# 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 and Related Fields, 2017, 7 (4) : 563-584. doi: 10.3934/mcrf.2017021
##### References:
 [1] A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities Gauthier-Villars, Paris, 1982. [2] J. N. Capenter, Does option compensation increase managarial risk appetite?, The Journal of Finance, 50 (2000), 2311-2331. [3] C. Ceci and B. Bassan, Mixed optimal stopping and stochastic control problems with semicontinuous final reward for diffusion processes, Stochastics and Stochastics Reports, 76 (2004), 323-337.  doi: 10.1080/10451120410001728436. [4] M. H. Chang, T. Pang and J. Yong, Optimal stopping problem for stochastic differential equations with random coefficients, SIAM Journal on Control and Optimization, 48 (2009), 941-971.  doi: 10.1137/070705726. [5] S. Dayanik and I. Karatzas, On the optimal stopping problem for one-dimensional diffusions, Stochastic Processes and their Applications, 107 (2003), 173-212.  doi: 10.1016/S0304-4149(03)00076-0. [6] R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-7146-6. [7] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions 2nd edition. Springer-Verlag, New York, 2006. [8] A. Friedman, Parabolic variational inequalities in one space dimension and smoothness of the free boundary, Journal of Functional Analysis, 18 (1975), 151-176.  doi: 10.1016/0022-1236(75)90022-1. [9] A. Friedman, Variational Principles and Free Boundary Problems John Wiley and Sons, New York, 1982. [10] V. Henderson, Valuing the option to invest in an incomplete market, Mathematics and Financial Economics, 1 (2007), 103-128.  doi: 10.1007/s11579-007-0005-z. [11] V. Henderson and D. Hobson, An explicit solution for an optimal stopping/optimal control problem which models an asset sale, The Annals of Applied Probability, 18 (2008), 1681-1705.  doi: 10.1214/07-AAP511. [12] X. Jian, X. Li and F. Yi, Optimal investment with stopping in finite horizon Journal of Inequalities and Applications 2014 (2014), 44pp. doi: 10.1186/1029-242X-2014-432. [13] I. Karatzas and S. G. Kou, Hedging American contingent claims with constrained portfolios, Finance and Stochastics, 2 (1998), 215-258.  doi: 10.1007/s007800050039. [14] I. Karatzas and D. Ocone, A leavable bounded-velocity stochastic control problem, Stochastic Processes and their Applications, 99 (2002), 31-51.  doi: 10.1016/S0304-4149(01)00157-0. [15] I. Karatzas and W. D. Sudderth, Control and stopping of a diffusion process on an interval, The Annals of Applied Probability, 9 (1999), 188-196.  doi: 10.1214/aoap/1029962601. [16] I. Karatzas and H. Wang, Utility maximization with discretionary stopping, SIAM Journal on Control and Optimization, 39 (2000), 306-329.  doi: 10.1137/S0363012998346323. [17] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R. I. , 1968. [18] X. Li and Z. Y. Wu, Reputation entrenchment or risk minimization? Early stop and investor-manager agency conflict in fund management, Journal of Risk Finance, 9 (2008), 125-150. [19] X. Li and Z. Y. Wu, Corporate risk management and investment decisions, Journal of Risk Finance, 10 (2009), 155-168.  doi: 10.1108/15265940910938233. [20] X. Li and X. Y. Zhou, Continuous-time mean-variance efficiency: The 80% rule, The Annals of Applied Probability, 16 (2006), 1751-1763.  doi: 10.1214/105051606000000349. [21] O. A. Oleinik and E. V. Radkevie, Second Order Equations with Nonnegative Characteristic Form American Mathematical Society, Rhode Island and Plenum Press, New York, 1973. [22] G. Peskir and A. Shiryaev, Optimal Stopping and Free-Boundary Problems 2nd edition. Birkhäuser Verlag, Berlin, 2006. [23] H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89500-8. [24] P. A. Samuelson, Rational theory of warrant pricing. With an appendix by H. P. McKean, A free boundary problem for the heat equation arising from a problem in mathematical economics, Industrial Management Review, 6 (1965), 13-31. [25] A. Shiryaev, Z. Q. Xu and X. Y. Zhou, Thou shalt buy and hold, Quantitative Finance, 8 (2008), 765-776.  doi: 10.1080/14697680802563732. [26] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

show all references

##### References:
 [1] A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities Gauthier-Villars, Paris, 1982. [2] J. N. Capenter, Does option compensation increase managarial risk appetite?, The Journal of Finance, 50 (2000), 2311-2331. [3] C. Ceci and B. Bassan, Mixed optimal stopping and stochastic control problems with semicontinuous final reward for diffusion processes, Stochastics and Stochastics Reports, 76 (2004), 323-337.  doi: 10.1080/10451120410001728436. [4] M. H. Chang, T. Pang and J. Yong, Optimal stopping problem for stochastic differential equations with random coefficients, SIAM Journal on Control and Optimization, 48 (2009), 941-971.  doi: 10.1137/070705726. [5] S. Dayanik and I. Karatzas, On the optimal stopping problem for one-dimensional diffusions, Stochastic Processes and their Applications, 107 (2003), 173-212.  doi: 10.1016/S0304-4149(03)00076-0. [6] R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-7146-6. [7] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions 2nd edition. Springer-Verlag, New York, 2006. [8] A. Friedman, Parabolic variational inequalities in one space dimension and smoothness of the free boundary, Journal of Functional Analysis, 18 (1975), 151-176.  doi: 10.1016/0022-1236(75)90022-1. [9] A. Friedman, Variational Principles and Free Boundary Problems John Wiley and Sons, New York, 1982. [10] V. Henderson, Valuing the option to invest in an incomplete market, Mathematics and Financial Economics, 1 (2007), 103-128.  doi: 10.1007/s11579-007-0005-z. [11] V. Henderson and D. Hobson, An explicit solution for an optimal stopping/optimal control problem which models an asset sale, The Annals of Applied Probability, 18 (2008), 1681-1705.  doi: 10.1214/07-AAP511. [12] X. Jian, X. Li and F. Yi, Optimal investment with stopping in finite horizon Journal of Inequalities and Applications 2014 (2014), 44pp. doi: 10.1186/1029-242X-2014-432. [13] I. Karatzas and S. G. Kou, Hedging American contingent claims with constrained portfolios, Finance and Stochastics, 2 (1998), 215-258.  doi: 10.1007/s007800050039. [14] I. Karatzas and D. Ocone, A leavable bounded-velocity stochastic control problem, Stochastic Processes and their Applications, 99 (2002), 31-51.  doi: 10.1016/S0304-4149(01)00157-0. [15] I. Karatzas and W. D. Sudderth, Control and stopping of a diffusion process on an interval, The Annals of Applied Probability, 9 (1999), 188-196.  doi: 10.1214/aoap/1029962601. [16] I. Karatzas and H. Wang, Utility maximization with discretionary stopping, SIAM Journal on Control and Optimization, 39 (2000), 306-329.  doi: 10.1137/S0363012998346323. [17] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R. I. , 1968. [18] X. Li and Z. Y. Wu, Reputation entrenchment or risk minimization? Early stop and investor-manager agency conflict in fund management, Journal of Risk Finance, 9 (2008), 125-150. [19] X. Li and Z. Y. Wu, Corporate risk management and investment decisions, Journal of Risk Finance, 10 (2009), 155-168.  doi: 10.1108/15265940910938233. [20] X. Li and X. Y. Zhou, Continuous-time mean-variance efficiency: The 80% rule, The Annals of Applied Probability, 16 (2006), 1751-1763.  doi: 10.1214/105051606000000349. [21] O. A. Oleinik and E. V. Radkevie, Second Order Equations with Nonnegative Characteristic Form American Mathematical Society, Rhode Island and Plenum Press, New York, 1973. [22] G. Peskir and A. Shiryaev, Optimal Stopping and Free-Boundary Problems 2nd edition. Birkhäuser Verlag, Berlin, 2006. [23] H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89500-8. [24] P. A. Samuelson, Rational theory of warrant pricing. With an appendix by H. P. McKean, A free boundary problem for the heat equation arising from a problem in mathematical economics, Industrial Management Review, 6 (1965), 13-31. [25] A. Shiryaev, Z. Q. Xu and X. Y. Zhou, Thou shalt buy and hold, Quantitative Finance, 8 (2008), 765-776.  doi: 10.1080/14697680802563732. [26] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.
$\beta\geq\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)\geq 0$
$\beta>\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)< 0$
$\beta<\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)> 0$
$\beta<\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)\leq 0$, 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 $\beta=\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)< 0$
$\beta<\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)> 0$
 [1] T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks and Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675 [2] S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial and Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155 [3] Xiaoshan Chen, Xun Li, Fahuai Yi. Optimal stopping investment with non-smooth utility over an infinite time horizon. Journal of Industrial and Management Optimization, 2019, 15 (1) : 81-96. doi: 10.3934/jimo.2018033 [4] Ugur G. Abdulla, Evan Cosgrove, Jonathan Goldfarb. On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems. Evolution Equations and Control Theory, 2017, 6 (3) : 319-344. doi: 10.3934/eect.2017017 [5] Wenqing Bao, Xianyi Wu, Xian Zhou. Optimal stopping problems with restricted stopping times. Journal of Industrial and Management Optimization, 2017, 13 (1) : 399-411. doi: 10.3934/jimo.2016023 [6] Nicholas Westray, Harry Zheng. Constrained nonsmooth utility maximization on the positive real line. Mathematical Control and Related Fields, 2015, 5 (3) : 679-695. doi: 10.3934/mcrf.2015.5.679 [7] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems and Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025 [8] Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete and Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71 [9] Yarui Duan, Pengcheng Wu, Yuying Zhou. Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022017 [10] Martin Brokate, Pavel Krejčí. Optimal control of ODE systems involving a rate independent variational inequality. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 331-348. doi: 10.3934/dcdsb.2013.18.331 [11] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems and Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307 [12] Andrzej Nowakowski. Variational analysis of semilinear plate equation with free boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3133-3154. doi: 10.3934/dcds.2015.35.3133 [13] Jakob Kotas. Optimal stopping for response-guided dosing. Networks and Heterogeneous Media, 2019, 14 (1) : 43-52. doi: 10.3934/nhm.2019003 [14] Cong Qin, Xinfu Chen. A new weak solution to an optimal stopping problem. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4823-4837. doi: 10.3934/dcdsb.2020128 [15] Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic type chemotaxis model. Kinetic and Related Models, 2015, 8 (4) : 667-684. doi: 10.3934/krm.2015.8.667 [16] Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799 [17] Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122 [18] Jinsen Guo, Yongwu Zhou, Baixun Li. The optimal pricing and service strategies of a dual-channel retailer under free riding. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2049-2076. doi: 10.3934/jimo.2021056 [19] Lorena Bociu, Lucas Castle, Kristina Martin, Daniel Toundykov. Optimal control in a free boundary fluid-elasticity interaction. Conference Publications, 2015, 2015 (special) : 122-131. doi: 10.3934/proc.2015.0122 [20] Jésus Ildefonso Díaz, Tommaso Mingazzini, Ángel Manuel Ramos. On the optimal control for a semilinear equation with cost depending on the free boundary. Networks and Heterogeneous Media, 2012, 7 (4) : 605-615. doi: 10.3934/nhm.2012.7.605

2020 Impact Factor: 1.284