Martingale approach to stochastic control with discretionary stopping

Applied Mathematics and Optimization

Volumn 53, Issue 2, 2006, Pages 163-184

  Karatzas, Ioannis ^a   Zamfirescu, Ingrid Mona ^b  

^a Och Spine at New York Presbyterian Hospitals   (United States)

^b CITY UNIVERSITY OF NEW YORK   (United States)

Author keywords

Doob Meyer decompositions; Martingales; Optimal stopping; Stochastic control

Indexed keywords

DISCRETE TIME CONTROL SYSTEMS; DYNAMIC PROGRAMMING; OPTIMAL CONTROL SYSTEMS;

DOOB-MEYER DECOMPOSITIONS; MARTINGALES; OPTIMAL STOPPING;

STOCHASTIC CONTROL SYSTEMS;

EID: 32944458368     PISSN: 00954616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00245-005-0841-2     Document Type: Article

References (36)

1. Beneš, V.E. (1970) Existence of optimal policies based on specified information, for a class of stochastic decision problems. SIAM Journal on Control & Optimization 8:179-188.
2. Beneš, V.E. (1971) Existence of optimal stochastic control laws. SIAM Journal on Control & Optimization 9:446-475.
3. Beneš, V.E. (1992) Some combined control and stopping problems. Paper presented at the CRM Workshop on Stochastic Systems, Montréal, November 1992.
4. Bensoussan, A., and Lions, J.L. (1982) Applications of Variational Inequalities in Stochastic Control. North-Holland, Amsterdam.
5. Ceci, C., and Basan, B. (2004) Mixed optimal stopping and stochastic control problems with semicintinuous final reward, for diffusion processes. Stochastics & Stochastics Reports 76:323-337.
6. Davis, M.H.A. (1973) On the existence of optimal policies in stochastic control. SIAM Journal on Control & Optimization 11:587-594.
7. Davis, M.H.A., and Varaiya, P.P. (1973) Dynamic programming conditions for partially observable stochastic systems. SIAM Journal on Control & Optimization 11:226-261.
8. Davis, M.H.A., and Zervos, M. (1994) A problem of singular stochastic control with discretionary stopping. Annals of Applied Probability 4:226-240.
9. Dubins, L.E., and Savage, L.J. (1976) How to Gamble If You Must: Inequalities for Stochastic Processes. Springer-Verlag, New York.
10. Duncan, T.E., and Varaiya, P.P. (1971) On the solutions of a stochastic control system. SIAM Journal on Control & Optimization 9:354-371.
11. El Karoui, N. (1981) Les Aspects Probabilistes du Contrôle Stochastique. Lecture Notes in Mathematics, vol. 876, pp. 73-238. Springer-Verlag, Berlin.
12. El Karoui, N., and Quenez, M.C. (1995) Dynamic programming and pricing of contingent claims in an incomplete market. SIAM Journal on Control & Optimization 33:29-66.
13. El Karoui, N., Nguyen, D.H., and Jeanblanc-Picqué, M. (1987) Compactification methods in the control of degenerate diffusions: existence of an optimal control. Stochastics 20:169-219.
14. Elliott, R.J. (1982) Stochastic Calculus and Applications. Springer-Verlag, New York.
15. Fleming, W.R., and Rishel, R.W. (1975) Deterministic and Stochastic Optimal Control. Springer-Verlag, New York.
16. Fleming, W.R., and Soner, H.M. (1993) Controlled Diffusion Processes and Viscosity Solutions. Springer-Verlag, New York.
17. Fujisaki, M., Kallianpur, G., and Kunita, H. (1972) Stochastic differential equations for the non-linear filtering problem. Osaka Journal of Mathematics 9:19-40.
18. Haussmann, U.G. (1986) A Stochastic Maximum Principle for Optimal Control of Diffusions. Longman Scientific and Technical, Harlow.
19. Haussmann, U.G., and Lepeltier, J.P. (1990) On the existence of optimal controls. SIAM Journal on Control & Optimization 28:851-902.
20. Kamizono, K., and Morimoto, H. (2002) On a combined control and stopping time game. Stochastics and Stochastics Reports 73:99-123.
21. Karatzas, I., and Kou, S.G. (1998) Hedging American contingent claims with constrained portfolios. Finance and Stochastics 3:215-258.
22. Karatzas, I., and Ocone, D.L. (2002) A leavable, bounded-velocity stochastic control problem. Stochastic Processes & Applications 99:31-51.
23. Karatzas, I., and Shreve, S.E. (1991) Brownian Motion and Stochastic Calculus, second edition. Springer-Verlag, New York.
24. Karatzas, I., and Shreve, S.E. (1998) Methods of Mathematical Finance. Springer-Verlag, New York.
25. Karatzas, I., and Sudderth, W.D. (1999) Control and stopping of a diffusion process on an interval. Annals of Applied Probability 9:188-196.
26. Karatzas, I., and Sudderth, W.D. (2001) The controller and stopper game for a linear diffusion. Annals of Probability 29:1111-1127.
27. Karatzas, I., and Wang, H. (2000) A barrier option of American type. Applied Mathematics & Optimization 42:259-280.
28. Karatzas, I., and Wang, H. (2001) Utility maximization with discretionary stopping. SIAM Journal on Control & Optimization 39:306-329.
29. Karatzas, I., Ocone, D.L., Wang, H., and Zervos, M. (2000) Finite-fuel singular control with discretionary stopping. Stochastics 71:1-50.
30. Krylov, N.V. (1980) Controlled Diffusion Processes. Springer-Verlag, New York.
31. Maitra, A., and Sudderth, W.D. (1996) Discrete Gambling and Stochastic Games. Springer-Verlag, New York.
32. Morimoto, H. (2003) Variational inequalities for combined control and stopping. SIAM Journal on Control & Optimization 42:686-708.
33. Peng, S. (1990) A general stochastic maximum principle for optimal control problems. SIAM Journal on Control & Optimization 28:966-979.
34. Peng, S. (1993) Backwards stochastic differrential equations and applications to optimal control. Applied Mathematics & Optimization 27:125-144.
35. Rishel, R.W. (1970) Necessary and sufficient dynamic programming conditions for continuous-time optimal control. SIAM Journal on Control & Optimization 8:559-571.
36. Rogers, L.C.G., and Williams, D. (1987) Diffusions, Markov Processes and Martingales: Vol. 2, Itô Calculus. Wiley, Chichester.