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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

 

Generalized differentiability with respect to the initial data of a flow generated by a stochastic equation with reflection


Author: A. Yu. Pilipenko
Translated by: O. I. Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 75 (2006).
Journal: Theor. Probability and Math. Statist. 75 (2007), 147-160
MSC (2000): Primary 60H10; Secondary 60J25, 60F25
Published electronically: January 25, 2008
MathSciNet review: 2321188
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \varphi_t(x)$, $ x\in\mathbb{R}^d_+$, be a solution of a stochastic differential equation in the half-space $ \mathbb{R}^d_+$ with normal reflection in the boundary; the solution starts from a point $ x$. We prove that the random mapping $ \varphi_t(\boldsymbol\cdot,\omega)$ is differentiable in the Sobolev sense for almost all $ \omega$. We obtain a stochastic equation for the derivative $ \nabla\varphi_t$.


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Additional Information

A. Yu. Pilipenko
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivs’ka Street, 3, 01601, Kyiv, Ukraine
Email: apilip@imath.kiev.ua

DOI: http://dx.doi.org/10.1090/S0094-9000-08-00721-7
PII: S 0094-9000(08)00721-7
Keywords: Stochastic equations with reflection, stochastic flows, Sobolev spaces
Received by editor(s): November 19, 2004
Published electronically: January 25, 2008
Additional Notes: Supported by the Ministry for Science and Education of Ukraine, project GP/F8/0086
Article copyright: © Copyright 2008 American Mathematical Society