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Sobolev weak solutions of the Hamilton--Jacobi--Bellman equations

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journal contribution
posted on 10.11.2016, 11:05 by Lifeng Wei, Zhen Wu, Huaizhong Zhao
This paper is concerned with the Sobolev weak solutions of the Hamilton-Jacobi-Bellman (HJB) equations. These equations are derived from the dynamic programming principle in the study of stochastic optimal control problems. Adopting the Doob-Meyer decomposition theorem as one of the main tools, we prove that the optimal value function is the unique Sobolev weak solution of the corresponding HJB equation. In the recursive optimal control problem, the cost function is described by the solution of a backward stochastic differential equation (BSDE). This problem has a practical background in economics and finance. We prove that the value function is the unique Sobolev weak solution of the related HJB equation by virtue of the nonlinear Doob-Meyer decomposition theorem introduced in the study of BSDEs.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

SIAM Journal on Control and Optimization

Volume

52

Issue

3

Pages

1499 - 1526

Citation

WEI, L., WU, Z. and ZHAO, H., 2014. Sobolev weak solutions of the Hamilton Jacobi Bellman equations. SIAM Journal on Control and Optimization, 52 (3), pp. 1499 - 1526

Publisher

© Society for Industrial and Applied Mathematics

Version

AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Acceptance date

29/01/2014

Publication date

2014-05-06

Notes

This article was published in the SIAM Journal on Control and Optimization [© SIAM] and the definitive version is available at: http://dx.doi.org/10.1137/120889174

ISSN

0363-0129

eISSN

1095-7138

Language

en

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