Singular control with state constraints on unbounded domain

Annals of Probability

Volumn 34, Issue 5, 2006, Pages 1864-1909

  Atar, Rami ^a   Budhiraja, Amarjit ^b  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

^b University of North Carolina at Chapel Hill   (United States)

Author keywords

Brownian control problems; Hamilton Jacobi Bellman equations; Singular control; Skorohod problem; State constraints; Stochastic networks; Viscosity solutions

Indexed keywords

EID: 33845509299     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117906000000359     Document Type: Article

References (33)

1. BARDI, M. and CAPUZZO-UOLCETTA, I. (1997). Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston. MR1484411
2. BENĚ, V. E., SHEPP, L. A. and WITSENHAUSEN, H. S. (1980/81). Some solvable stochastic control problems. Stochastics 4 39-83. MR0587428
3. BOHM, V. (1975). On the continuity of the optimal policy set for linear programs. SIAM J. Appl. Math. 28 303-306. MR0371390
4. BORKAR, V. (1989). Optimal Control of Diffusion Processes. Longman Scientific and Technical, Harlow. MR1005532
5. BRAMSON, M. and WILLIAMS, R. J. (2003). Two workload properties for Brownian networks. Queueing Systems 45 191-221. MR2024178
6. CAPUZZO-DOLCETTA, I. and LIONS, P. L. (1990). Hamilton-Jacobi equations with state constraints. Trans. AMS 318 643-683. MR0951880
7. CHOW, P. L., MENALDI, J. L. and ROBIN, M. (1985). Additive control of stochastic linear systems with finite horizon. SIAM J. Control Optim. 23 858-899. MR0809540
8. CRANDALL, M. G., ISHII, H. and LIONS, P. L. (1992). User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 1-67. MR1118699
9. CRANDALL, M. G. and LIONS, P. L. (1987). Remarks on the existence and uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations. Illinois J. Math. 31 665-688. MR0909790
10. DUFFIE, D., FLEMING, W., SONER, H. M. and ZARIPHOPOULOU, T. (1997). Hedging in incomplete markets with HARA utility. J. Econom. Dynam. Control 21 753-782. MR1455755
11. DUPUIS, P. and ISHII, H. (1991). On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stochastics Stochastics Rep. 35 31-62. MR1110990
12. DUPUIS, P. and RAMANAN, K. (1999). Convex duality and the Skorokhod problem. I, II. Probab. Theory Related Fields 2 153-195, 197-236. MR1720348
13. DURRETT, R. (1995). Probability: Theory and Examples, 2nd ed. Duxbury Press, N. Scituate, MA. MR1609153
14. EVANS, L. C. (1979). A second-order elliptic equation with gradient constraint. Comm. Partial Differential Equations 4 555-572. MR0529814
15. FLEMING, W. and SONER, H. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, New York. MR1199811
16. HARRISON, J. M. (1988). Brownian models of queueing networks with heterogeneous customer population. In Stochastic Differential Systems Stochastic Control Theory and Their Applications 147-186. Springer, New York. MR0934722
17. HARRISON, J. M. (2000). Brownian models of open processing networks: Canonical representation of workload. Ann. Appl. Probab. 10 75-103. MR1765204
18. HARRISON, J. M. and TAKSAR, M. I. (1983). Instantaneous control of Brownian motion. Math. Oper. Res. 8 439-453. MR0716123
19. HARRISON, J. M. and VAN MIEGHEM, J. A. (1997). Dynamic control of Brownian networks: State space collapse and equivalent workload formulation. Ann. Appl. Probab. 7 747-771. MR1459269
20. ISHII, H. (1989). On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Comm. Pure Appl. Math. 42 15-45. MR0973743
21. ISHII, H. and KOIKE, S. (1983). Boundary regularity and uniqueness for an elliptic equation with gradient constraint. Comm. Partial Differential Equations 8 317-346. MR0693645
22. ISHII, H. and LORETI, P. (2002). A class of optimal control problems with state constraint. Indiana Univ. Math. J. 51 1167-1196. MR1947872
23. KARATZAS, I. (1983). A class of singular stochastic control problems. Adv. in Appl. Probab. 15225-254. MR0698818
24. KARATZAS, I. and SHREVE, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York. MR1121940
25. KATSOULAKIS, M. A. (1994). Viscosity solutions of second order fully nonlinear elliptic equations with state constraints. Indiana Univ. Math. J. 43 493-519. MR1291526
26. LASRY, J. M. and LIONS, P. L. (1989). Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem. Math. Ann. 283 583-630. MR0990591
27. LIONS, P. L. (1982). Generalized solutions of Hamilton-Jacobi Equations. Pitman, Boston. MR0667669
28. MARTINS, L. F., SHREVE, S. E. and SONER, H. M. (1996). Heavy traffic convergence of a controlled multi-class queuing system. SIAM J. Control. Optim. 34 2133-2171. MR1416504
29. MENALDI, J. L. and ROBIN, M. (1983). On some cheap control problems for diffusion processes. Trans. AMS 278 771-802. MR0701523
30. ROCKAFELLAR, R. T. (1970). Convex Analysis. Princeton Univ. Press. MR0274683
31. SHREVE, S. E. and SONER, H. M. (1994). Optimal investment and consumption with transaction costs. Ann. Appl. Probab. 4 609-692. MR1284980
32. SONER, H. M. (1986). Optimal control with state constraint. I. SIAM J. Control. Optim. 24 552-561. MR0838056
33. WONG, E. (1971). Representation of martingales, quadratic variation and applications. SIAM J. Control Optim. 9 621-633. MR0307324