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
Journal:
Theor. Probability and Math. Statist. 75 (2007), 147-160
MSC (2000):
Primary 60H10; Secondary 60J25, 60F25
DOI:
https://doi.org/10.1090/S0094-9000-08-00721-7
Published electronically:
January 25, 2008
MathSciNet review:
2321188
Full-text PDF Free Access
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 $\nabla \varphi _t$.
References
- A. Yu. Pilipenko, Flows generated by stochastic equations with reflection, Random Oper. Stochastic Equations 12 (2004), no. 4, 385–392. MR 2108191, DOI https://doi.org/10.1163/1569397042722364
- A. Yu. Pilipenko, Properties of flows generated by stochastic equations with reflection, Ukraïn. Mat. Zh. 57 (2005), no. 8, 1069–1078 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 57 (2005), no. 8, 1262–1274. MR 2218469, DOI https://doi.org/10.1007/s11253-005-0260-1
- Hiroshi Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J. 9 (1979), no. 1, 163–177. MR 529332
- C. Stricker and M. Yor, Calcul stochastique dépendant d’un paramètre, Z. Wahrsch. Verw. Gebiete 45 (1978), no. 2, 109–133 (French). MR 510530, DOI https://doi.org/10.1007/BF00715187
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
- Nicolas Bouleau and Francis Hirsch, Dirichlet forms and analysis on Wiener space, De Gruyter Studies in Mathematics, vol. 14, Walter de Gruyter & Co., Berlin, 1991. MR 1133391
- Adam Jakubowski, A non-Skorohod topology on the Skorohod space, Electron. J. Probab. 2 (1997), no. 4, 21 pp.}, issn=1083-6489, review=\MR{1475862}, doi=10.1214/EJP.v2-18,.
References
- A. Yu. Pilipenko, Flows generated by stochastic equations with reflection, Random Oper. Stochastic Equations 12 (2004), no. 4, 389–396. MR 2108191 (2006a:60109)
- 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)
- 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)
- 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)
- P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York–London–Sydney–Toronto, 1968. MR 0233396 (38:1718)
- 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)
- A. Jakubowski, A non-Skorokhod topology on the Skorokhod space, Electron. J. Probab. 2 (1997), no. 4. MR 1475862 (98k:60046)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2000):
60H10,
60J25,
60F25
Retrieve articles in all journals
with MSC (2000):
60H10,
60J25,
60F25
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
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