2008/05/31 by Simon J. A. Malham, Anke Wiese · 13 citations
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Topics in Algebra #Concatenation (mathematics) #Exponential function #Lie algebra #Nonlinear system #Polynomial and algebraic computation #Series (stratigraphy) #Spectrum (functional analysis) #Stochastic differential equation #Stochastic processes and financial applications #Taylor series #Vector field #math.OC #math.PR #msc:16W30 #msc:60H10 #msc:60H35 #msc:65C30 #msc:81R60 #msc:93E03
paper · pdf · doi:10.1098/rspa.2009.0203
published in Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences 465(2112), 3729-3749 (Royal Society) · 23 pages, 4 figures
arxiv created 2009/04/17 · openalex publication_date 2009/09/21 · arxiv updated 2015/05/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We study solutions to nonlinear stochastic differential systems driven by a multi-dimensional Wiener process. A useful algorithm for strongly simulating such stochastic systems is the Castell–Gaines method, which is based on the exponential Lie series. When the diffusion vector fields commute, it has been proved that, at low orders, this method is more accurate in the mean-square error than corresponding stochastic Taylor methods. However, it has also been shown that when the diffusion vector fields do not commute, this is not true for strong order one methods. Here, we prove that when there is no drift, and the diffusion vector fields do not commute, the exponential Lie series is usurped by the sinh-log series. In other words, the mean-square error associated with a numerical method based on the sinh-log series is always smaller than the corresponding stochastic Taylor error, in fact to all orders. Our proof uses the underlying Hopf algebra structure of these series, and a two-alphabet associative algebra of shuffle and concatenation operations. We illustrate the benefits of the proposed series in numerical studies.