The Annals of Probability

On time-inhomogeneous controlled diffusion processes in domains

Hongjie Dong and N. V. Krylov

Full-text: Open access


Time-inhomogeneous controlled diffusion processes in both cylindrical and noncylindrical domains are considered. Bellman’s principle and its applications to proving the continuity of value functions are investigated.

Article information

Ann. Probab., Volume 35, Number 1 (2007), 206-227.

First available in Project Euclid: 19 March 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 93E20: Optimal stochastic control 90C40: Markov and semi-Markov decision processes

Principle of optimality Bellman’s principle Bellman’s equations continuity of value functions


Dong, Hongjie; Krylov, N. V. On time-inhomogeneous controlled diffusion processes in domains. Ann. Probab. 35 (2007), no. 1, 206--227. doi:10.1214/009117906000000395.

Export citation


  • Bensoussan, A. and Lions, J.-L. (1978). Aplications des inéquations variationnelles en contrôle stochastique. Dunod, Paris.
  • Borkar, V. S. (1989). Optimal Control of Diffusion Processes. Longman, Harlow, Essex, UK.
  • 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. To appear.
  • Fleming, W. H. and Soner, H. M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, New York.
  • Krylov, N. V. (1977). Controlled Diffusion Processes. Nauka, Moscow. (In Russian.) [English translation by A. B. Aries, (1980).]
  • 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.
  • Krylov, N. V. (1999). On Kolmogorov's equations for finite dimensional diffusions. CIME Courses. Lecture Notes Math. 1715 1–63. Springer, Berlin.
  • 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.
  • 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.
  • Kurtz, T. (1987). Martingale problems for controlled processes. Stochastic Modelling and Filtering. Lecture Notes in Control and Inform. Sci. 91 75–90. Springer, Berlin.
  • Lions, P.-L. (1983). Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations. I. Comm. Partial Differential Equations 8 1101–1174.