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
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, 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, 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, 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.

[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, 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.

[13]

I. Karatzas, S. G. Kou, Hedging American contingent claims with constrained portfolios, Finance and Stochastics, 2 (1998), 215-258. doi: 10.1007/s007800050039.

[14]

I. Karatzas, 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, 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, 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, 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, Z. Y. Wu, Corporate risk management and investment decisions, Journal of Risk Finance, 10 (2009), 155-168. doi: 10.1108/15265940910938233.

[20]

X. Li, 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.

[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, 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.

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, 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, 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, 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.

[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, 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.

[13]

I. Karatzas, S. G. Kou, Hedging American contingent claims with constrained portfolios, Finance and Stochastics, 2 (1998), 215-258. doi: 10.1007/s007800050039.

[14]

I. Karatzas, 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, 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, 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, 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, Z. Y. Wu, Corporate risk management and investment decisions, Journal of Risk Finance, 10 (2009), 155-168. doi: 10.1108/15265940910938233.

[20]

X. Li, 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.

[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, 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.

Figure 4.1.  $\beta\geq\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)\geq 0$
Figure 4.2.  $\beta>\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)< 0$
Figure 4.3.  $\beta<\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)> 0$
Figure 4.4.  $\beta<\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)\leq 0$, or $\beta=\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)< 0$
Figure 2.1.  $\varphi(x)$
Figure 3.1.  $\psi(y)$
Figure 5.1.  $\beta\geq\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)\geq 0$
Figure 5.2.  $\beta>\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)< 0$
Figure 5.4.  $\beta<\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)\leq 0$, or $\beta=\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)< 0$
Figure 5.3.  $\beta<\frac{a^2}{2}\frac{\gamma}{1-\gamma}+r\gamma$, $\Psi (k)> 0$
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