Conditional distributions, exchangeable particle systems, and stochastic partial differential equations

Crisan, Dan; Kurtz, Thomas G.; Lee, Yoonjung

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 3, pp. 946-974.

On considère des équations aux dérivées partielles stochastiques (EDPS) dont les solutions sont des processus à valeurs dans les mesures de probabilité. Des processus à valeurs mesures de ce type apparaissent naturellement comme des mesures de De Finetti de systèmes infinis de particules échangeables et comme solutions de problèmes de filtrage. En particulier nous considérons un modèle de détermination du prix d’un actif par une famille de traders en compétition. L’évaluation de chaque trader sur l’actif est donnée par la solution d’une équation différentielle stochastique et ce système infini d’EDSs, supposé échangeable, est couplé par un bruit commun et par les prix des actifs. Dans le cadre le plus simple à un seul actif, le prix d’équilibre du marché à tout temps $t$ est donné par un quantile de la mesure de De Finetti déterminé par les évaluations du trader individuel. Dans le cadre à plusieurs actifs, les prix sont donnés essentiellement par la solution d’un problème d’attribution introduit par Shapley et Shubik. L’existence de solutions pour le système échangeable infini est obtenue par un argument d’approximation qui nécessite la dépendance continue des distributions des prix par rapport à la mesure de De Finetti associée. Ceci est vrai si la mesure de De Finetti donne une masse positive à tout ouvert non-vide. La solution de l’EDPS satisfaite par la mesure de De Finetti peut être interprétée comme la distribution conditionnelle de la solution d’une seule EDS donnée par le bruit commun et par le processus du prix. Sous des conditions faibles de non-dégénérescence des coefficients de l’EDS, on montre que la distribution conditionnelle donne une masse positive à tout ouvert non-vide, et sous des conditions légèrement plus fortes, on prouve qu’elle est absolument continue par rapport à la mesure de Lebesgue avec une densité strictement positive. Les résultats sur la distribution conditionnelle constituent la contribution technique principale et ils peuvent être aussi utilisés pour étudier les propriétés de la solution de l’équation de filtrage non-linéaire dans un cadre où le bruit du signal et celui de l’observation sont corrélés.

Stochastic partial differential equations (SPDEs) whose solutions are probability-measure-valued processes are considered. Measure-valued processes of this type arise naturally as de Finetti measures of infinite exchangeable systems of particles and as the solutions for filtering problems. In particular, we consider a model of asset price determination by an infinite collection of competing traders. Each trader’s valuations of the assets are given by the solution of a stochastic differential equation, and the infinite system of SDEs, assumed to be exchangeable, is coupled through a common noise process and through the asset prices. In the simplest, single asset setting, the market clearing price at any time $t$ is given by a quantile of the de Finetti measure determined by the individual trader valuations. In the multi-asset setting, the prices are essentially given by the solution of an assignment game introduced by Shapley and Shubik. Existence of solutions for the infinite exchangeable system is obtained by an approximation argument that requires the continuous dependence of the prices on the determining de Finetti measures which is ensured if the de Finetti measures charge every open set. The solution of the SPDE satisfied by the de Finetti measures can be interpreted as the conditional distribution of the solution of a single stochastic differential equation given the common noise and the price process. Under mild nondegeneracy conditions on the coefficients of the stochastic differential equation, the conditional distribution is shown to charge every open set, and under slightly stronger conditions, it is shown to be absolutely continuous with respect to Lebesgue measure with strictly positive density. The conditional distribution results are the main technical contribution and can also be used to study the properties of the solution of the nonlinear filtering equation within a framework that allows for the signal noise and the observation noise to be correlated.

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