d'allocation optimale appliqué au cas d'un nouveau type de place de trading: les {\em dark pools}. Ces places proposent un prix d'achat (ou de vente) certain, mais n'assurent pas le volume délivré. Le but est alors d'exécuter le maximum de la quantité souhaitée sur ces places. Ceci mène à la construction d'un algorithme stochastique sous contraintes à l'aide du Lagrangien que nous étudions dans les cadres d'innovations i.i.d. et moyennisantes. Le chapitre suivant présente un algorithme d'optimisation pour trouver la meilleure distance de placement d'ordres limites: il s'agit de minimiser le coût d'exécution d'une quantité donnée. Ceci mène à la construction d'un algorithme stochastique sous contraintes avec projection. Pour assurer l'existence et l'unicité de l'équilibre, des critères suffisants sur certains paramètres du modèle sont obtenus à l'aide d'un principe de monotonie opposée pour les diffusions unidimensionnelles. Le chapitre suivant porte sur l'implicitation et la calibration de paramètres dans des modèles financiers. La première technique mène à un algorithme de recherche de zéro et la seconde à une méthode de gradient stochastique. Nous illustrons ces deux techniques par des exemples d'applications sur 3 modèles: le modèle de Black-Scholes, le modèle de Merton et le modèle pseudo-CEV. Enfin le dernier chapitre porte sur l'application des algorithmes stochastiques dans le cadre de modèles d'urnes aléatoires utilisés en essais cliniques. A l'aide des méthodes de l'EDO et de l'EDS, nous retrouvons les résultats de consistance (convergence p.s.) et de normalité asymptotique (TCL) de Bai et Hu mais sous des hypothèses plus faibles sur les matrices génératrices. Nous étudions aussi un modèle ''multi-bras'' pour lequel nous retrouvons le résultat de convergence p.s. et nous montrons un nouveau résultat de normalité asymptotique par simple application du TCL pour les algorithmes stochastiques

Collecting Netherlandish Tapestry in Germany during the 16th century

The luxury industry of tapestry weaving has long been associated with the rich and powerful rulers of the European countries. The huge collection of Flemish tapestries amassed by the Habsburg is now well documented. Stimulated by the example of the monarchs, the leading nobility were also important patrons of the workshops in the Low Countries. Some collections are now a little better known but not fully investigated, such as those of Frederick the Wise, Elector of Saxony (r. 1486-1525); Günther XLI, Count of Schwarzburg-Arnstadt (r. 1552-1583); or Albrecht V, Duke of Bavaria (r. 1550-1579). Another keen clients of the Netherlandish workshops were the Counts Palatine and Electors of the Rhine (especially Otto-Henry and Frederick III), whose made substantial purchases of Netherlandish tapestries to decorate their residences at Neuburg and Heidelberg. In this paper, I will determine what survives and is acceptable as secure evidence for the presence of Flemish tapestry in Germany. I will exploit that evidence to give an overview of the different types of owners and their collections, the character of the works they commissioned or bought, their means of collecting, their agents (both German and Flemish), their likely reasons or intentions in collecting, and the use to which they put Flemish tapestry. Our approach will provide new perspectives for further research