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 Izv. Akad. Nauk SSSR Ser. Mat., 1989, Volume 53, Issue 1, Pages 66–96 (Mi izv1162)

Smoothness of the value function for a controlled diffusion process in a domain

N. V. Krylov

Abstract: The author gives conditions on the behavior of a controlled diffusion process within and on the boundary of a domain that are sufficient for the value function to have two bounded generalized derivatives and to satisfy the Bellman equation. These conditions are almost necessary even for uncontrolled diffusion processes, and at the same time they encompass, for example, the heat equation in a disc and the Monge-Ampère equation in a convex domain.
Bibliography: 24 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1990, 34:1, 65–95

Bibliographic databases:

UDC: 517.9
MSC: Primary 60J60, 58G32, 35J25; Secondary 35K20, 60H10

Citation: N. V. Krylov, “Smoothness of the value function for a controlled diffusion process in a domain”, Izv. Akad. Nauk SSSR Ser. Mat., 53:1 (1989), 66–96; Math. USSR-Izv., 34:1 (1990), 65–95

Citation in format AMSBIB
\Bibitem{Kry89} \by N.~V.~Krylov \paper Smoothness of the value function for a~controlled diffusion process in a~domain \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1989 \vol 53 \issue 1 \pages 66--96 \mathnet{http://mi.mathnet.ru/izv1162} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=992979} \zmath{https://zbmath.org/?q=an:0701.93054} \transl \jour Math. USSR-Izv. \yr 1990 \vol 34 \issue 1 \pages 65--95 \crossref{https://doi.org/10.1070/IM1990v034n01ABEH000603} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. N. V. Krylov, “On the first quasiderivatives of solutions of Ito stochastic equations”, Russian Acad. Sci. Izv. Math., 40:2 (1993), 377–403
2. Yu. A. Alkhutov, “$L_p$-solubility of the Dirichlet problem for the heat equation in non-cylindrical domains”, Sb. Math., 193:9 (2002), 1243–1279
3. Hongjie Dong, “About smoothness of solutions of the heat equations in closed, smooth space-time domains”, Comm Pure Appl Math, 58:6 (2005), 799
4. Nina Ivochkina, Neil Trudinger, Xu-Jia Wang, “The Dirichlet Problem for Degenerate Hessian Equations”, Communications in Partial Differential Equations, 29:1-2 (2005), 219
5. St. Petersburg Math. J., 19:1 (2008), 1–13
6. Yan P., Zhong L., Wen X., Tang A., “Fabrication of Cu2O/Tio2/Sepiolite Electrode For Effectively Detecting of H2O2”, J. Electroanal. Chem., 827 (2018), 1–9
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