CUBATURE ON WIENER SPACE FOR MCKEAN-VLASOV SDES WITH SMOOTH SCALAR INTERACTION

• 成果类型: Article
• 署名作者: Crisan, Dan; McMurray, Eamon
• 署名单位: Imperial College London
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1407
• 发表日期: 2019
• 页码: 130-177
• 关键词: particle method; simulation; bounds

摘要

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 Hormander 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. JA, 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.