2013/04/30 by Georg A. Gottwald, Ian Melbourne · 53 citations
Computer Science · Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Differential equation #Discrete time and continuous time #Multiplicative function #Multiplicative noise #Skew #Stochastic calculus #Stochastic differential equation #Stochastic process #Stochastic processes and financial applications #math.DS #stochastic dynamics and bifurcation
paper · pdf · doi:10.1098/rspa.2013.0201
published in Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences 469(2156), 20130201 (Royal Society) · The updated version contains a minor correction in the appendix, and removes an unnecessary large deviation assumption
openalex publication_date 2013/05/29 · arxiv created 2015/04/29 · arxiv updated 2015/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A recent paper of Melbourne & Stuart (2011 A note on diffusion limits of chaotic skew product flows. Nonlinearity 24 , 1361–1367 (doi:10.1088/0951-7715/24/4/018)) gives a rigorous proof of convergence of a fast–slow deterministic system to a stochastic differential equation with additive noise. In contrast to other approaches, the assumptions on the fast flow are very mild. In this paper, we extend this result from continuous time to discrete time. Moreover, we show how to deal with one-dimensional multiplicative noise. This raises the issue of how to interpret certain stochastic integrals; it is proved that the integrals are of Stratonovich type for continuous time and neither Stratonovich nor Itô for discrete time. We also provide a rigorous derivation of super-diffusive limits where the stochastic differential equation is driven by a stable Lévy process. In the case of one-dimensional multiplicative noise, the stochastic integrals are of Marcus type both in the discrete and continuous time contexts.