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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 |
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 and Management Optimization,
2019, 15
(1)
: 81-96.
doi: 10.3934/jimo.2018033
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References:
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Entrepreneurial decisions on effort and project with a nonconcave objective function, Mathematics of Operations Research, 40 (2015), 901-914.
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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. |
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J. N. Carpenter,
Does option compensation increase managarial risk appetite?, The Journal of Finance, 50 (2000), 2311-2331.
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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. |
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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.
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| [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. |
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