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January 1999 On the Spatial Asymptotic Behavior of Stochastic Flows in Euclidean Space
Peter Imkeller, Michael Scheutzow
Ann. Probab. 27(1): 109-129 (January 1999). DOI: 10.1214/aop/1022677255


We study asymptotic growth rates of stochastic flows on $\mathbf{R}^d$ and their derivatives with respect to the spatial parameter under Lipschitz conditions on the local characteristics of the generating semimartingales. In a first step these conditions are seen to imply moment inequalities for the flow $\phi$ of the form $$E \sup_{0 \le t \le T}|\phi_{0t} (x) - \phi_{0t} (y)|^p\le|x-y|^p\exp(cp^2)\quad\text{for all $p\ge 1$.}$$ In a second step we deduce the growth rates from an integrated version of these moment inequalities, using the continuity lemma of Garsia, Rodemich and Rumsey. We provide two examples to show that our results are sharp.


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Peter Imkeller. Michael Scheutzow. "On the Spatial Asymptotic Behavior of Stochastic Flows in Euclidean Space." Ann. Probab. 27 (1) 109 - 129, January 1999.


Published: January 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0942.60053
MathSciNet: MR1681138
Digital Object Identifier: 10.1214/aop/1022677255

Primary: 34F05 , 60H10
Secondary: 60G17 , 60G48

Keywords: GRR lemma , modulus of continuity , Semimartingale , spatial growth rate , Stochastic differential equation , stochastic flow

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.27 • No. 1 • January 1999
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