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Time-discretization for controlled Markov processes. I: General approximation results. (English) Zbl 0874.93094
[Part II, ibid. 32, No. 2, 139-158 (1996), is reviewed below.]
The authors prove a convergence theorem for approximation of a continuous-time controlled Markov process by a discrete-time controlled Markov process. The theorem is analogous to the well-known Lax-Richtmeyer theorem which states that a consistent and stable difference method converges.
MSC:
93E20 Optimal stochastic control
93C57 Sampled-data control/observation systems
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References:
[1] A. Bensoussan M. Robin: On the convergence of the discrete time dynamic programming equation for general semi-groups. SIAM J. Control Optim. 20 (1982), 1, 722-746. · Zbl 0488.93062
[2] N. Christopeit: Discrete approximation of continuous time stochastic control systems. SIAM J. Control Optim. 21 (1983), 1, 17-40. · Zbl 0508.93065
[3] B. T. Doshi: Optimal control of the service rate in an \(M|G|1\)-queueing system. Adv. in Appl. Probab. 10 (1978), 682-701. · Zbl 0381.60086
[4] W. H. Fleming R. W. Rishel: Deterministic and Stochastic Optimal Control. Springer Verlag, Berlin 1975. · Zbl 0323.49001
[5] I. I. Gihman A. V. Skorohod: Controlled Stochastic Processes. Springer Verlag, Berlin 1979. · Zbl 0404.60061
[6] U. G. Haussmann: A discrete approximation to optimal stochastic. Analysis and Optimization of Stochastic Systems, Academic Press, London 1980, pp. 229-241. · Zbl 0476.93082
[7] A. Hordijk F. A. Van der Duyn Schouten: Average optimal policies in Markov decision drift processes with applications to queueing and replacement model. Adv. in Appl. Probab. 15 (1983), 274-303. · Zbl 0517.90091
[8] A. Hordijk F. A. Van der Duyn Schouten: Discretization and weak convergence in Markov decision drift processes. Math. Oper. Res. 9 (1984), 1, 112-141. · Zbl 0531.90097
[9] A. Hordijk F. A. Van der Duyn Schouten: Markov decision drift processes; Conditions for optimality obtained by discretization. Math. Oper. Res. 10 (1985), 160-173. · Zbl 0573.90099
[10] A. Hordijk F. A. Van der Duyn Schouten: On the optimality of \((s,S)\)-policies in continuous review inventory models. SIAM J. Appl. Math. 46 (1986), 912-929. · Zbl 0648.90021
[11] G. M. Koole: Stochastic Scheduling and Dynamic Programming. Ph.D. Thesis, University of Leiden 1992. · Zbl 0851.90065
[12] T. G. Kurtz: Extensions of Trotter’s operator semigroup approximations theorems. J. Funct. Anal. 3 (1969), 111-132. · Zbl 0174.18401
[13] H. J. Kushner: Probability Methods for Approximation in Stochastic Control and for Elliptic Equations. Academic Press, New York 1977. · Zbl 0547.93076
[14] P. D. Lax R. D. Richtmeyer: Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math. 9 (1956), 267-293. · Zbl 0072.08903
[15] T. Meis U. Marcowitz: Numerical Solution of Partial Differential Equations. Springer Verlag, Berlin 1981. · Zbl 0446.65045
[16] H. J. Plum: Impulsive and continuously acting control of jump processes – Time discretization. Stochastics and Stochastic Reports 36 (1991), 163-192. · Zbl 0739.60076
[17] R. Rishel: Necessary and sufficient dynamic programming conditions for continuous time stochastic optimal control. SIAM J. Control 8 (1970), 4, 559-571. · Zbl 0206.45804
[18] R. Rishel: Controls optimal from the toward and dynamic programming for systems of controlled jump processes. Math. Programming Study 6 (1976), 125-153.
[19] F. A. Van der Duyn Schouten: Markov Decision Processes with Continuous Time Parameter. Mathematical Centre Tract 164, Amsterdam 1983. · Zbl 0519.90052
[20] N. M. Van Dijk: Controlled Markov Processes; Time Discretization/Networks of Queues. Ph.D. Thesis, University of Leiden 1983.
[21] N. M. Van Dijk: On the finite horizon Bellman equation for controlled Markov jump models with unbounded characteristics: existence and approximations. Stochastic Process. Appl. 28 (1988), 141-157. · Zbl 0645.93072
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