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Probability measure-valued polynomial diffusions. (English) Zbl 1467.60062

Summary: We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming-Viot process is a particular example. The defining property of finite dimensional polynomial processes considered in [C. Cuchiero et al., Finance Stoch. 16, No. 4, 711–740 (2012; Zbl 1270.60079); D. Filipović and M. Larsson, Finance Stoch. 20, No. 4, 931–972 (2016; Zbl 1386.60237)] is transferred to this infinite dimensional setting. This leads to a representation of conditional marginal moments via a finite dimensional linear PDE, whose spatial dimension corresponds to the degree of the moment. As a result, the tractability of finite dimensional polynomial processes are preserved in this setting. We also obtain a representation of the corresponding extended generators, and prove well-posedness of the associated martingale problems. In particular, uniqueness is obtained from the duality relationship with the PDEs mentioned above.

MSC:

60J68 Superprocesses
60G57 Random measures
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References:

[1] A. Ahdida and A. Alfonsi. A mean-reverting SDE on correlation matrices. Stochastic Processes and their Applications, 123(4):1472-1520, 2013. · Zbl 1271.65014
[2] D. Bakry and M. Émery. Diffusions hypercontractives. In Séminaire de Probabilités XIX 1983/84, pages 177-206. Springer, 1985. · Zbl 0561.60080
[3] C. Beck, S. Becker, P. Grohs, N. Jaafari, and A. Jentzen. Solving stochastic differential equations and Kolmogorov equations by means of deep learning. PreprintarXiv:1806.00421, 2018. · Zbl 1490.65006
[4] D. P. Bertsekas. Nonlinear Programming. Athena Scientific, 1995. · Zbl 0935.90037
[5] A. Boumezoued, H. Hardy, N. El Karoui, and S. Arnold. Cause-of-death mortality: What can be learned from population dynamics? Insurance: Mathematics and Economics, 78:301-315, 2018. · Zbl 1400.91242
[6] P. Courrège. Sur la forme intégro-différentielle des opérateurs de \(C^{\infty }_k\) dans \(C\) satisfaisant au principe du maximum. Séminaire Brelot-Choquet-Deny (Théorie du Potentiel), 10(2), 1965. · Zbl 0155.17402
[7] C. Cuchiero. Polynomial processes in stochastic portfolio theory. Stochastic processes and their applications, Forthcoming, 10.1016/j.spa.2018.06.007, 2018.
[8] C. Cuchiero, M. Keller-Ressel, and J. Teichmann. Polynomial processes and their applications to mathematical finance. Finance and Stochastics, 16:711-740, 2012. · Zbl 1270.60079
[9] C. Cuchiero, M. Larsson, and S. Svaluto-Ferro. Polynomial jump-diffusions on the unit simplex. Annals of Applied Probability, 28(4):2451-2500, 2018. · Zbl 1402.60105
[10] G. Da Prato and H. Frankowska. Invariance of stochastic control systems with deterministic arguments. Journal of Differential Equations, 200(1):18-52, 2004. · Zbl 1066.93049
[11] D. Dawson. Measure-valued Markov processes. In École d’Été de Probabilités de Saint-Flour XXI-1991, Lecture Notes in Mathematics, pages 1-260. Springer, 1993. · Zbl 0799.60080
[12] D. Dawson and K. Hochberg. Wandering Random Measures in the Fleming-Viot Model. Annals of Probability, 10(3):554-580, 1982. · Zbl 0492.60045
[13] D. Duffie, D. Filipović, and W. Schachermayer. Affine processes and applications in finance. Annals of applied probability, pages 984-1053, 2003. · Zbl 1048.60059
[14] A. Etheridge. Some Mathematical Models from Population Genetics: École D’Été de Probabilités de Saint-Flour XXXIX-2009. Lecture Notes in Mathematics. Springer, 2011. · Zbl 1320.92003
[15] S. N. Ethier and T. G. Kurtz. The Infinitely-Many-Alleles Model with Selection as a Measure-Valued Diffusion, pages 72-86. Springer Berlin Heidelberg, Berlin, Heidelberg, 1987. · Zbl 0644.92011
[16] S. N. Ethier and T. G. Kurtz. Fleming-Viot processes in population genetics. SIAM Journal on Control and Optimization, 31(2):345-386, 1993. · Zbl 0774.60045
[17] S. N. Ethier and T.G. Kurtz. Markov Processes: Characterization and Convergence. Wiley Series in Probability and Statistics. Wiley, 2 edition, 2005.
[18] R. Fernholz. Stochastic Portfolio Theory. Applications of Mathematics. Springer-Verlag, New York, 2002. · Zbl 1049.91067
[19] R. Fernholz and I. Karatzas. Relative arbitrage in volatility-stabilized markets. Annals of Finance, 1(2):149-177, 2005. · Zbl 1233.91244
[20] R. Fernholz and I. Karatzas. Stochastic portfolio theory: an overview. Handbook of numerical analysis, 15:89-167, 2009. · Zbl 1180.91267
[21] D. Filipović and M. Larsson. Polynomial diffusions and applications in finance. Finance and Stochastics, 20(4):931-972, 2016. · Zbl 1386.60237
[22] D. Filipović and M. Larsson. Polynomial jump-diffusion models. ArXiv e-prints, 2017. URL https://arxiv.org/abs/1711.08043. · Zbl 1386.60237
[23] W. H. Fleming and M. Viot. Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J., 28(5):817-843, 1979. · Zbl 0444.60064
[24] W. Hoh. Pseudo differential operators generating Markov processes. Habilitationsschrift, Univeristät Bielefeld, 1998. · Zbl 0922.47045
[25] M. Kimura. Diffusion models in population genetics. Journal of Applied Probability, 1(2):177-232, 1964. · Zbl 0134.38103
[26] A. Klenke. Probability Theory: A Comprehensive Course. Universitext. Springer London, 2 edition, 2013. · Zbl 1295.60001
[27] P. Kotelenez and T. Kurtz. Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type. Probability Theory and Related Fields, 146(1):189, 2008.
[28] T. Kurtz and J. Xiong. Particle representations for a class of nonlinear SPDEs. Stochastic Processes and their Applications, 83, 1999. · Zbl 0996.60071
[29] E. Regazzini, A. Guglielmi, and G. Di Nunno. Theory and numerical analysis for exact distributions of functionals of a Dirichlet process. Ann. Statist., 30(5):1376-1411, 2002. · Zbl 1018.62011
[30] E. Regazzini, A. Lijoi, and I. Prünster. Distributional results for means of normalized random measures with independent increments. Ann. Statist., 31(2):560-585, 2003. Dedicated to the memory of Herbert E. Robbins. · Zbl 1068.62034
[31] M. Shkolnikov. Large volatility-stabilized markets. Stochastic Processes and their Applications, 123(1):212-228, 2013. · Zbl 1288.60092
[32] J. Vaillancourt. On the existence of random McKean-Vlasov limits for triangular arrays of exchangeable diffusions. Stochastic Analysis and Applications, 6(4):431-446, 1988. · Zbl 0677.60107
[33] J. Vaillancourt. Interacting Fleming-Viot processes. Stochastic processes and their applications, 36(1):45-57, 1990. · Zbl 0729.92017
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