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
Global existence and uniqueness for a hyperbolic system with free boundary
Tong Yang Fahuai Yi
Discrete & Continuous Dynamical Systems - A 2001, 7(4): 763-780 doi: 10.3934/dcds.2001.7.763
In this paper, we consider a $2\times 2$ hyperbolic system originates from the theory of phase dynamics. This one-phase problem can be obtained by using the Catteneo-Fourier law which is a variant of the standard Fourier law in one dimensional space. A new classical existence and uniqueness result is established by some a priori estimates using the characteristic method. The convergence of the solutions to the one of classical Stefan problems is also obtained.
keywords: Stefan problem classical solution. Hyperbolic system
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

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