Bounded variation singular stochastic control and dynkin game

SIAM Journal on Control and Optimization

Volumn 44, Issue 4, 2005, Pages 1289-1321

  Boetius, Frederik ^a  

^a KPMG AG   (Germany)

Author keywords

Backward stochastic differential equation; Comparison theorem for SDE; Dynkin game; Nonlinear cost functional; Optimal stopping; Pathwise construction; Singular stochastic control

Indexed keywords

BOUNDARY VALUE PROBLEMS; DIFFERENTIAL EQUATIONS; ESTIMATION; OPTIMAL CONTROL SYSTEMS; PROBLEM SOLVING; THEOREM PROVING;

BACKWARD STOCHASTIC DIFFERENTIAL EQUATION (BSDE); COMPARISON THEOREM FOR SDE; DYNKIN GAMES; NONLINEAR COST FUNCTIONALS; OPTIMAL STOPPING; PATHWISE CONSTRUCTION; SINGULAR STOCHASTIC CONTROL;

STOCHASTIC CONTROL SYSTEMS;

EID: 33746046784     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012903429049     Document Type: Article

References (68)

1. L. H. R. ALVAREZ, Optimal exit and valuation under demand uncertainty: A real options approach, European J. Oper. Res., 114 (1999), pp. 320-329.
2. L. H. R. ALVAREZ, Singular stochastic control, linear diffusions, and optimal stopping: A class of solvable problems, SIAM J. Control Optim., 39 (2001), pp. 1697-1710.
3. L. H. R. ALVAREZ AND RUNE STENBACKA, Adoption of uncertain multi-stage technology projects: A real options approach, J. Math. Econom., 35 (2001), pp. 71-97.
4. F. M. BALDURSSON, Singular stochastic control and optimal stopping, Stochastics, 21 (1987), pp. 1-40.
5. F. M. BALDURSSON AND I. KARATZAS, Irreversible investment and industry equilibrium, Finance Stoch., 1 (1997), pp. 69-89.
6. J. S. BARAS, R. J. ELLIOT, AND M. KOHLMANN, The partially observed stochastic minimum principle, SIAM J. Control Optim., 27 (1989), pp. 1279-1292.
7. J. A. BATHER AND H. CHERNOFF, Sequential decisions in the control of a spaceship, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 3, (1966), University of California Press, Berkeley, CA, 1967, pp. 181-207.
8. P. BAXENDALE, Stability and equilibrium properties of stochastic flows of diffeomorphisms, in Diffusion Processes and Related Problems in Analysis, Vol. 2, Progr. Probab. 27, M. A. Pinsky and V. Wihsturz, eds., Birkhäuser Boston, Boston, 1992, pp. 3-35.
9. C. BENDER, Rückwärtsstochastische Differentialgleichungen und Anwendungen bei der Bewertung von Finanzderivaten, Diplomarbeit, Universität Konstanz, Konstanz, Germany, 2002.
10. C. BENDER. AND M. KOHLMANN, BSDEs with stochastic Lipschitz condition, CoFE Discussion Paper 00-08, http://cofe.uni-konstanz.de (2000).
11. V. E. EENEŠ, L. A. SHEPP, AND H. S. WITSENHAUSEN, Some solvable stochastic control problems, Stochastics, 4 (1980), pp. 39-83.
12. A. BENSOUSSAN AND J. L. LIONS, Applications of Variational Inequalities in Stochastic Control, North-Holland, Amsterdam, New York, 1982.
13. F. E. BENTH, K. H. KARLSEN, AND K. REIKVAM, Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution, Finance Stoch., 5 (2001), pp. 447-467.
14. J. M. BISMUT, Sur un problème de Dynkin, Wahrscheinlichkeitstheorie und Verw. Giebiete, 39 (1977), pp. 31-53.
15. J. M. BISMUT, Temps d'arrêt optimal, quasi-temps d'arrêt et retournement du temps, Ann. Probab., 7 (1979), pp. 933-963.
16. F. BOETIUS, Entry-exit problems and bounded variation singular stochastic control, working paper, 2003.
17. F. BOETIUS, Singular Stochastic Control and Its Relations to Dynkin Game and Entry-Exit Problems, Ph.D. thesis, Universität Konstanz, Konstanz, Germany, 2003.
18. F. BOETIUS AND M. KOHLMANN, Connections between optimal stopping and singular stochastic control, Stochastic Process Appl., 77 (1998), pp. 253-281.
19. A. CADENILLAS, Consumption-investment problems with transaction costs: Survey and open problems, Math. Methods Oper. Res., 51 (2000), pp. 43-68.
20. P.-L. CHOW, J.-L. MENALDI, AND M. ROBIN, Additive control of stochastic linear systems with finite horizon, SIAM J. Control Optim., 23 (1985), pp. 858-899.
21. J. CVITANIĆ AND I. KARATZAS, Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab., 24 (1996), pp. 2024-2056.
22. R. W. R. DARLING, Isotropic stochastic flows, in Diffusion Processes and Related Problems in Analysis, Vol. 2, Progr. Probab. 27, M. A. Pinsky and V. Wihsturz, eds., Birkhäuser Boston, Boston, 1992, pp. 75-94.
23. M. H. A. DAVIS AND I. KARATZAS, A deterministic approach to optimal stopping, in Probability, Statistics, and Optimization, F. P. Kelly, ed., Wiley, New York, 1994, pp. 455-466.
24. M. H. A. DAVIS AND A. NORMAN, Portfolio selection with transaction costs, Math. Oper. Res., 15 (1990), pp. 676-713.
25. A. K. DIXIT AND R. S. PINDYCK, Investment under Uncertainty, Princeton University Press, Princeton, NJ, 1994.
26. D. DUFFIE AND L. G. EPSTEIN, Stochastic differential utility, Econometrica, 60 (1992), pp. 353-394.
27. E. B. DYNKIN, Game variant of a problem on optimal stopping, Soviet Math. Dokl., 10 (1969), pp. 270-274.
28. N. EL KAROUI, C. KAPOUDJIAN, E. PARDOUX, S. PENG, AND M. C. QUENEZ, Reflected solutions of backward SDE's, and related obstacle problems for PDE's, Ann. Probab., 25 (1997), pp. 702-737.
29. N. EL KAROUI AND I. KARATZAS, Probabilistic aspects of finite-fuel, reflected follower problems, Acta Appl. Math., 11 (1988), pp. 223-258.
30. N. EL KAROUI AND I. KARATZAS, A new approach to the Skorohod problem, and its applications, Stochastics Stochastics Rep., 34 (1991), pp. 57-82.
31. N. EL KAROUI, S. PENG, AND M. C. QUENEZ, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), pp. 1-71.
32. N. EL KAROUI AND M. C. QUENEZ, Non-linear pricing theory and backward stochastic differential equations, in Financial Mathematics (Bressanone, Italy, 1996), Lecture Notes in Math. 1656, W. Runggaldier, ed., Springer-Verlag, Berlin, 1997, pp. 191-246.
33. W. H. FLEMING AND H. M. SONER, Controlled Markov Processes and Viscosity Solutions, Appl. Math. 25, Springer-Verlag, New York, 1993.
34. A. FRIEDMAN, Stochastic Differential Equations and Applications, Vol. 2, Academic Press, New York, London, 1976.
35. D. FUDENBERG AND J. TIROLE, Game Theory, MIT Press, Cambridge, MA, 1991.
36. S. HAMADENE AND J.-P. LEPELTIER, Backward equations, stochastic control and zero-sum stochastic differential games, Stochastics Stochastics Rep., 54 (1995), pp. 221-231.
37. S. HAMADENE AND J.-P. LEPELTIER, Zero-sum stochastic differential games and backward equations, Systems Control Lett., 24 (1995), pp. 259-263.
38. S. HAMADENE AND J.-P. LEPELTIER, Reflected BSDEs and mixed game problem, Stochastic Process Appl., 85 (2000), pp. 177-188.
39. S. HAMADENE, J.-P. LEPELTIER, AND Z. Wu, Infinite horizon reflected backward stochastic differential equations and applications in mixed control and game problems, Probab. Math. Statist., 19 (1999), pp. 211-234.
40. U. G. HAUSSMANN AND W. SUO, Singular optimal stochastic controls I: Existence, SIAM J. Control Optim., 33 (1995), pp. 916-936.
41. B. HøJGAARD AND M. TAKSAR, Optimal risk control for a large corporation in the presence of returns on investments, Finance Stoch., 5 (2001), pp. 527-547.
42. I. KARATZAS, A class of singular stochastic control problems, Adv. in Appl. Probab., 15 (1983), pp. 225-254.
43. I. KARATZAS, Probabilistic aspects of finite fuel stochastic control, Proc. Natl. Acad. Sci. USA, 82 (1985), pp. 5579-5581.
44. I. KARATZAS AND S. E. SHREVE, Connections between optimal stopping and singular stochastic control I. Monotone follower problems, SIAM J. Control Optim., 22 (1984), pp. 856-877.
45. I. KARATZAS AND S. E. SHREVE, Connections between optimal stopping and singular stochastic control II. Reflected follower problems, SIAM J. Control Optim., 23 (1985), pp. 433-451.
46. I. KARATZAS AND S. E. SHREVE, Equivalent models for finite-fuel stochastic control, Stochastics, 18 (1986), pp. 245-276.
47. I. KARATZAS AND S. E. SHREVE, Brownian Motion and Stochastic Calculus, Grad. Texts in Math. 113, 3rd ed., Springer-Verlag, New York, 1994.
48. I. KARATZAS AND S. E. SHREVE, Methods of Mathematical Finance, Appl. Math. 39, Springer-Verlag, New York, 1998.
49. I. KARATZAS AND H. WANG, Connections between bounded-variation control and Dynkin games, in Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001, pp. 353-362.
50. Y. KIFER, Game options, Finance Stoch., 4 (2000), pp. 443-463.
51. J. KOMLÓS, A generalization of a problem of Steinhaus, Acta Math. Acad. Sci. Hungar., 18 (1967), pp. 217-229.
52. L. KRUK, Optimal policies for n-dimensional singular stochastic control problems. Part I: The Skorohod problem, SIAM J. Control Optim., 38 (2000), pp. 1603-1622.
53. L. KRUK, Optimal policies for n-dimensional singular stochastic control problems. Part II: The radially symmetric case. Ergodic control, SIAM J. Control Optim., 39 (2000), pp. 635-659.
54. H. J. KUSHNER, Control and optimal control of assemble to order manufacturing systems under heavy traffic, Stochastics Stochastics Rep., 66 (1999), pp. 233-27.
55. J. MA AND J. YONG, Dynamic programming for multidimensional stochastic control problems, Acta Math. Sin. (Engl. Ser.), 15 (1999), pp. 485-506.
56. J. MA AND J. YONG, Forward-backward Stochastic Differential Equations and Their Applications, Lecture Notes in Math. 1702, Springer-Verlag, Berlin, 1999.
57. R. MCDONALD AND D. SIEGEL, The value of waiting to invest, Quart. J. Econom., 101 (1986), pp. 707-727.
58. H. P. MCKEAN, JR., Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics, Ind. Management Rev., 6 (1965), pp. 32-39.
59. J. L. MENALDI AND M. I. TAKSAR, Optimal correction problem of a multidimensional stochastic system, Automatica J. IFAC, 25 (1989), pp. 223-232.
60. S. PENG, A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation, Stochastics Stochastics Rep., 38 (1992), pp. 119-134.
61. S. PENG, Monotonie limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type, Probab. Theory Related Fields, 113 (1999), pp. 473-499.
62. P. FROTTER, Stochastic Integration and Differential Equations, Appl. Math. 21, Springer-Verlag, Berlin, 1990.
63. P. A. SAMUELSON, Rational theory of warrant pricing, Ind. Mangement Rev., 6 (1965), pp. 13-31.
64. U. SCHMOCK, S. E. SHREVE, AND U. WYSTUP, Valuation of exotic options under shortselling constraints, Finance Stoch., 6 (2002), pp. 143-172.
65. A. N. SHIRYAEV, Optimal Stopping Rules, Springer-Verlag, Berlin, Heidelberg, New York, 1978.
66. H. M. SONER AND S. E. SHREVE, Regularity of the value function for a two-dimensional singular stochastic control problem, SIAM J. Control Optim., 27 (1989), pp. 876-907.
67. M. TAKSAR, Average optimal singular control and a related stopping problem, Math. Oper. Res., 10 (1985), pp. 63-81.
68. J. YONG AND X. Y. ZHOU, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.