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January  2019, 15(1): 81-96. doi: 10.3934/jimo.2018033

## Optimal stopping investment with non-smooth utility over an infinite time horizon

 1 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China 3 School of Finance, Guangdong University of Foreign Studies, Guangzhou 510420, China

Received  March 2017 Revised  October 2017 Published  January 2019 Early access  April 2018

Fund Project: This work was partially supported by Research Grants Council of Hong Kong under grant 519913, 15224215 and 15255416; NNSF of China (No. 11601163, No.11471276, No.11771158); NSF Guangdong Province of China (No.2016A030313448, No.2015A030313574, No.2017A030313397); The Humanities and Social Science Research Foundation of the Ministry of Education of China (No.15YJAZH051).

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.

Citation: 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
##### References:
 [1] A. Bensoussan, A. Cadenillas and H. K. Koo, Entrepreneurial decisions on effort and project with a nonconcave objective function, Mathematics of Operations Research, 40 (2015), 901-914.  doi: 10.1287/moor.2014.0702. [2] A. Berger and G. F. Udell, The economics of small business finance: The roles of private equity and debt markets in the financial growth cycle, Journal of Banking and Finance, 22 (1998), 613-673.  doi: 10.2139/ssrn.137991. [3] J. N. Carpenter, Does option compensation increase managarial risk appetite?, The Journal of Finance, 50 (2000), 2311-2331. [4] S. Carter, C. Mason and S. Tagg, Lifting the barriers to growth in UK small businesses: The FSB biennial membership survey, Federation of Small Businesses, London, 2004. [5] 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. [6] M. H. bChang, 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. [7] K. J. Choi, H. K. Koo and D. Y. Kwak, Optimal stopping of active portfolio management, Annals of Economics and Finance, 5 (2004), 93-126. [8] J. Chua, J. Chrisman, F. Kellermanns and Z. Wu, Family involvement and new venture debt financing, Journal of Business Venturing, 26 (2011), 472-488.  doi: 10.1016/j.jbusvent.2009.11.002. [9] D. Cumming and U. Walz, Private equity returns and disclosure around the world, Journal of International Business Studies, 41 (2010), 727-754. [10] 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. [11] R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets, Springer-Verlag, New York, 1999. [12] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993. [13] 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. [14] X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.  doi: 10.1137/S0363012900378504. [15] 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. [16] G. Peskir and A. Shiryaev, Optimal Stopping and Free-Boundary Problems, 2nd edition. Birkhäuser Verlag, Berlin, 2006. [17] A. Shiryaev, Z. Q. Xu and X. Y. Zhou, Thou shalt buy and hold, Quantitative Finance, 8 (2008), 765-776.  doi: 10.1080/14697680802563732. [18] J. Sparrow and P. Bentley, Decision tendencies of entrepreneurs and small business risk management practices, Risk Management, 2 (2000), 17-26.  doi: 10.1057/palgrave.rm.8240037. [19] G. L. Xu and S. E. Shreve, A duality method for optimal consumption and investment under short-selling prohibition: Ⅱ. constant market coefficients, Annals of Applied Probability, 2 (1992), 314-328.  doi: 10.1214/aoap/1177005706. [20] 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, A. Cadenillas and H. K. Koo, Entrepreneurial decisions on effort and project with a nonconcave objective function, Mathematics of Operations Research, 40 (2015), 901-914.  doi: 10.1287/moor.2014.0702. [2] A. Berger and G. F. Udell, The economics of small business finance: The roles of private equity and debt markets in the financial growth cycle, Journal of Banking and Finance, 22 (1998), 613-673.  doi: 10.2139/ssrn.137991. [3] J. N. Carpenter, Does option compensation increase managarial risk appetite?, The Journal of Finance, 50 (2000), 2311-2331. [4] S. Carter, C. Mason and S. Tagg, Lifting the barriers to growth in UK small businesses: The FSB biennial membership survey, Federation of Small Businesses, London, 2004. [5] 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. [6] M. H. bChang, 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. [7] K. J. Choi, H. K. Koo and D. Y. Kwak, Optimal stopping of active portfolio management, Annals of Economics and Finance, 5 (2004), 93-126. [8] J. Chua, J. Chrisman, F. Kellermanns and Z. Wu, Family involvement and new venture debt financing, Journal of Business Venturing, 26 (2011), 472-488.  doi: 10.1016/j.jbusvent.2009.11.002. [9] D. Cumming and U. Walz, Private equity returns and disclosure around the world, Journal of International Business Studies, 41 (2010), 727-754. [10] 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. [11] R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets, Springer-Verlag, New York, 1999. [12] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993. [13] 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. [14] X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.  doi: 10.1137/S0363012900378504. [15] 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. [16] G. Peskir and A. Shiryaev, Optimal Stopping and Free-Boundary Problems, 2nd edition. Birkhäuser Verlag, Berlin, 2006. [17] A. Shiryaev, Z. Q. Xu and X. Y. Zhou, Thou shalt buy and hold, Quantitative Finance, 8 (2008), 765-776.  doi: 10.1080/14697680802563732. [18] J. Sparrow and P. Bentley, Decision tendencies of entrepreneurs and small business risk management practices, Risk Management, 2 (2000), 17-26.  doi: 10.1057/palgrave.rm.8240037. [19] G. L. Xu and S. E. Shreve, A duality method for optimal consumption and investment under short-selling prohibition: Ⅱ. constant market coefficients, Annals of Applied Probability, 2 (1992), 314-328.  doi: 10.1214/aoap/1177005706. [20] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.
The free boundaries $x^*$ and $\bar x$ change when $\alpha$ changes
The free boundaries $x^*$ and $\bar x$ change when $\alpha$ changes
The free boundaries $x^*$ and $\bar x$ change when $K$ changes
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