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September  2015, 5(3): 475-488. doi: 10.3934/mcrf.2015.5.475

Continuous-time portfolio selection under ambiguity

Hanqing Jin 1, and  Xun Yu Zhou 1,

1. 

Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom, United Kingdom

Received  August 2014 Revised  November 2014 Published  July 2015

In a financial market, the appreciation rates of stocks are statistically difficult to estimate, and typically only some confidence intervals in which the rates reside can be estimated. In this paper we study continuous-time portfolio selection under ambiguity, in the sense that the appreciation rates are only known to be in a certain convex closed set and the portfolios are allowed to be based on only the historical stocks prices. We formulate the problem in both the expected utility and the mean--variance frameworks, and derive robust portfolios explicitly for both models.
Keywords: Portfolio selection, continuous time, mean--variance, robust portfolio., ambiguity, Sharpe ratio, expected utility.
Mathematics Subject Classification: Primary: 91G10, 93E20; Secondary: 90B5.
Citation: Hanqing Jin, Xun Yu Zhou. Continuous-time portfolio selection under ambiguity. Mathematical Control & Related Fields, 2015, 5 (3) : 475-488. doi: 10.3934/mcrf.2015.5.475
References:
[1]

A. Ben-Tal and A. Nemirovski, Robust convex optimization, Math. Oepr. Res., 23 (1998), 796-805. doi: 10.1287/moor.23.4.769.  Google Scholar

[2]

S. Boyd, L. ElGhaoui, E. Feron and B. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, 15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. doi: 10.1137/1.9781611970777.  Google Scholar

[3]

T. R. Bielecki, H. Jin, S. R. Pliska and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Math. Finance, 15 (2005), 213-244. doi: 10.1111/j.0960-1627.2005.00218.x.  Google Scholar

[4]

Z. Chen and L. Epstein, Ambiguity, risk, and asset returns in continuous time, Econometrica, 70 (2002), 1403-1443. doi: 10.1111/1468-0262.00337.  Google Scholar

[5]

V. K. Chopra and W. T. Ziemba, The effect of errors in means, variances and covariances on optimal portfolio choice, J. Portfolio Management, (1993), 6-11. doi: 10.3905/jpm.1993.409440.  Google Scholar

[6]

I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior, Journal of Mathematical Economics, 18 (1989), 141-153. doi: 10.1016/0304-4068(89)90018-9.  Google Scholar

[7]

D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Mathematics of Operations Research, 28 (2003), 1-38. doi: 10.1287/moor.28.1.1.14260.  Google Scholar

[8]

A. Gundel, Robust utility maximization for complete and incomplete market models, Finance Stochast, 9 (2005), 151-176. doi: 10.1007/s00780-004-0148-1.  Google Scholar

[9]

L. P. Hanse and T. J. Sargent, Robust control and model uncertainty, The American Economic Review, 91 (2001), 60-66. Google Scholar

[10]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.  Google Scholar

[11]

J. E. Ingersoll, Theory of Financial Decision Making, Rowman and Littlefield, Savage, MD, 1987. Google Scholar

[12]

H. Jin, Z. Q. Xu and X. Y. Zhou, A convex stochastic optimization problem arising from portfolio selection, Math. Finance, 18 (2008), 171-183. doi: 10.1111/j.1467-9965.2007.00327.x.  Google Scholar

[13]

I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer-Verlag, New York, 1998. doi: 10.1007/b98840.  Google Scholar

[14]

R. W. Klein and V. S. Bawa, The effect of estimation risk on optimal portfolio choice, J. Financial Economics, 3 (1976), 215-231. doi: 10.1016/0304-405X(76)90004-0.  Google Scholar

[15]

P. Lakner, Utility maximization with partial information, Stochastic Processes and their Applications, 56 (1995), 247-273. doi: 10.1016/0304-4149(94)00073-3.  Google Scholar

[16]

H. Markowitz, Portfolio selection, J. Finance, 7 (1952), 77-91. Google Scholar

[17]

C. Skiadas, Robust control and recursive utility, Finance Stochast., 7 (2003), 475-489. doi: 10.1007/s007800300100.  Google Scholar

[18]

X. Y. Zhou and D. Li, Continuous time mean-variance portfolio selection: A stochastic LQ framework, Appl. Math. Optim., 42 (2000), 19-33. doi: 10.1007/s002450010003.  Google Scholar

show all references

References:
[1]

A. Ben-Tal and A. Nemirovski, Robust convex optimization, Math. Oepr. Res., 23 (1998), 796-805. doi: 10.1287/moor.23.4.769.  Google Scholar

[2]

S. Boyd, L. ElGhaoui, E. Feron and B. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, 15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. doi: 10.1137/1.9781611970777.  Google Scholar

[3]

T. R. Bielecki, H. Jin, S. R. Pliska and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Math. Finance, 15 (2005), 213-244. doi: 10.1111/j.0960-1627.2005.00218.x.  Google Scholar

[4]

Z. Chen and L. Epstein, Ambiguity, risk, and asset returns in continuous time, Econometrica, 70 (2002), 1403-1443. doi: 10.1111/1468-0262.00337.  Google Scholar

[5]

V. K. Chopra and W. T. Ziemba, The effect of errors in means, variances and covariances on optimal portfolio choice, J. Portfolio Management, (1993), 6-11. doi: 10.3905/jpm.1993.409440.  Google Scholar

[6]

I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior, Journal of Mathematical Economics, 18 (1989), 141-153. doi: 10.1016/0304-4068(89)90018-9.  Google Scholar

[7]

D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Mathematics of Operations Research, 28 (2003), 1-38. doi: 10.1287/moor.28.1.1.14260.  Google Scholar

[8]

A. Gundel, Robust utility maximization for complete and incomplete market models, Finance Stochast, 9 (2005), 151-176. doi: 10.1007/s00780-004-0148-1.  Google Scholar

[9]

L. P. Hanse and T. J. Sargent, Robust control and model uncertainty, The American Economic Review, 91 (2001), 60-66. Google Scholar

[10]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.  Google Scholar

[11]

J. E. Ingersoll, Theory of Financial Decision Making, Rowman and Littlefield, Savage, MD, 1987. Google Scholar

[12]

H. Jin, Z. Q. Xu and X. Y. Zhou, A convex stochastic optimization problem arising from portfolio selection, Math. Finance, 18 (2008), 171-183. doi: 10.1111/j.1467-9965.2007.00327.x.  Google Scholar

[13]

I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer-Verlag, New York, 1998. doi: 10.1007/b98840.  Google Scholar

[14]

R. W. Klein and V. S. Bawa, The effect of estimation risk on optimal portfolio choice, J. Financial Economics, 3 (1976), 215-231. doi: 10.1016/0304-405X(76)90004-0.  Google Scholar

[15]

P. Lakner, Utility maximization with partial information, Stochastic Processes and their Applications, 56 (1995), 247-273. doi: 10.1016/0304-4149(94)00073-3.  Google Scholar

[16]

H. Markowitz, Portfolio selection, J. Finance, 7 (1952), 77-91. Google Scholar

[17]

C. Skiadas, Robust control and recursive utility, Finance Stochast., 7 (2003), 475-489. doi: 10.1007/s007800300100.  Google Scholar

[18]

X. Y. Zhou and D. Li, Continuous time mean-variance portfolio selection: A stochastic LQ framework, Appl. Math. Optim., 42 (2000), 19-33. doi: 10.1007/s002450010003.  Google Scholar

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