The Tree-Grid Method with Control-Independent Stencil

Kossaczký, Igor; Ehrhardt, Mattias; Günther, Michael

  • Proceedings of Equadiff 14, Publisher: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing(Bratislava), page 79-88

Abstract

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The Tree-Grid method is a novel explicit convergent scheme for solving stochastic control problems or Hamilton-Jacobi-Bellman equations with one space dimension. One of the characteristics of the scheme is that the stencil size is dependent on space, control and possibly also on time. Because of the dependence on the control variable, it is not trivial to solve the optimization problem inside the method. Recently, this optimization part was solved by brute-force testing of all permitted controls. In this paper, we present a simple modification of the Tree-Grid scheme leading to a control-independent stencil. Under such modification an optimal control can be found analytically or with the Fibonacci search algorithm.

How to cite

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Kossaczký, Igor, Ehrhardt, Mattias, and Günther, Michael. "The Tree-Grid Method with Control-Independent Stencil." Proceedings of Equadiff 14. Bratislava: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing, 2017. 79-88. <http://eudml.org/doc/294934>.

@inProceedings{Kossaczký2017,
abstract = {The Tree-Grid method is a novel explicit convergent scheme for solving stochastic control problems or Hamilton-Jacobi-Bellman equations with one space dimension. One of the characteristics of the scheme is that the stencil size is dependent on space, control and possibly also on time. Because of the dependence on the control variable, it is not trivial to solve the optimization problem inside the method. Recently, this optimization part was solved by brute-force testing of all permitted controls. In this paper, we present a simple modification of the Tree-Grid scheme leading to a control-independent stencil. Under such modification an optimal control can be found analytically or with the Fibonacci search algorithm.},
author = {Kossaczký, Igor, Ehrhardt, Mattias, Günther, Michael},
booktitle = {Proceedings of Equadiff 14},
keywords = {Tree-Grid Method, Hamilton-Jacobi-Bellman equation, Stochastic control problem, Fibonacci algorithm},
location = {Bratislava},
pages = {79-88},
publisher = {Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing},
title = {The Tree-Grid Method with Control-Independent Stencil},
url = {http://eudml.org/doc/294934},
year = {2017},
}

TY - CLSWK
AU - Kossaczký, Igor
AU - Ehrhardt, Mattias
AU - Günther, Michael
TI - The Tree-Grid Method with Control-Independent Stencil
T2 - Proceedings of Equadiff 14
PY - 2017
CY - Bratislava
PB - Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing
SP - 79
EP - 88
AB - The Tree-Grid method is a novel explicit convergent scheme for solving stochastic control problems or Hamilton-Jacobi-Bellman equations with one space dimension. One of the characteristics of the scheme is that the stencil size is dependent on space, control and possibly also on time. Because of the dependence on the control variable, it is not trivial to solve the optimization problem inside the method. Recently, this optimization part was solved by brute-force testing of all permitted controls. In this paper, we present a simple modification of the Tree-Grid scheme leading to a control-independent stencil. Under such modification an optimal control can be found analytically or with the Fibonacci search algorithm.
KW - Tree-Grid Method, Hamilton-Jacobi-Bellman equation, Stochastic control problem, Fibonacci algorithm
UR - http://eudml.org/doc/294934
ER -

References

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  1. Ferguson, D.E., Fibonaccian searching, . Communications of the ACM, 3(12):648, 1960. MR0115851
  2. Forsyth, P.A., HASH(0x20dc550), Labahn, G., Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance, . Journal of Computational Finance, 11(2):1, 2007. 
  3. Kilianová, S., HASH(0x20f2568), Ševčovič, D., A transformation method for solving the Hamilton-Jacobi-Bellman equation for a constrained dynamic stochastic optimal allocation problem, . The ANZIAM Journal, 55(01):14–38, 2013. MR3144202
  4. Kossaczký, I., Ehrhardt, M., HASH(0x20f4660), Günther, M., A new convergent explicit Tree-Grid method for HJB equations in one space dimension, . Preprint 17/06, University of Wuppertal, to appear in Numerical Mathematics: Theory, Methods and Applications, 2017. MR3844119
  5. Kushner, H., HASH(0x20f4f60), Dupuis, P.G., Numerical methods for stochastic control problems in continuous time, , volume 24. Springer Science & Business Media, 2013. MR1217486
  6. Wang, J., HASH(0x20f7808), Forsyth, P.A., Maximal use of central differencing for Hamilton-Jacobi-Bellman PDEs in finance, . SIAM Journal on Numerical Analysis, 46(3):1580–1601, 2008. MR2391007
  7. Yong, Jiongmin, HASH(0x20f8108), Zhou, Xun Yu, Stochastic controls: Hamiltonian systems and HJB equations, , volume 43. Springer Science & Business Media, 1999. MR1696772

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