Dong, Hongjie; Krylov, N. V. On time-inhomogeneous controlled diffusion processes in domains. (English) Zbl 1127.93062 Ann. Probab. 35, No. 1, 206-227 (2007). Summary: The first part of this article is devoted to quite an old subject in the theory of controlled diffusion processes, namely deriving Bellman’s principle (also called the principle of optimality) for processes controlled up to the first exit time from bounded domains. Our main results depend heavily on the validity of Bellman’s principle. Later on, these results are used to obtain sharp results concerning when the value functions for time-homogeneous processes satisfy the corresponding Bellman’s equations. Cited in 6 Documents MSC: 93E20 Optimal stochastic control 90C40 Markov and semi-Markov decision processes 49L20 Dynamic programming in optimal control and differential games Keywords:principle of optimality; Bellman’s principle; Bellman’s equations; continuity of value functions × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Bensoussan, A. and Lions, J.-L. (1978). Aplications des inéquations variationnelles en contrôle stochastique . Dunod, Paris. · Zbl 0411.49002 [2] Borkar, V. S. (1989). Optimal Control of Diffusion Processes. Longman, Harlow, Essex, UK. · Zbl 0669.93065 [3] Dong, H. and Krylov, N. V. (2006). On the rate of convergence of finite-difference approximations for parabolic Bellman equations with Lipschitz coefficients in cylindrical domains. Appl. Math. Optim. · Zbl 1127.65068 · doi:10.1007/s00245-007-0879-4 [4] Fleming, W. H. and Soner, H. M. (1993). Controlled Markov Processes and Viscosity Solutions . Springer, New York. · Zbl 0773.60070 [5] Krylov, N. V. (1977). Controlled Diffusion Processes . Nauka, Moscow. (In Russian.) [English translation by A. B. Aries, (1980).] · Zbl 0513.60043 [6] Krylov, N. V. (1981). On controlled diffusion processes with unbounded coefficients. Izv. Akad. Nauk SSSR Matem. 45 734–759. (In Russian.) English translation Math. USSR Izvestija (1982) 19 41–64. · Zbl 0497.93060 [7] Krylov, N. V. (1999). On Kolmogorov’s equations for finite dimensional diffusions. CIME Courses. Lecture Notes Math. 1715 1–63. Springer, Berlin. · Zbl 0943.60070 · doi:10.1007/BFb0092417 [8] Krylov, N. V. (2000). On the rate of convergence of finite-difference approximations for Bellman’s equations with variable coefficients. Probab. Theory Related Fields. 117 1–16. · Zbl 0971.65081 · doi:10.1007/s004409900044 [9] Krylov, N. V. (2002). Adapting some ideas from stochastic control theory to studying the heat equation in closed smooth domain. Appl. Math Optim. 46 231–261. · Zbl 1030.35093 · doi:10.1007/s00245-002-0745-3 [10] Kurtz, T. (1987). Martingale problems for controlled processes. Stochastic Modelling and Filtering . Lecture Notes in Control and Inform. Sci. 91 75–90. Springer, Berlin. · Zbl 0642.93065 · doi:10.1007/BFb0009051 [11] Lions, P.-L. (1983). Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations. I. Comm. Partial Differential Equations 8 1101–1174. · Zbl 0716.49022 · doi:10.1080/03605308308820297 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.