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Cubature on Wiener space for McKean-Vlasov SDEs with smooth scalar interaction. (English) Zbl 1418.60094
Summary: We present two cubature on Wiener space algorithms for the numerical solution of McKean-Vlasov SDEs with smooth scalar interaction. First, we consider a method introduced in [P. E. Chaudru de Raynal and C. A. Garcia Trillos, Stochastic Processes Appl. 125, No. 6, 2206–2255 (2015; Zbl 1320.65015)] under a uniformly elliptic assumption and extend the analysis to a uniform strong Hörmander assumption. Then we introduce a new method based on Lagrange polynomial interpolation. The analysis hinges on sharp gradient to time-inhomogeneous parabolic PDEs bounds. These bounds may be of independent interest. They extend the classical results of S. Kusuoka and D. Stroock [J. Fac. Sci., Univ. Tokyo, Sect. I A 32, 1–76 (1985; Zbl 0568.60059)] and S. Kusuoka [J. Math. Sci., Tokyo 10, No. 2, 261–277 (2003; Zbl 1031.60048)] further developed in [the first author et al., J. Funct. Anal. 263, No. 10, 3024–3101 (2012; Zbl 1262.35064); J. Funct. Anal. 268, No. 7, 1928–1971 (2015; Zbl 1333.60147); Lect. Notes Math. 2081, 203–316 (2013; Zbl 1273.65011)] and, more recently, in [the authors, Probab. Theory Relat. Fields 171, No. 1–2, 97–148 (2018; Zbl 1393.60074)]. Both algorithms are tested through two numerical examples.

60H30 Applications of stochastic analysis (to PDEs, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus
34F05 Ordinary differential equations and systems with randomness
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
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[1] Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. · Zbl 0543.33001
[2] Antonelli, F. and Kohatsu-Higa, A. (2002). Rate of convergence of a particle method to the solution of the McKean–Vlasov equation. Ann. Appl. Probab.12 423–476. · Zbl 1015.60048
[3] Bayer, C. and Friz, P. K. (2013). Cubature on Wiener space: Pathwise convergence. Appl. Math. Optim.67 261–278. · Zbl 1276.60069
[4] Bossy, M. (2005). Some stochastic particle methods for nonlinear parabolic PDEs. In ESAIM: Proceedings15 18–57. EDP Sciences. · Zbl 1090.65008
[5] Bossy, M. and Talay, D. (1997). A stochastic particle method for the McKean–Vlasov and the Burgers equation. Math. Comp.66 157–192. · Zbl 0854.60050
[6] Cardaliaguet, P. (2012). Notes on mean field games (from P.-L. Lyons’ lectures at Collège de France). https://www.ceremade.dauphine.fr/ cardaliaguet/MFG20130420.pdf.
[7] Cattiaux, P. and Mesnager, L. (2002). Hypoelliptic non-homogeneous diffusions. Probab. Theory Related Fields123 453–483. · Zbl 1009.60058
[8] Chassagneux, J.-F., Crisan, D. and Delarue, F. (2014). A Probabilistic approach to classical solutions of the master equation for large population equilibria. ArXiv e-prints. Available at arXiv:1411.3009.
[9] Crisan, D. and Delarue, F. (2012). Sharp derivative bounds for solutions of degenerate semi-linear partial differential equations. J. Funct. Anal.263 3024–3101. · Zbl 1262.35064
[10] Crisan, D. and Ghazali, S. (2007). On the convergence rates of a general class of weak approximations of SDEs. In Stochastic Differential Equations: Theory and Applications. Interdiscip. Math. Sci.2 221–248. World Sci. Publ., Hackensack, NJ. · Zbl 1156.60052
[11] Crisan, D., Kurtz, T. G. and Lee, Y. (2014). Conditional distributions, exchangeable particle systems, and stochastic partial differential equations. Ann. Inst. Henri Poincaré Probab. Stat.50 946–974. · Zbl 1306.60086
[12] Crisan, D., Litterer, C. and Lyons, T. (2015). Kusuoka–Stroock gradient bounds for the solution of the filtering equation. J. Funct. Anal.268 1928–1971. · Zbl 1333.60147
[13] Crisan, D. and Manolarakis, K. (2012). Solving backward stochastic differential equations using the cubature method: Application to nonlinear pricing. SIAM J. Financial Math.3 534–571. · Zbl 1259.65005
[14] Crisan, D. and Manolarakis, K. (2014). Second order discretization of backward SDEs and simulation with the cubature method. Ann. Appl. Probab.24 652–678. · Zbl 1303.60046
[15] Crisan, D., Manolarakis, K. and Nee, C. (2013). Cubature methods and applications. In Paris–Princeton Lectures on Mathematical Finance 2013. Lecture Notes in Math.2081 203–316. Springer, Cham. · Zbl 1273.65011
[16] Crisan, D. and McMurray, E. (2016). Smoothing properties of McKean–Vlasov SDEs. Probab. Theory Related Fields171 97–148. · Zbl 1393.60074
[17] Crisan, D. and Ortiz-Latorre, S. (2013). A Kusuoka–Lyons–Victoir particle filter. In Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.469 20130076. · Zbl 1371.60069
[18] de Raynal, P. C. and Trillos, C. G. (2015). A cubature based algorithm to solve decoupled Mckean–Vlasov forward–backward stochastic differential equations. Stochastic Process. Appl.125 2206–2255. · Zbl 1320.65015
[19] Gyurkó, L. G. and Lyons, T. J. (2010). Efficient and practical implementations of cubature on Wiener space. In Stochastic Analysis 2010. 73–111. Springer, Berlin.
[20] Jourdain, B., Méléard, S. and Woyczynski, W. A. (2008). Nonlinear SDEs driven by Lévy processes and related PDEs. ALEA Lat. Am. J. Probab. Math. Stat.4 1–29. · Zbl 1162.60327
[21] Jourdain, B., Méléard, S. and Woyczynski, W. A. (2008). Nonlinear SDEs driven by Lévy processes and related PDEs. ALEA Lat. Am. J. Probab. Math. Stat.4 1–29. · Zbl 1162.60327
[22] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin. · Zbl 0752.60043
[23] Kohatsu-Higa, A. and Ogawa, S. (1997). Weak rate of convergence for an Euler scheme of nonlinear SDE’s. Monte Carlo Methods Appl.3 327–345. · Zbl 0890.65147
[24] Kolokoltsov, V. N. (2004). Hydrodynamic limit of coagulation-fragmentation type models of \(k\)-nary interacting particles. J. Stat. Phys.115 1621–1653. · Zbl 1157.82370
[25] Kolokoltsov, V. N. (2010). Nonlinear Markov Processes and Kinetic Equations. Cambridge Tracts in Mathematics182. Cambridge Univ. Press, Cambridge. · Zbl 1222.60003
[26] Kotelenez, P. M. and Kurtz, T. G. (2010). Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type. Probab. Theory Related Fields146 189–222. · Zbl 1189.60123
[27] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics24. Cambridge Univ. Press, Cambridge. · Zbl 0743.60052
[28] Kurtz, T. G. and Protter, P. E. (1996). Weak convergence of stochastic integrals and differential equations. In Probabilistic Models for Nonlinear Partial Differential Equations. Lecture Notes in Math.1627 1–41. Springer, Berlin. · Zbl 0862.60041
[29] Kusuoka, S. (2001). Approximation of expectation of diffusion process and mathematical finance. In Taniguchi Conference on Mathematics Nara ’98. Adv. Stud. Pure Math.31 147–165. Math. Soc. Japan, Tokyo. · Zbl 1028.60052
[30] Kusuoka, S. (2003). Malliavin calculus revisited. J. Math. Sci. Univ. Tokyo10 261–277. · Zbl 1031.60048
[31] Kusuoka, S. and Stroock, D. (1985). Applications of the Malliavin calculus. II. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math.32 1–76. · Zbl 0568.60059
[32] Kusuoka, S. and Stroock, D. (1987). Applications of the Malliavin calculus. III. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math.34 391–442. · Zbl 0633.60078
[33] Lee, W. and Lyons, T. (2015). The adaptive patched cubature filter and its implementation. arXiv preprint. Available at arXiv:1509.04239. · Zbl 1343.60042
[34] Lions, P.-L. (2014) Cours au collège de france. Available at http://www.college-de-france.fr/site/pierre-louis-lions/seminar-2014-11-14-11h15.htm.
[35] Litterer, C. and Lyons, T. (2011). Introducing cubature to filtering. In The Oxford Handbook of Nonlinear Filtering 768–796. Oxford Univ. Press, Oxford. · Zbl 1237.93168
[36] Lyons, T. and Victoir, N. (2004). Cubature on Wiener space. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.460 169–198. · Zbl 1055.60049
[37] Morale, D., Capasso, V. and Oelschläger, K. (2005). An interacting particle system modelling aggregation behavior: From individuals to populations. J. Math. Biol.50 49–66. · Zbl 1055.92046
[38] Ninomyia, S. and Victoir, N. (2008). Weak approximation scheme of stochastic differential equations and applications to derivatives pricing. Appl. Math. Finance. 15 107–121.
[39] Ogawa, S. (1995). Some problems in the simulation of nonlinear diffusion processes. Math. Comput. Simulation38 217–223. · Zbl 0824.60064
[40] Ricketson, L. F. (2015). A multilevel Monte Carlo method for a class of McKean–Vlasov processes. Available at https://arxiv.org/abs/1508.02299.
[41] Teichmann, J. (2006). Calculating the Greeks by cubature formulae. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.462 647–670. · Zbl 1149.91317
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