Crisan Dan et Mcmurray Eamon, Cubature on Wiener space for McKean.Vlasov SDEs with smooth scalar interaction, Annals of applied probability , 29(1), 2019, pp. 130-177
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 [Stochastic Process. Appl. 125 (2015) 2206.2255] 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 Kusuoka and Stroock [J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 32 (1985) 1.76] and Kusuoka [J. Math. Sci. Univ. Tokyo 10 (2003) 261.277] further developed in [J. Funct. Anal. 263 (2012) 3024.3101; J. Funct. Anal. 268 (2015) 1928.1971; Cubature Methods and Applications (2013), Springer, Cham] and, more recently, in [Probab. Theory Related Fields 171 (2016) 97.148].Both algorithms are tested through two numerical examples.