Cubature on Wiener space for McKean–Vlasov SDEs with smooth scalar interaction

Abstract

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.