A stochastic control problem and related free boundaries in finance
Chonghu Guan Xun Li Zuo Quan Xu Fahuai Yi
Mathematical Control & Related Fields 2017, 7(4): 563-584 doi: 10.3934/mcrf.2017021

In this paper, we investigate an optimal stopping problem (mixed with stochastic controls) for a manager whose utility is nonsmooth and nonconcave over a finite time horizon. The paper aims to develop a new methodology, which is significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature, so as to figure out the manager's best strategies. The problem is first reformulated into a free boundary problem with a fully nonlinear operator. Then, by means of a dual transformation, it is further converted into a free boundary problem with a linear operator, which can be consequently tackled by the classical method. Finally, using the inverse transformation, we obtain the properties of the optimal trading strategy and the optimal stopping time for the original problem.

keywords: Parabolic variational inequality free boundary nonsmooth utility optimal stopping dual transformation
On an exact penalty function method for semi-infinite programming problems
Cheng Ma Xun Li Ka-Fai Cedric Yiu Yongjian Yang Liansheng Zhang
Journal of Industrial & Management Optimization 2012, 8(3): 705-726 doi: 10.3934/jimo.2012.8.705
In this paper, we study a new exact and smooth penalty function for semi-infinite programming problems with continuous inequality constraints. Through this exact penalty function, we can transform a semi-infinite programming problem into an unconstrained optimization problem. We find that, under some reasonable conditions when the penalty parameter is sufficiently large, the local minimizer of this penalty function is the local minimizer of the primal problem. Moreover, under some mild assumptions, the local exactness property is explored. The numerical results demonstrate that it is an effective and promising approach for solving constrained semi-infinite programming problems.
keywords: smooth and exact penalty function nonsmooth optimization Constrained semi-infinite programming problem extended Managasarian-Fromovitz constraint qualification.
Optimal Sharpe ratio in continuous-time markets with and without a risk-free asset
Haixiang Yao Zhongfei Li Xun Li Yan Zeng
Journal of Industrial & Management Optimization 2017, 13(3): 1273-1290 doi: 10.3934/jimo.2016072

In this paper, we investigate a continuous-time mean-variance portfolio selection model with only risky assets and its optimal Sharpe ratio in a new way. We obtain closed-form expressions for the efficient investment strategy, the efficient frontier and the optimal Sharpe ratio. Using these results, we further prove that (ⅰ) the efficient frontier with only risky assets is significantly different from the one with inclusion of a risk-free asset and (ⅱ) inclusion of a risk-free asset strictly enhances the optimal Sharpe ratio. Also, we offer an explicit expression for the enhancement of the optimal Sharpe ratio. Finally, we test our theory results using an empirical analysis based on real data of Chinese equity market. Out-of-sample analyses shed light on advantages of our theoretical results established.

keywords: Continuous-time mean-variance model efficient investment strategy efficient frontier sharpe ratio Hamilton-Jacobi-Bellman equation
A mean-field formulation for multi-period asset-liability mean-variance portfolio selection with probability constraints
Xianping Wu Xun Li Zhongfei Li
Journal of Industrial & Management Optimization 2018, 14(1): 249-265 doi: 10.3934/jimo.2017045

This paper is concerned with studying an optimal multi-period asset-liability mean-variance portfolio selection with probability constraints using mean-field formulation without embedding technique. We strictly derive its analytical optimal strategy and efficient frontier. Numerical examples shed light on efficiency and accuracy of our method when dealing with this class of multi-period non-separable mean-variance portfolio selection problems.

keywords: Mean-field formulation multi-period portfolio selection asset-liability management probability constraints optimal strategy
A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon
Jianhui Huang Xun Li Jiongmin Yong
Mathematical Control & Related Fields 2015, 5(1): 97-139 doi: 10.3934/mcrf.2015.5.97
A linear-quadratic (LQ, for short) optimal control problem is considered for mean-field stochastic differential equations with constant coefficients in an infinite horizon. The stabilizability of the control system is studied followed by the discussion of the well-posedness of the LQ problem. The optimal control can be expressed as a linear state feedback involving the state and its mean, through the solutions of two algebraic Riccati equations. The solvability of such kind of Riccati equations is investigated by means of semi-definite programming method.
keywords: MF-stabilizability Riccati equation. Mean-field stochastic differential equation linear-quadratic optimal control
Optimal stopping investment with non-smooth utility over an infinite time horizon
Xiaoshan Chen Xun Li Fahuai Yi
Journal of Industrial & Management Optimization 2019, 15(1): 81-96 doi: 10.3934/jimo.2018033

This study addresses an investment problem facing a venture fund manager who has a non-smooth utility function. The theoretical model characterizes an absolute performance-based compensation package. Technically, the research methodology features stochastic control and optimal stopping by formulating a free-boundary problem with a nonlinear equation, which is transferred to a new one with a linear equation. Numerical results based on simulations are presented to better illustrate this practical investment decision mechanism.

keywords: Optimal stopping optimal investment non-smooth utility dual transformation free boundary

Year of publication

Related Authors

Related Keywords

[Back to Top]