The Annals of Applied Probability

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

Dan Crisan and Eamon McMurray

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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.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 1 (2019), 130-177.

Dates
Received: August 2016
Revised: January 2018
First available in Project Euclid: 5 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1544000427

Digital Object Identifier
doi:10.1214/18-AAP1407

Mathematical Reviews number (MathSciNet)
MR3910002

Zentralblatt MATH identifier
07039123

Subjects
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

Keywords
Cubature on Wiener space Lagrange polynomial interpolation McKean–Vlasov SDEs Kusuoka–Stroock functions

Citation

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


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