Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Dan Crisan, Thomas G. Kurtz, and Yoonjung Lee

Full-text: Open access

Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 3 (2014), 946-974.

Dates
First available in Project Euclid: 20 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1403277004

Digital Object Identifier
doi:10.1214/13-AIHP543

Mathematical Reviews number (MathSciNet)
MR3224295

Zentralblatt MATH identifier
1306.60086

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60G09: Exchangeability 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 60J25: Continuous-time Markov processes on general state spaces

Keywords
Exchangeable systems Conditional distributions Stochastic partial differential equations Quantile processes Filtering equations Measure-valued processes Auction based pricing Assignment games

Citation

Crisan, Dan; Kurtz, Thomas G.; Lee, Yoonjung. Conditional distributions, exchangeable particle systems, and stochastic partial differential equations. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 3, 946--974. doi:10.1214/13-AIHP543. https://projecteuclid.org/euclid.aihp/1403277004


Export citation

References

  • [1] M. T. Barlow. A diffussion model for electricity prices. Math. Finance 12 (2002) 287–298.
  • [2] J.-M. Bismut. 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.
  • [3] J.-M. Bismut and D. Michel. Diffusions conditionnelles. I. Hypoellipticité partielle. J. Funct. Anal. 44 (1981) 174–211. ISSN 0022-1236.
  • [4] M. Chaleyat-Maurel. 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.
  • [5] M. Chaleyat-Maurel and D. Michel. Hypoellipticity theorems and conditional laws. Z. Wahrsch. Verw. Gebiete 65 (1984) 573–597. ISSN 0044-3719.
  • [6] M. Chaleyat-Maurel and D. Michel. 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.
  • [7] G. Demange, D. Gale and M. Sotomayor. 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] S. N. Ethier and T. G. Kurtz. 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.
  • [9] H. Föllmer and M. Schweizer. 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.
  • [10] H. Föllmer, W. Cheung and M. A. H. Dempster. 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.
  • [11] R. Frey and A. Stremme. Market volatility and feedback effects from dynamic hedging. Math. Finance 7 (1997) 351–374.
  • [12] U. Horst. Financial price fluctuations in a stock market model with many interacting agents. Econom. Theory 25 (2005) 917–932.
  • [13] A. Ichikawa. 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.
  • [14] P. M. Kotelenez and T. G. Kurtz. 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. Available at http://dx.doi.org/10.1007/s00440-008-0188-0.
  • [15] N. V. Krylov. Filtering equations for partially observable diffusion processes with Lipschitz continuous coefficients. In Oxford Handbook of Nonlinear Filtering. Oxford Univ. Press, Oxford, 2010.
  • [16] H. Kunita. 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.
  • [17] T. G. Kurtz. 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.
  • [18] T. G. Kurtz and P. E. Protter. 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.
  • [19] T. G. Kurtz and J. Xiong. Particle representations for a class of nonlinear SPDEs. Stochastic Process. Appl. 83 (1999) 103–126. ISSN 0304-4149.
  • [20] T. G. Kurtz and J. Xiong. 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.
  • [21] S. Kusuoka and D. Stroock. The partial Malliavin calculus and its application to nonlinear filtering. Stochastics 12 (1984) 83–142.
  • [22] Y. Lee. 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] E. Lenglart, D. Lépingle and M. Pratelli. 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.
  • [24] D. Nualart and M. Zakai. 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.
  • [25] L. S. Shapley and M. Shubik. The assignment game. I. The core. Internat. J. Game Theory 1 (1972) 111–130. ISSN 0020-7276.
  • [26] K. R. Sircar and G. Papanicolaou. 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.