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November  2011, 16(4): 1039-1053. doi: 10.3934/dcdsb.2011.16.1039

Dimension reduction and Mutual Fund Theorem in maximin setting for bond market

Nikolai Dokuchaev 1,

1. 

Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, Western Australia, 6845, Australia

Received  October 2010 Revised  March 2011 Published  August 2011

We study optimal investment problem for a continuous time stochastic market model. The risk-free rate, the appreciation rates, and the volatility of the stocks are all random; they are not necessary adapted to the driving Brownian motion, their distributions are unknown, and they are supposed to be currently observable. To cover fixed income management problems, we assume that the number of risky assets can be larger than the number of driving Brownian motion. The optimal investment problem is stated as a problem with a maximin performance criterion to ensure that a strategy is found such that the minimum of expected utility over all possible parameters is maximal. We show that Mutual Fund Theorem holds for this setting. We found also that a saddle point exists and can be found via minimization over a single scalar parameter.
Keywords: minimax problems, Mutual fund Theorem, Continuous time market, saddle point, fixed income management., optimal portfolio.
Mathematics Subject Classification: Primary: 93E20, 91G10; Secondary: 91G8.
Citation: Nikolai Dokuchaev. Dimension reduction and Mutual Fund Theorem in maximin setting for bond market. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1039-1053. doi: 10.3934/dcdsb.2011.16.1039
References:
[1]

T. R. Bielecki and S. R. Pliska, Risk sensitive control with applications to fixed income portfolio management,, "European Congress of Mathematics, Vol. II", 202 (2001), 331.   Google Scholar

[2]

M. J. Brennan, The role of learning in dynamic portfolio decisions,, European Finance Review, 1 (1998), 295.  doi: 10.1023/A:1009725805128.  Google Scholar

[3]

J. Cvitanić, Minimizing expected loss of hedging in incomplete and constrained markets,, SIAM J. of Control and Optimization, 38 (2000), 1050.  doi: 10.1137/S036301299834185X.  Google Scholar

[4]

J. Cvitanić and I. Karatzas, On dynamic measures of risk,, Finance and Stochastics, 3 (1999), 451.   Google Scholar

[5]

N. Dokuchaev, Maximin investment problems for discounted and total wealth,, IMA Journal Management Mathematics, 19 (2008), 63.  doi: 10.1093/imaman/dpm031.  Google Scholar

[6]

N. Dokuchaev, "Mathematical Finance: Core Theory, Problems, and Statistical Algorithms,", Routledge, (2007).  doi: 10.4324/9780203964729.  Google Scholar

[7]

N. Dokuchaev, Saddle points for maximin investment problems with observable but non-predictable parameters: Solution via heat equation,, IMA J. Management Mathematics, 17 (2006), 257.  doi: 10.1093/imaman/dpi041.  Google Scholar

[8]

N. G. Dokuchaev, Optimal solution of investment problems via linear parabolic equations generated by Kalman filter,, SIAM J. of Control and Optimization, 44 (2005), 1239.  doi: 10.1137/S036301290342557X.  Google Scholar

[9]

N. G. Dokuchaev and U. Haussmann, Optimal portfolio selection and compression in an incomplete market,, Quantitative Finance, 1 (2001), 336.  doi: 10.1088/1469-7688/1/3/305.  Google Scholar

[10]

N. G. Dokuchaev and K. L. Teo, "A Duality Approach to an Optimal Investment Problem with Unknown and Nonobservable Parameters,", Department of Applied Mathematics, (1998).   Google Scholar

[11]

N. G. Dokuchaev and K. L. Teo, Optimal hedging strategy for a portfolio investment problem with additional constraints,, Dynamics of Continuous, 7 (2000), 385.   Google Scholar

[12]

I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance,", Applications of Mathematics (New York), 39 (1998).   Google Scholar

[13]

A. Khanna and M. Kulldorff, A generalization of the mutual fund theorem,, Finance and Stochastics, 3 (1999), 167.  doi: 10.1007/s007800050056.  Google Scholar

[14]

S. Komuro and H. Konno, Empirical studies on internationally diversified investment using a stock-bond integrated model,, Journal of Industrial and Management Optimization, 1 (2005), 433.   Google Scholar

[15]

D. Lambertone and B. Lapeyre, "Introduction to Stochastic Calculus Applied to Finance,", Chapman & Hall, (1996).   Google Scholar

[16]

Libin Mou and Jiongmin Yong, Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method,, Journal of Industrial and Management Optimization, 2 (2006), 95.   Google Scholar

[17]

M. Rutkowski, Self-financing trading strategies for sliding, rolling-horizon, and consol bonds,, Mathematical Finance, 9 (1999), 361.  doi: 10.1111/1467-9965.00074.  Google Scholar

[18]

W. Schachermayer, M. Sîrbu and E. Taflin, In which financial markets do mutual fund theorems hold true?,, Finance and Stochastics, 13 (2009), 49.  doi: 10.1007/s00780-008-0072-x.  Google Scholar

[19]

M. Yaari, The dual theory of choice under risk,, Econometrica, 55 (1987), 95.  doi: 10.2307/1911158.  Google Scholar

show all references

References:
[1]

T. R. Bielecki and S. R. Pliska, Risk sensitive control with applications to fixed income portfolio management,, "European Congress of Mathematics, Vol. II", 202 (2001), 331.   Google Scholar

[2]

M. J. Brennan, The role of learning in dynamic portfolio decisions,, European Finance Review, 1 (1998), 295.  doi: 10.1023/A:1009725805128.  Google Scholar

[3]

J. Cvitanić, Minimizing expected loss of hedging in incomplete and constrained markets,, SIAM J. of Control and Optimization, 38 (2000), 1050.  doi: 10.1137/S036301299834185X.  Google Scholar

[4]

J. Cvitanić and I. Karatzas, On dynamic measures of risk,, Finance and Stochastics, 3 (1999), 451.   Google Scholar

[5]

N. Dokuchaev, Maximin investment problems for discounted and total wealth,, IMA Journal Management Mathematics, 19 (2008), 63.  doi: 10.1093/imaman/dpm031.  Google Scholar

[6]

N. Dokuchaev, "Mathematical Finance: Core Theory, Problems, and Statistical Algorithms,", Routledge, (2007).  doi: 10.4324/9780203964729.  Google Scholar

[7]

N. Dokuchaev, Saddle points for maximin investment problems with observable but non-predictable parameters: Solution via heat equation,, IMA J. Management Mathematics, 17 (2006), 257.  doi: 10.1093/imaman/dpi041.  Google Scholar

[8]

N. G. Dokuchaev, Optimal solution of investment problems via linear parabolic equations generated by Kalman filter,, SIAM J. of Control and Optimization, 44 (2005), 1239.  doi: 10.1137/S036301290342557X.  Google Scholar

[9]

N. G. Dokuchaev and U. Haussmann, Optimal portfolio selection and compression in an incomplete market,, Quantitative Finance, 1 (2001), 336.  doi: 10.1088/1469-7688/1/3/305.  Google Scholar

[10]

N. G. Dokuchaev and K. L. Teo, "A Duality Approach to an Optimal Investment Problem with Unknown and Nonobservable Parameters,", Department of Applied Mathematics, (1998).   Google Scholar

[11]

N. G. Dokuchaev and K. L. Teo, Optimal hedging strategy for a portfolio investment problem with additional constraints,, Dynamics of Continuous, 7 (2000), 385.   Google Scholar

[12]

I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance,", Applications of Mathematics (New York), 39 (1998).   Google Scholar

[13]

A. Khanna and M. Kulldorff, A generalization of the mutual fund theorem,, Finance and Stochastics, 3 (1999), 167.  doi: 10.1007/s007800050056.  Google Scholar

[14]

S. Komuro and H. Konno, Empirical studies on internationally diversified investment using a stock-bond integrated model,, Journal of Industrial and Management Optimization, 1 (2005), 433.   Google Scholar

[15]

D. Lambertone and B. Lapeyre, "Introduction to Stochastic Calculus Applied to Finance,", Chapman & Hall, (1996).   Google Scholar

[16]

Libin Mou and Jiongmin Yong, Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method,, Journal of Industrial and Management Optimization, 2 (2006), 95.   Google Scholar

[17]

M. Rutkowski, Self-financing trading strategies for sliding, rolling-horizon, and consol bonds,, Mathematical Finance, 9 (1999), 361.  doi: 10.1111/1467-9965.00074.  Google Scholar

[18]

W. Schachermayer, M. Sîrbu and E. Taflin, In which financial markets do mutual fund theorems hold true?,, Finance and Stochastics, 13 (2009), 49.  doi: 10.1007/s00780-008-0072-x.  Google Scholar

[19]

M. Yaari, The dual theory of choice under risk,, Econometrica, 55 (1987), 95.  doi: 10.2307/1911158.  Google Scholar

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