A multidimensional singular stochastic control problem on a finite time horizon

Marcin Boryc; Łukasz Kruk

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2015)

  • Volume: 69, Issue: 1
  • ISSN: 0365-1029

Abstract

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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.

How to cite

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Marcin Boryc, and Łukasz Kruk. "A multidimensional singular stochastic control problem on a finite time horizon." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 69.1 (2015): null. <http://eudml.org/doc/289816>.

@article{MarcinBoryc2015,
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.},
author = {Marcin Boryc, Łukasz Kruk},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Singular stochastic control; generalized derivative; HJB equation; optimal control},
language = {eng},
number = {1},
pages = {null},
title = {A multidimensional singular stochastic control problem on a finite time horizon},
url = {http://eudml.org/doc/289816},
volume = {69},
year = {2015},
}

TY - JOUR
AU - Marcin Boryc
AU - Łukasz Kruk
TI - A multidimensional singular stochastic control problem on a finite time horizon
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2015
VL - 69
IS - 1
SP - null
AB - 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.
LA - eng
KW - Singular stochastic control; generalized derivative; HJB equation; optimal control
UR - http://eudml.org/doc/289816
ER -

References

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  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. 

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