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 Mat. Sb. (N.S.), 1979, Volume 109(151), Number 1(5), Pages 146–164 (Mi msb2362)

Some new results in the theory of controlled diffusion processes

N. V. Krylov

Abstract: The validity of the Bellman equation for the payoff function for a controlled random diffusion process is proved. The main difference between the results in this article and those known earlier on the same theme is that here no assumptions whatever are made on the nondegeneracy of the controlled process. A theorem on the uniqueness of the solution of the Bellman equation is proved as well. The Bellman equation is examined in a lattice of measures; the derivatives of functions on which it is studied are understood as measures.
Bibliography: 13 titles.

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English version:
Mathematics of the USSR-Sbornik, 1980, 37:1, 133–149

Bibliographic databases:

UDC: 519.2+517.9
MSC: Primary 49C10, 60J60; Secondary 35K65

Citation: N. V. Krylov, “Some new results in the theory of controlled diffusion processes”, Mat. Sb. (N.S.), 109(151):1(5) (1979), 146–164; Math. USSR-Sb., 37:1 (1980), 133–149

Citation in format AMSBIB
\Bibitem{Kry79} \by N.~V.~Krylov \paper Some new results in the theory of controlled diffusion processes \jour Mat. Sb. (N.S.) \yr 1979 \vol 109(151) \issue 1(5) \pages 146--164 \mathnet{http://mi.mathnet.ru/msb2362} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=538554} \zmath{https://zbmath.org/?q=an:0436.93055|0404.93052} \transl \jour Math. USSR-Sb. \yr 1980 \vol 37 \issue 1 \pages 133--149 \crossref{https://doi.org/10.1070/SM1980v037n01ABEH001946} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1980KN98200009} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. V. Krylov, “On controlled diffusion processes with unbounded coefficients”, Math. USSR-Izv., 19:1 (1982), 41–64
2. N. V. Krylov, “On control of a diffusion process up to the time of first exit from a region”, Math. USSR-Izv., 19:2 (1982), 297–313
3. N. V. Krylov, “Boundedly nonhomogeneous elliptic and parabolic equations”, Math. USSR-Izv., 20:3 (1983), 459–492
4. P. L. Lions, “On the Hamilton–Jacobi–Bellman equations”, Acta Appl Math, 1:1 (1983), 17
5. N. V. Krylov, “Boundedly nonhomogeneous elliptic and parabolic equations in a domain”, Math. USSR-Izv., 22:1 (1984), 67–97
6. B. L. Rozovskii, “Filtering, Smoothing and Prediction of Degenerate Diffusion Processes. Backward Equations”, Theory Probab. Appl, 28:4 (1984), 762
7. N. V. Krylov, “An Approach to Controlled Diffusion Processes”, Theory Probab Appl, 31:4 (1987), 604
8. Karoui Nicole el, Nguyen Du'hŪŪ, Jeanblanc-Picqué Monique, “Compactification methods in the control of degenerate diffusions: existence of an optimal control”, Stochastics, 20:3 (1987), 169
9. N. V. Krylov, “On control of diffusion processes on a surface in Euclidean space”, Math. USSR-Sb., 65:1 (1990), 185–203
10. N. V. Krylov, “Smoothness of the value function for a controlled diffusion process in a domain”, Math. USSR-Izv., 34:1 (1990), 65–95
11. Fleming W., Vermes D., “Convex Duality Approach to the Optimal-Control of Diffusions”, SIAM J. Control Optim., 27:5 (1989), 1136–1155
12. Nosovskij G., “Nonlinear Potentials for Hamilton–Jacobi–Bellman Equations”, Acta Appl. Math., 30:2 (1993), 101–123
13. Nosovskij G., “Nonlinear Potentials for Hamilton–Jacobi–Bellman Equations .2.”, Acta Appl. Math., 46:1 (1997), 29–48
14. Zhou W., “A Probabilistic Approach to Interior Regularity of Fully Nonlinear Degenerate Elliptic Equations in Smooth Domains”, Appl. Math. Optim., 67:3 (2013), 419–452
15. Zhou W., “The Quasiderivative Method for Derivative Estimates of Solutions to Degenerate Elliptic Equations”, Stoch. Process. Their Appl., 123:8 (2013), 3064–3099
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