2016/05/26 by Anis Al Gerbi, Benjamin Jourdain, Gerbi, Anis Al +3
Business, Management and Accounting · Economics, Econometrics and Finance · Mathematics · #Brownian motion #Commutative property #Computer science #Convergence (economics) #Credit Risk and Financial Regulations #Distribution (mathematics) #FOS: Mathematics #Field (mathematics) #Financial Literacy, Pension, Retirement Analysis #Financial Markets and Investment Strategies #Geometry #Limit (mathematics) #Limiting #Mathematical analysis #Mathematics #Order (exchange) #Probability (math.PR) #Pure mathematics #Rate of convergence #Scheme (mathematics) #Stochastic processes and financial applications #Vector field #Weak convergence #math.PR
paper · pdf · doi:10.48550/arxiv.1605.08270
arXiv admin note: text overlap with arXiv:1601.05268
arxiv created 2016/05/26 · openalex publication_date 2016/05/26 · arxiv updated 2016/05/27 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
In a previous work, we proved strong convergence with order 1 of the Ninomiya-Victoir scheme XNV with time step T/N to the solution X of the limiting SDE when the Brownian vector fields commute. In this paper, we prove that the normalized error process N (X - XNV) converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields and the drift vector field. This result ensures that the strong convergence rate is actually 1 when the Brownian vector fields commute, but at least one of them does not commute with the drift vector field. When all the vector fields commute the limit vanishes. Our result is consistent with the fact that the Ninomiya-Victoir scheme solves the SDE in this case.