Conditional distributions, exchangeable particle systems, and stochastic partial differential equations. (English. French summary) Zbl 1306.60086
The authors study stochastic partial differential equations whose solutions are probability-measure-valued processes. 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 the present paper, the authors consider a financial model of asset pricing with an infinite collection of competing traders. Each trader’s valuation of the assets is modeled by an SDE, and the infinite system of exchangeable SDEs is coupled through a common noise process and through the asset prices. The solution of the system can be interpreted as the conditional distribution of the solution of a single SDE given the common noise and the price process. Under certain assumptions, this conditional distribution is proved to be absolutely continuous with respect to Lebesgue measure with a strictly positive density.
In the present paper, the authors consider a financial model of asset pricing with an infinite collection of competing traders. Each trader’s valuation of the assets is modeled by an SDE, and the infinite system of exchangeable SDEs is coupled through a common noise process and through the asset prices. The solution of the system can be interpreted as the conditional distribution of the solution of a single SDE given the common noise and the price process. Under certain assumptions, this conditional distribution is proved to be absolutely continuous with respect to Lebesgue measure with a strictly positive density.
Reviewer: Dora Seleši (Novi Sad)
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60G09 | Exchangeability for stochastic processes |
| 60G35 | Signal detection and filtering (aspects of stochastic processes) |
| 60J25 | Continuous-time Markov processes on general state spaces |
| 91B25 | Asset pricing models (MSC2010) |
| 91B26 | Auctions, bargaining, bidding and selling, and other market models |
Keywords:
stochastic partial differential equations; exchangeable systems; conditional distributions; quantile processes; filtering equations; measure-valued processes; auction based pricing; assignment gamesReferences:
| [1] | M. T. Barlow. A diffussion model for electricity prices. Math. Finance 12 (2002) 287-298. · Zbl 1046.91041 · doi:10.1111/j.1467-9965.2002.tb00125.x |
| [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. . Available at http://dx.doi.org.ezproxy.library.wisc.edu/10.1007/BF00531428 . · Zbl 0445.60049 · doi: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. · Zbl 0475.60061 · doi:10.1016/0022-1236(81)90010-0 |
| [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. · Zbl 1227.60053 |
| [5] | M. Chaleyat-Maurel and D. Michel. Hypoellipticity theorems and conditional laws. Z. Wahrsch. Verw. Gebiete 65 (1984) 573-597. ISSN 0044-3719. · Zbl 0524.35028 · doi:10.1007/BF00531840 |
| [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. · Zbl 0674.60043 · doi:10.1007/BFb0083934 |
| [7] | G. Demange, D. Gale and M. Sotomayor. Multi-item auctions. Journal of Political Economy 94 (1986) 863-872. ISSN 00223808. Available at . |
| [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. · Zbl 0592.60049 |
| [9] | H. Föllmer and M. Schweizer. A microeconomic approach to diffusion models for stock prices. Math. Finance 3 (1993) 1-23. ISSN 1467-9965. . Available at http://dx.doi.org/10.1111/j.1467-9965.1993.tb00035.x . · Zbl 0884.90027 · doi: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. · Zbl 0822.90022 · doi:10.1098/rsta.1994.0057 |
| [11] | R. Frey and A. Stremme. Market volatility and feedback effects from dynamic hedging. Math. Finance 7 (1997) 351-374. · Zbl 1020.91023 · doi:10.1111/1467-9965.00036 |
| [12] | U. Horst. Financial price fluctuations in a stock market model with many interacting agents. Econom. Theory 25 (2005) 917-932. · Zbl 1134.91426 · doi:10.1007/s00199-004-0500-x |
| [13] | A. Ichikawa. Some inequalities for martingales and stochastic convolutions. Stoch. Anal. Appl. 4 (1986) 329-339. ISSN 0736-2994. Available at . · Zbl 0622.60066 · doi: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. . Available at http://dx.doi.org/10.1007/s00440-008-0188-0 . · Zbl 1189.60123 · doi: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. · Zbl 1387.35149 |
| [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. · Zbl 0554.60066 |
| [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. · doi:10.1007/BFb0007058 |
| [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. · Zbl 0862.60042 · doi:10.1007/BFb0093181 |
| [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. · Zbl 0996.60071 · doi:10.1016/S0304-4149(99)00024-1 |
| [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. · Zbl 0991.60053 · doi:10.1007/978-1-4612-0167-0_13 |
| [21] | S. Kusuoka and D. Stroock. The partial Malliavin calculus and its application to nonlinear filtering. Stochastics 12 (1984) 83-142. · Zbl 0567.60046 · doi:10.1080/17442508408833296 |
| [22] | Y. Lee. Modeling the random demand curve for stock: An interacting particle representation approach. Ph.D. thesis, Univ. Wisconsin-Madison, 2004. Available at . |
| [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. · Zbl 0427.60042 |
| [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. . Available at http://dx.doi.org/10.1007/BFb0083986 . · Zbl 0738.60055 · doi: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. · Zbl 0236.90078 · doi:10.1007/BF01753437 |
| [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 . · Zbl 1009.91023 · doi:10.1080/135048698334727 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
