A singular stochastic control problem in n dimensions with timedependent coefficients on a finite time horizon is considered. We show that the value function for this problem is a generalized solution of the corresponding HJB equation with locally bounded second derivatives with respect to the space variables and the first derivative with respect to time. Moreover, we prove that an optimal control exists and is unique
@article{bwmeta1.element.doi-10_1515_umcsmath-2015-0011,
author = {Marcin Boryc and \L ukasz Kruk},
title = {A multidimensional singular stochastic control problem on a finite time horizon},
journal = {Annales UMCS, Mathematica},
volume = {68},
year = {2015},
pages = {23-57},
zbl = {1317.93268},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_umcsmath-2015-0011}
}
Marcin Boryc; Łukasz Kruk. A multidimensional singular stochastic control problem on a finite time horizon. Annales UMCS, Mathematica, Tome 68 (2015) pp. 23-57. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_umcsmath-2015-0011/
[1] Budhiraja, A., Ross, K., Existence of optimal controls for singular control problems with state constraints, Ann. Appl. Probab. 16, No. 4 (2006), 2235-2255. | Zbl 1118.49008
[2] Chow, P. L., Menaldi, J. L., Robin, M., Additive control of stochastic linear systems with finite horizon, SIAM J. Control Optim. 23, No.6 (1985), 858-899. | Zbl 0587.93068
[3] Dufour, F., Miller, B., Singular stichastic control problems, SIAM J. Control Optim. 43, No. 2 (2004), 708-730. | Zbl 1101.93084
[4] Evans, L. C., Partial Differential Equations, American Mathematical Society, Providence, RI, 1998. | Zbl 0902.35002
[5] Fleming, W. H., Soner, H. M., Controlled Markov Processes and Viscosity Solutions, Springer, New York, 2006. | Zbl 1105.60005
[6] Haussman, U. G., Suo, W., Singular optimal stochastic controls. I. Existence, SIAM J. Control Optim. 33, No. 3 (1995), 916-936. | Zbl 0925.93958
[7] Karatzas, I., Shreve, S. E., Brownian Motion and Stochastic Calculus, Springer- Verlag, New York, 1988. | Zbl 0638.60065
[8] Kruk, Ł., Optimal policies for n-dimensional singular stochastic control problems, Part I: The Skorokhod problem, SIAM J. Control Optim. 38, No. 5 (2000), 1603-1622. | Zbl 0990.93134
[9] Kruk, Ł., Optimal policies for n-dimensional singular stochastic control problems, Part II: The radially symmetric case. Ergodic control, SIAM J. Control Optim. 39, No. 2 (2000), 635-659. | Zbl 0990.93135
[10] Krylov, N. V., Controlled Diffusion Processes, Springer-Verlag, New York, 1980. | Zbl 0459.93002
[11] Menaldi, J. L., Taksar, M. I., Optimal correction problem of a multidimensional stochastic system, Automatica J. IFAC 25, No. 2 (1989), 223-232. | Zbl 0685.93079
[12] Rudin, W., Functional Analysis, McGraw-Hill Book Company, New York, 1991.
[13] Rudin, W., Principles of Mathematical Analysis, McGraw-Hill Book Company, New York, 1976.
[14] Soner, H. M., Shreve, S. E., Regularity of the value function for a two-dimensional singular stochastic control problem, SIAM J. Control Optim. 27 (1989), 876-907. | Zbl 0685.93076
[15] Soner, H. M., Shreve, S. E., A free boundary problem related to singular stochastic control, Applied stochastic analysis (London, 1989), Stochastics Monogr. 5, Gordon and Breach, New York, 1991, 265-301.
[16] Soner, H. M., Shreve, S. E., A free boundary problem related to singular stochastic control: the parabolic case, Comm. Partial Differential Equations 16 (1991), 373-424. | Zbl 0746.35058
[17] S. A. Williams, P. L. Chow and J. L. Menaldi, Regularity of the free boundary in singular stochastic control, J. Differential Equations 111 (1994), 175-201. | Zbl 0930.93086
[18] http://en.wikipedia.org/wiki/Gronwall's_inequality, 24.09.2013.