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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[16]

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

[17]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[16]

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

[17]

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

[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.

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