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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.

##### MSC:
 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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