Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach

In this paper, we study the indefinite linear-quadratic (LQ) stochastic optimal control problem for stochastic differential equations (SDEs) with jump diffusions and random coefficients driven by both the Brownian motion and the (compensated) Poisson process. In our problem setup, the coefficients i...

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Autores principales:	Jun Moon, Jin-Ho Chung
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	stochastic systems with jump diffusions and random coefficients completion of squares method stochastic HJB equation with jump diffusions verification theorem Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/6c80e97f68b94e3b936ccc2741ba7d87
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spelling	oai:doaj.org-article:6c80e97f68b94e3b936ccc2741ba7d872021-11-25T18:17:11ZIndefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach10.3390/math92229182227-7390https://doaj.org/article/6c80e97f68b94e3b936ccc2741ba7d872021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2918https://doaj.org/toc/2227-7390In this paper, we study the indefinite linear-quadratic (LQ) stochastic optimal control problem for stochastic differential equations (SDEs) with jump diffusions and random coefficients driven by both the Brownian motion and the (compensated) Poisson process. In our problem setup, the coefficients in the SDE and the objective functional are allowed to be random, and the jump-diffusion part of the SDE depends on the state and control variables. Moreover, the cost parameters in the objective functional need not be (positive) definite matrices. Although the solution to this problem can also be obtained through the stochastic maximum principle or the dynamic programming principle, our approach is simple and direct. In particular, by using the Itô-Wentzell’s formula, together with the integro-type stochastic Riccati differential equation (ISRDE) and the backward SDE (BSDE) with jump diffusions, we obtain the equivalent objective functional that is quadratic in control <i>u</i> under the positive definiteness condition, where the approach is known as the completion of squares method. Then the explicit optimal solution, which is linear in state characterized by the ISRDE and the BSDE jump diffusions, and the associated optimal cost are derived by eliminating the quadratic term of <i>u</i> in the equivalent objective functional. We also verify the optimality of the proposed solution via the verification theorem, which requires solving the stochastic HJB equation, a class of stochastic partial differential equations with jump diffusions.Jun MoonJin-Ho ChungMDPI AGarticlestochastic systems with jump diffusions and random coefficientscompletion of squares methodstochastic HJB equation with jump diffusionsverification theoremMathematicsQA1-939ENMathematics, Vol 9, Iss 2918, p 2918 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	stochastic systems with jump diffusions and random coefficients completion of squares method stochastic HJB equation with jump diffusions verification theorem Mathematics QA1-939
spellingShingle	stochastic systems with jump diffusions and random coefficients completion of squares method stochastic HJB equation with jump diffusions verification theorem Mathematics QA1-939 Jun Moon Jin-Ho Chung Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach
description	In this paper, we study the indefinite linear-quadratic (LQ) stochastic optimal control problem for stochastic differential equations (SDEs) with jump diffusions and random coefficients driven by both the Brownian motion and the (compensated) Poisson process. In our problem setup, the coefficients in the SDE and the objective functional are allowed to be random, and the jump-diffusion part of the SDE depends on the state and control variables. Moreover, the cost parameters in the objective functional need not be (positive) definite matrices. Although the solution to this problem can also be obtained through the stochastic maximum principle or the dynamic programming principle, our approach is simple and direct. In particular, by using the Itô-Wentzell’s formula, together with the integro-type stochastic Riccati differential equation (ISRDE) and the backward SDE (BSDE) with jump diffusions, we obtain the equivalent objective functional that is quadratic in control <i>u</i> under the positive definiteness condition, where the approach is known as the completion of squares method. Then the explicit optimal solution, which is linear in state characterized by the ISRDE and the BSDE jump diffusions, and the associated optimal cost are derived by eliminating the quadratic term of <i>u</i> in the equivalent objective functional. We also verify the optimality of the proposed solution via the verification theorem, which requires solving the stochastic HJB equation, a class of stochastic partial differential equations with jump diffusions.
format	article
author	Jun Moon Jin-Ho Chung
author_facet	Jun Moon Jin-Ho Chung
author_sort	Jun Moon
title	Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach
title_short	Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach
title_full	Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach
title_fullStr	Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach
title_full_unstemmed	Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach
title_sort	indefinite linear-quadratic stochastic control problem for jump-diffusion models with random coefficients: a completion of squares approach
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/6c80e97f68b94e3b936ccc2741ba7d87
work_keys_str_mv	AT junmoon indefinitelinearquadraticstochasticcontrolproblemforjumpdiffusionmodelswithrandomcoefficientsacompletionofsquaresapproach AT jinhochung indefinitelinearquadraticstochasticcontrolproblemforjumpdiffusionmodelswithrandomcoefficientsacompletionofsquaresapproach
_version_	1718411386895204352

Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach

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