A multidimensional singular stochastic control problem on a finite time horizon

Marcin Boryc, Łukasz Kruk

Abstract


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

Full Text:

PDF

References


Budhiraja, A., Ross, K., Existence of optimal controls for singular control problems with state constraints, Ann. Appl. Probab. 16, No. 4 (2006), 2235–2255.

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.

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

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

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

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

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

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.

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.

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

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

Rudin, W., Functional Analysis, McGraw-Hill Book Company, New York, 1991.

Rudin, W., Principles of Mathematical Analysis, McGraw-Hill Book Company, New York, 1976.

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.

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.

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.

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.

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




DOI: http://dx.doi.org/10.17951/a.2015.69.1.23
Date of publication: 2015-11-30 09:21:11
Date of submission: 2015-09-03 12:17:00


Statistics


Total abstract view - 1074
Downloads (from 2020-06-17) - PDF - 709

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2015 Marcin Boryc, Łukasz Kruk