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 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 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.
@article{AIHPB_2014__50_3_946_0,
author = {Crisan, Dan and Kurtz, Thomas G. and Lee, Yoonjung},
title = {Conditional distributions, exchangeable particle systems, and stochastic partial differential equations},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
volume = {50},
year = {2014},
pages = {946-974},
doi = {10.1214/13-AIHP543},
mrnumber = {3224295},
zbl = {06340414},
language = {en},
url = {http://dml.mathdoc.fr/item/AIHPB_2014__50_3_946_0}
}
Crisan, Dan; Kurtz, Thomas G.; Lee, Yoonjung. Conditional distributions, exchangeable particle systems, and stochastic partial differential equations. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) pp. 946-974. doi : 10.1214/13-AIHP543. http://gdmltest.u-ga.fr/item/AIHPB_2014__50_3_946_0/
[1] . A diffussion model for electricity prices. Math. Finance 12 (2002) 287-298. | Zbl 1046.91041
[2] . Martingales, the Malliavin calculus and hypoellipticity under general Hörmander's conditions. Z. Wahrsch. Verw. Gebiete 56 (1981) 469-505. ISSN 0044-3719. DOI:10.1007/BF00531428. Available at http://dx.doi.org.ezproxy.library.wisc.edu/10.1007/BF00531428. | MR 621660 | Zbl 0445.60049
[3] and . Diffusions conditionnelles. I. Hypoellipticité partielle. J. Funct. Anal. 44 (1981) 174-211. ISSN 0022-1236. | MR 642916 | Zbl 0475.60061
[4] . Malliavin calculus applications to the study of nonlinear filtering. In The Oxford Handbook of Nonlinear Filtering 195-231. D. Crisan and B. Rozovsky (Eds). Oxford Univ. Press, Oxford, 2011. | MR 2884597 | Zbl 1227.60053
[5] and . Hypoellipticity theorems and conditional laws. Z. Wahrsch. Verw. Gebiete 65 (1984) 573-597. ISSN 0044-3719. | MR 736147 | Zbl 0524.35028
[6] and . The support of the density of a filter in the uncorrelated case. In Stochastic Partial Differential Equations and Applications, II (Trento, 1988). Lecture Notes in Math. 1390 33-41. Springer, Berlin, 1989. | MR 1019591 | Zbl 0674.60043
[7] , and . Multi-item auctions. Journal of Political Economy 94 (1986) 863-872. ISSN 00223808. Available at http://www.eecs.harvard.edu/~parkes/cs286r/spring02/papers/dgs86.pdf.
[8] and . Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York, 1986. ISBN 0-471-08186-8. | MR 838085 | Zbl 1089.60005
[9] and . A microeconomic approach to diffusion models for stock prices. Math. Finance 3 (1993) 1-23. ISSN 1467-9965. DOI:10.1111/j.1467-9965.1993.tb00035.x. Available at http://dx.doi.org/10.1111/j.1467-9965.1993.tb00035.x. | Zbl 0884.90027
[10] , and . Stock price fluctuation as a diffusion in a random environment [and discussion]. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 347 (1994) 471-483. | MR 1407254 | Zbl 0822.90022
[11] and . Market volatility and feedback effects from dynamic hedging. Math. Finance 7 (1997) 351-374. | MR 1482708 | Zbl 1020.91023
[12] . Financial price fluctuations in a stock market model with many interacting agents. Econom. Theory 25 (2005) 917-932. | MR 2209541 | Zbl 1134.91426
[13] . Some inequalities for martingales and stochastic convolutions. Stoch. Anal. Appl. 4 (1986) 329-339. ISSN 0736-2994. Available at http://www.informaworld.com/10.1080/07362998608809094. | MR 857085 | Zbl 0622.60066
[14] and . Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type. Probab. Theory Related Fields 146 (2010) 189-222. ISSN 0178-8051. DOI:10.1007/s00440-008-0188-0. | MR 2550362 | Zbl 1189.60123
[15] . Filtering equations for partially observable diffusion processes with Lipschitz continuous coefficients. In Oxford Handbook of Nonlinear Filtering. Oxford Univ. Press, Oxford, 2010. | MR 2884596 | Zbl 1223.93112
[16] . Stochastic differential equations and stochastic flows of diffeomorphisms. In École d'été de probabilités de Saint-Flour, XII-1982. Lecture Notes in Math. 1097 143-303. Springer, Berlin, 1984. | MR 876080 | Zbl 0554.60066
[17] . Averaging for martingale problems and stochastic approximation. In Applied Stochastic Analysis (New Brunswick, NJ, 1991). Lecture Notes in Control and Inform. Sci. 177 186-209. Springer, Berlin, 1992. | MR 1169928
[18] and . Weak convergence of stochastic integrals and differential equations. II. Infinite-dimensional case. In Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995). Lecture Notes in Math. 1627 197-285. Springer, Berlin, 1996. | MR 1431303 | Zbl 0844.00022
[19] and . Particle representations for a class of nonlinear SPDEs. Stochastic Process. Appl. 83 (1999) 103-126. ISSN 0304-4149. | MR 1705602 | Zbl 0996.60071
[20] and . Numerical solutions for a class of SPDEs with application to filtering. In Stochastics in Finite and Infinite Dimensions. Trends Math. 233-258. Birkhäuser Boston, Boston, MA, 2001. | MR 1797090 | Zbl 0991.60053
[21] and . The partial Malliavin calculus and its application to nonlinear filtering. Stochastics 12 (1984) 83-142. | MR 747781 | Zbl 0567.60046
[22] . Modeling the random demand curve for stock: An interacting particle representation approach. Ph.D. thesis, Univ. Wisconsin-Madison, 2004. Available at http://www.people.fas.harvard.edu/~lee48/research.html.
[23] , and . Présentation unifiée de certaines inégalités de la théorie des martingales. In Seminar on Probability, XIV (Paris, 1978/1979) (French). Lecture Notes in Math. 784 26-52. Springer, Berlin, 1980. With an appendix by Lenglart. | Numdam | MR 580107 | Zbl 0427.60042
[24] and . The partial Malliavin calculus. In Séminaire de Probabilités, XXIII. Lecture Notes in Math. 1372 362-381. Springer, Berlin, 1989. DOI:10.1007/BFb0083986. Available at http://dx.doi.org/10.1007/BFb0083986. | Numdam | MR 1022924 | Zbl 0738.60055
[25] and . The assignment game. I. The core. Internat. J. Game Theory 1 (1972) 111-130. ISSN 0020-7276. | MR 311290 | Zbl 0236.90078
[26] and . General Black-Scholes models accounting for increased market volatility from hedging strategies. Appl. Math. Finance 5 (1998) 45-82. Available at http://www.informaworld.com/10.1080/135048698334727. | Zbl 1009.91023