# American Institute of Mathematical Sciences

December  2017, 7(4): 563-584. doi: 10.3934/mcrf.2017021

## A stochastic control problem and related free boundaries in finance

 1 School of Mathematics, Jiaying University, Meizhou 514015, China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China 3 School of Finance, Guangdong University of Foreign Studies, Guangzhou, China

* Corresponding author

Received  August 2016 Revised  December 2016 Published  September 2017

Fund Project: The first author is supported by NNSF of China (No.11626117 and No.11601163), NSF of Guangdong Province of China (No.2016A030307008). The second author is supported by Research Grants Council of Hong Kong under grants 519913,15224215 and 15255416. The third author is supported by NSFC (No.11471276) and Research Grants Council of Hong Kong (No.15204216 and No.15202817). The fourth author is supported by NNSF of China (No.11371155); NSF Guangdong Province of China (No.2016A030313448 and No.2015A030313574); The Humanities and Social Science Research Foundation of the Ministry of Education of China (No.15YJAZH051)

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.

Citation: Chonghu Guan, Xun Li, Zuo Quan Xu, Fahuai Yi. A stochastic control problem and related free boundaries in finance. Mathematical Control & Related Fields, 2017, 7 (4) : 563-584. doi: 10.3934/mcrf.2017021
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##### References:
$\beta\geq\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)\geq 0$
$\beta>\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)< 0$
$\beta<\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)> 0$
$\beta<\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)\leq 0$, or $\beta=\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)< 0$
$\varphi(x)$
$\psi(y)$
$\beta\geq\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)\geq 0$
$\beta>\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)< 0$
$\beta<\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)\leq 0$, or $\beta=\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)< 0$
$\beta<\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)> 0$
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