The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 29, Number 1 (2019), 130-177.
Cubature on Wiener space for McKean–Vlasov SDEs with smooth scalar interaction
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.
Ann. Appl. Probab., Volume 29, Number 1 (2019), 130-177.
Received: August 2016
Revised: January 2018
First available in Project Euclid: 5 December 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60H10: Stochastic ordinary differential equations [See also 34F05] 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03] 35K55: Nonlinear parabolic equations 35K65: Degenerate parabolic equations
Crisan, Dan; McMurray, Eamon. Cubature on Wiener space for McKean–Vlasov SDEs with smooth scalar interaction. Ann. Appl. Probab. 29 (2019), no. 1, 130--177. doi:10.1214/18-AAP1407. https://projecteuclid.org/euclid.aoap/1544000427