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Theory of Probability and Mathematical Statistics

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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: 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$.

References [Enhancements On Off] (What's this?)

  • 1. A. Yu. Pilipenko, Flows generated by stochastic equations with reflection, Random Oper. Stochastic Equations 12 (2004), no. 4, 389-396. MR 2108191 (2006a:60109)
  • 2. A. Yu. Pilipenko, Properties of flows generated by stochastic equations with reflection, Ukrain. Mat. Zh. 57 (2005), no. 8, 1069-1078; English transl. in Ukrainian Math. J. 57 (2005), no. 8, 1262-1274. MR 2218469 (2007g:60066)
  • 3. H. Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J. 9 (1979), no. 1, 163-177. MR 529332 (80k:60075)
  • 4. C. Striker and M. Yor, Calcul stochastique dépendant d'un paramètre, Z. Wahrsch. Verw. Gebiete 45 (1978), no. 2, 109-133. MR 510530 (80f:60047)
  • 5. P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York-London-Sydney-Toronto, 1968. MR 0233396 (38:1718)
  • 6. N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, de Gruyter Studies in Mathematics, vol. 14, Walter de Gruyter & Co., Berlin, 1991. MR 1133391 (93e:60107)
  • 7. A. Jakubowski, A non-Skorokhod topology on the Skorokhod space, Electron. J. Probab. 2 (1997), no. 4. MR 1475862 (98k:60046)

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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

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

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