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2025 , Vol. 41 >Issue 2: 375 - 399

DOI: https://doi.org/10.1007/s10255-024-1157-9

ARTICLES

Constrained Stochastic Recursive Linear Quadratic Optimal Control Problems and Application to Finance

  • Liang-quan ZHANG ,
  • Qing ZHOU
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  • 1 School of Mathematics, Renmin University of China, Beijing 100872, China;
    2 School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

收稿日期: 2022-04-29

  录用日期: 2024-07-24

  网络出版日期: 2025-03-28

基金资助

L. Zhang acknowledges the financial support partly by the National Nature Science Foundation of China (Grant No. 12171053, 11701040, 11871010 &61871058) and the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (No. 23XNKJ05). Q. Zhou acknowledges the financial support partly by the National Nature Science Foundation of China (Grant No. 11871010, 11971040,) and the Fundamental Research Funds for the Central Universities (No. 2019XD-A11).

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Constrained Stochastic Recursive Linear Quadratic Optimal Control Problems and Application to Finance

  • Liang-quan ZHANG ,
  • Qing ZHOU
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  • 1 School of Mathematics, Renmin University of China, Beijing 100872, China;
    2 School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

Received date: 2022-04-29

  Accepted date: 2024-07-24

  Online published: 2025-03-28

Supported by

L. Zhang acknowledges the financial support partly by the National Nature Science Foundation of China (Grant No. 12171053, 11701040, 11871010 &61871058) and the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (No. 23XNKJ05). Q. Zhou acknowledges the financial support partly by the National Nature Science Foundation of China (Grant No. 11871010, 11971040,) and the Fundamental Research Funds for the Central Universities (No. 2019XD-A11).

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摘要

In this paper, we focus on a control-constrained stochastic LQ optimal control problem via backward stochastic differential equation (BSDE in short) with deterministic coefficients. One of the significant features in this framework, in contrast to the classical LQ issue, embodies that the admissible control set needs to satisfy more than the square integrability. By introducing two kinds of new generalized Riccati equations, we are able to announce the explicit optimal control and the solution to the corresponding H-J-B equation. A linear quadratic recursive utility portfolio optimization problem in the financial engineering is discussed as an explicitly illustrated example of the main result with short-selling prohibited. Feasibility of the mean-variance portfolio selection problem via BSDE for a financial market is characterized, and associated efficient portfolios are given in a closed form.

关键词： efficient portfolio; forward-backward stochastic differential equations; mean-variance portfolio selection; Riccati equation; recursive utilities; stochastic linear quadratic optimal control

本文引用格式

Liang-quan ZHANG , Qing ZHOU . Constrained Stochastic Recursive Linear Quadratic Optimal Control Problems and Application to Finance[J]. 应用数学学报(英文), 2025 , 41(2) : 375 -399 . DOI: 10.1007/s10255-024-1157-9

Abstract

In this paper, we focus on a control-constrained stochastic LQ optimal control problem via backward stochastic differential equation (BSDE in short) with deterministic coefficients. One of the significant features in this framework, in contrast to the classical LQ issue, embodies that the admissible control set needs to satisfy more than the square integrability. By introducing two kinds of new generalized Riccati equations, we are able to announce the explicit optimal control and the solution to the corresponding H-J-B equation. A linear quadratic recursive utility portfolio optimization problem in the financial engineering is discussed as an explicitly illustrated example of the main result with short-selling prohibited. Feasibility of the mean-variance portfolio selection problem via BSDE for a financial market is characterized, and associated efficient portfolios are given in a closed form.

Key words： efficient portfolio; forward-backward stochastic differential equations; mean-variance portfolio selection; Riccati equation; recursive utilities; stochastic linear quadratic optimal control

参考文献

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