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A spatial version of the Itô–Stratonovich correction
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Hairer, Martin and Maas, Jan (2012) A spatial version of the Itô–Stratonovich correction. The Annals of Probability, Vol.40 (No.4). pp. 1675-1714. doi:10.1214/11-AOP662 ISSN 0091-1798.
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Official URL: http://dx.doi.org/10.1214/11-AOP662
Abstract
We consider a class of stochastic PDEs of Burgers type in spatial dimension 1, driven by space–time white noise. Even though it is well known that these equations are well posed, it turns out that if one performs a spatial discretization of the nonlinearity in the “wrong” way, then the sequence of approximate equations does converge to a limit, but this limit exhibits an additional correction term.
This correction term is proportional to the local quadratic cross-variation (in space) of the gradient of the conserved quantity with the solution itself. This can be understood as a consequence of the fact that for any fixed time, the law of the solution is locally equivalent to Wiener measure, where space plays the role of time. In this sense, the correction term is similar to the usual Itô–Stratonovich correction term that arises when one considers different temporal discretizations of stochastic ODEs.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | The Annals of Probability | ||||
Publisher: | Institute of Mathematical Statistics | ||||
ISSN: | 0091-1798 | ||||
Official Date: | 2012 | ||||
Dates: |
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Volume: | Vol.40 | ||||
Number: | No.4 | ||||
Page Range: | pp. 1675-1714 | ||||
DOI: | 10.1214/11-AOP662 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published |
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