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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

Xiaoshan Chen 1, , Xun Li 2, and  Fahuai Yi 3,

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  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.

Keywords: Optimal stopping, optimal investment, non-smooth utility, dual transformation, free boundary.
Mathematics Subject Classification: Primary: 34R35; Secondary: 91B28, 93E20.
Citation: Xiaoshan Chen, Xun Li, Fahuai Yi. Optimal stopping investment with non-smooth utility over an infinite time horizon. Journal of Industrial & 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.  Google Scholar

[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.  Google Scholar

[3]

J. N. Carpenter, Does option compensation increase managarial risk appetite?, The Journal of Finance, 50 (2000), 2311-2331.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[9]

D. Cumming and U. Walz, Private equity returns and disclosure around the world, Journal of International Business Studies, 41 (2010), 727-754.   Google Scholar

[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.  Google Scholar

[11]

R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets, Springer-Verlag, New York, 1999.  Google Scholar

[12]

W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

G. Peskir and A. Shiryaev, Optimal Stopping and Free-Boundary Problems, 2nd edition. Birkhäuser Verlag, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

J. N. Carpenter, Does option compensation increase managarial risk appetite?, The Journal of Finance, 50 (2000), 2311-2331.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[9]

D. Cumming and U. Walz, Private equity returns and disclosure around the world, Journal of International Business Studies, 41 (2010), 727-754.   Google Scholar

[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.  Google Scholar

[11]

R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets, Springer-Verlag, New York, 1999.  Google Scholar

[12]

W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

G. Peskir and A. Shiryaev, Optimal Stopping and Free-Boundary Problems, 2nd edition. Birkhäuser Verlag, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.  Google Scholar

Figure .  The free boundaries $x^*$ and $\bar x$ change when $\alpha$ changes
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Figure .  The free boundaries $x^*$ and $\bar x$ change when $\alpha$ changes
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Figure .  The free boundaries $x^*$ and $\bar x$ change when $K$ changes
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