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

Dan Crisan; Thomas G. Kurtz; Yoonjung Lee

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 3, page 946-974
  • ISSN: 0246-0203

Abstract

top
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.

How to cite

top

Crisan, Dan, Kurtz, Thomas G., and Lee, Yoonjung. "Conditional distributions, exchangeable particle systems, and stochastic partial differential equations." Annales de l'I.H.P. Probabilités et statistiques 50.3 (2014): 946-974. <http://eudml.org/doc/271963>.

@article{Crisan2014,
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.},
author = {Crisan, Dan, Kurtz, Thomas G., Lee, Yoonjung},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {exchangeable systems; conditional distributions; stochastic partial differential equations; quantile processes; filtering equations; measure-valued processes; auction based pricing; assignment games},
language = {eng},
number = {3},
pages = {946-974},
publisher = {Gauthier-Villars},
title = {Conditional distributions, exchangeable particle systems, and stochastic partial differential equations},
url = {http://eudml.org/doc/271963},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Crisan, Dan
AU - Kurtz, Thomas G.
AU - Lee, Yoonjung
TI - Conditional distributions, exchangeable particle systems, and stochastic partial differential equations
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 3
SP - 946
EP - 974
AB - 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.
LA - eng
KW - exchangeable systems; conditional distributions; stochastic partial differential equations; quantile processes; filtering equations; measure-valued processes; auction based pricing; assignment games
UR - http://eudml.org/doc/271963
ER -

References

top
  1. [1] M. T. Barlow. A diffussion model for electricity prices. Math. Finance12 (2002) 287–298. Zbl1046.91041
  2. [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. Zbl0445.60049MR621660
  3. [3] J.-M. Bismut and D. Michel. Diffusions conditionnelles. I. Hypoellipticité partielle. J. Funct. Anal. 44 (1981) 174–211. ISSN 0022-1236. Zbl0475.60061MR642916
  4. [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. Zbl1227.60053MR2884597
  5. [5] M. Chaleyat-Maurel and D. Michel. Hypoellipticity theorems and conditional laws. Z. Wahrsch. Verw. Gebiete 65 (1984) 573–597. ISSN 0044-3719. Zbl0524.35028MR736147
  6. [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. Zbl0674.60043MR1019591
  7. [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. [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. Zbl1089.60005MR838085
  9. [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. Zbl0884.90027
  10. [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. Zbl0822.90022MR1407254
  11. [11] R. Frey and A. Stremme. Market volatility and feedback effects from dynamic hedging. Math. Finance7 (1997) 351–374. Zbl1020.91023MR1482708
  12. [12] U. Horst. Financial price fluctuations in a stock market model with many interacting agents. Econom. Theory25 (2005) 917–932. Zbl1134.91426MR2209541
  13. [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. Zbl0622.60066MR857085
  14. [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. Zbl1189.60123MR2550362
  15. [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. Zbl1223.93112MR2884596
  16. [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. Zbl0554.60066MR876080
  17. [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. MR1169928
  18. [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. Zbl0844.00022MR1431303
  19. [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. Zbl0996.60071MR1705602
  20. [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. Zbl0991.60053MR1797090
  21. [21] S. Kusuoka and D. Stroock. The partial Malliavin calculus and its application to nonlinear filtering. Stochastics12 (1984) 83–142. Zbl0567.60046MR747781
  22. [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. [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. Zbl0427.60042MR580107
  24. [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. Zbl0738.60055MR1022924
  25. [25] L. S. Shapley and M. Shubik. The assignment game. I. The core. Internat. J. Game Theory 1 (1972) 111–130. ISSN 0020-7276. Zbl0236.90078MR311290
  26. [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. Zbl1009.91023

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.