Geodesic
A multidimensional singular stochastic control problem on a finite time horizon
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 69 (2015) no. 1.

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: Singular stochastic control, generalized derivative, HJB equation, optimal control 
@article{AUM_2015_69_1_a0,
     author = {Boryc, Marcin and Kruk, {\L}ukasz},
     title = {A multidimensional singular stochastic control problem on a finite time horizon},
     journal = {Annales Universitatis Mariae Curie-Sk{\l}odowska. Mathematica },
     publisher = {mathdoc},
     volume = {69},
     number = {1},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUM_2015_69_1_a0/}
}
TY  - JOUR
AU  - Boryc, Marcin
AU  - Kruk, Łukasz
TI  - A multidimensional singular stochastic control problem on a finite time horizon
JO  - Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
PY  - 2015
VL  - 69
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AUM_2015_69_1_a0/
LA  - en
ID  - AUM_2015_69_1_a0
ER  - 
%0 Journal Article
%A Boryc, Marcin
%A Kruk, Łukasz
%T A multidimensional singular stochastic control problem on a finite time horizon
%J Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
%D 2015
%V 69
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AUM_2015_69_1_a0/
%G en
%F AUM_2015_69_1_a0
Boryc, Marcin; Kruk, Łukasz. A multidimensional singular stochastic control problem on a finite time horizon. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 69 (2015) no. 1. http://geodesic.mathdoc.fr/item/AUM_2015_69_1_a0/

[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.

[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.

[3] Dufour, F., Miller, B., Singular stichastic control problems, SIAM J. Control Optim. 43, No. 2 (2004), 708–730.

[4] Evans, L. C., Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.

[5] Fleming, W. H., Soner, H. M., Controlled Markov Processes and Viscosity Solutions, Springer, New York, 2006.

[6] Haussman, U. G., Suo, W., Singular optimal stochastic controls. I. Existence, SIAM J. Control Optim. 33, No. 3 (1995), 916–936.

[7] Karatzas, I., Shreve, S. E., Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1988.

[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.

[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.

[10] Krylov, N. V., Controlled Diffusion Processes, Springer-Verlag, New York, 1980.

[11] Menaldi, J. L., Taksar, M. I., Optimal correction problem of a multidimensional stochastic system, Automatica J. IFAC 25, No. 2 (1989), 223–232.

[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.

[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.

[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.

[18] http://en.wikipedia.org/wiki/Gronwall’s inequality, 24.09.2013.