Homogenization for deterministic maps and multiplicative noise
By Georg A. Gottwald, Ian Melbourne. Proceedings of the Royal Society A, 2013
89 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Finance, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/0a7d4d4b817fce1b599998c534351c3782438021
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In Computational Theory and Mathematics
In Finance
In Statistical and Nonlinear Physics
- Zemer Kosloff
- A. Korepanov
- Ian Melbourne
- Georg A. Gottwald
- Charalampos Skokos
- Alexander I. Bufetov
- E. Rio
Citations in the corpus
- Martingale–coboundary decomposition for families of dynamical systems (2016, background): “In [27], a homogenization theorem was proved for fast-slow systems of the form”
- A Spectral Approach for Quenched Limit Theorems for Random Expanding Dynamical Systems (2017, unknown): “[7,13,23–25,35,42], and another interesting set of related results on limit theorems occur in the setting of homogenisation [22,30,31].”
- Data-driven non-Markovian closure models (2014, unknown): “Such a situation is expected to occur when the separation of time scales between the observed and unobserved variables is not as sharp as required, for instance, in applying stochastic homogenization techniques [100]; see however the recent works [101, 102] for milder assumptions.”