Constrained Utility Deviation-Risk Optimization and Time-consistent HJB Equation

SIAM Journal on Control and Optimization, 2020, Vol. 58, No. 2 : pp. 866-894

30 Pages Posted: 30 May 2019 Last revised: 1 Apr 2020

See all articles by Jiawen Gu

Jiawen Gu

Southern University of Science and Technology

Shijing Si

Duke University

Harry Zheng

Imperial College London - Mathematical Finance

Date Written: March 23, 2020

Abstract

In this paper we propose a unified utility deviation-risk model which covers both utility maximization and mean-variance analysis as special cases. We derive the time-consistent Hamilton-Jacobi-Bellman (HJB) equation for the equilibrium value function and significantly reduce the number of state variables, which makes the HJB equation derived in this paper much easier to solve than the extended HJB equation in the literature. We illustrate the usefulness of the time-consistent HJB equation with several examples which recover the known results in the literature and go beyond, including a mean-variance model with stochastic volatility dependent risk aversion, a utility deviation-risk model with state dependent risk aversion and control constraint, and a constrained portfolio selection model. The numerical and statistical tests show that the utility and deviation-risk have a significant impact on the equilibrium control strategy and the distribution of the terminal wealth.

Keywords: utility deviation-risk optimization; stochastic risk aversion; incomplete market; control constraint; time-consistent dynamic programming equation

Suggested Citation

Gu, Jiawen and Si, Shijing and Zheng, Harry, Constrained Utility Deviation-Risk Optimization and Time-consistent HJB Equation (March 23, 2020). SIAM Journal on Control and Optimization, 2020, Vol. 58, No. 2 : pp. 866-894, Available at SSRN: https://ssrn.com/abstract=3384245 or http://dx.doi.org/10.2139/ssrn.3384245

Jiawen Gu (Contact Author)

Southern University of Science and Technology ( email )

No 1088, xueyuan Rd.
Xili, Nanshan District
Shenzhen, Guangdong 518055
China

Shijing Si

Duke University ( email )

100 Fuqua Drive
Durham, NC 27708-0204
United States

Harry Zheng

Imperial College London - Mathematical Finance ( email )

United Kingdom

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