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

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Preface

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

Abstract
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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 and 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, (eds. C. Casacuberta, et al.) (Barcelona, 2000), Progr. Math., 202, Birkhäuser, Basel, (2001), 331-345.
[2]	M. J. Brennan, The role of learning in dynamic portfolio decisions, European Finance Review, 1 (1998), 295-306. doi: 10.1023/A:1009725805128.
[3]	J. Cvitanić, Minimizing expected loss of hedging in incomplete and constrained markets, SIAM J. of Control and Optimization, 38 (2000), 1050-1066. doi: 10.1137/S036301299834185X.
[4]	J. Cvitanić and I. Karatzas, On dynamic measures of risk, Finance and Stochastics, 3 (1999), 451-482.
[5]	N. Dokuchaev, Maximin investment problems for discounted and total wealth, IMA Journal Management Mathematics, 19 (2008), 63-74. doi: 10.1093/imaman/dpm031.
[6]	N. Dokuchaev, "Mathematical Finance: Core Theory, Problems, and Statistical Algorithms," Routledge, (2007), 209 pp. doi: 10.4324/9780203964729.
[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), 257-276. doi: 10.1093/imaman/dpi041.
[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), 1239-1258. doi: 10.1137/S036301290342557X.
[9]	N. G. Dokuchaev and U. Haussmann, Optimal portfolio selection and compression in an incomplete market, Quantitative Finance, 1 (2001), 336-345. doi: 10.1088/1469-7688/1/3/305.
[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, Hong Kong Polytechnic University, Working Paper, 1998.
[11]	N. G. Dokuchaev and K. L. Teo, Optimal hedging strategy for a portfolio investment problem with additional constraints, Dynamics of Continuous, Discrete and Impulsive Systems, 7 (2000), 385-404.
[12]	I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance," Applications of Mathematics (New York), 39, Springer-Verlag, New York, 1998.
[13]	A. Khanna and M. Kulldorff, A generalization of the mutual fund theorem, Finance and Stochastics, 3 (1999), 167-185. doi: 10.1007/s007800050056.
[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), 433-442.
[15]	D. Lambertone and B. Lapeyre, "Introduction to Stochastic Calculus Applied to Finance," Chapman & Hall, London, 1996.
[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), 95-117.
[17]	M. Rutkowski, Self-financing trading strategies for sliding, rolling-horizon, and consol bonds, Mathematical Finance, 9 (1999), 361-385. doi: 10.1111/1467-9965.00074.
[18]	W. Schachermayer, M. Sîrbu and E. Taflin, In which financial markets do mutual fund theorems hold true?, Finance and Stochastics, 13 (2009), 49-77. doi: 10.1007/s00780-008-0072-x.
[19]	M. Yaari, The dual theory of choice under risk, Econometrica, 55 (1987), 95-115. doi: 10.2307/1911158.

show all references

References:

[1]	T. R. Bielecki and S. R. Pliska, Risk sensitive control with applications to fixed income portfolio management, (eds. C. Casacuberta, et al.) (Barcelona, 2000), Progr. Math., 202, Birkhäuser, Basel, (2001), 331-345.
[2]	M. J. Brennan, The role of learning in dynamic portfolio decisions, European Finance Review, 1 (1998), 295-306. doi: 10.1023/A:1009725805128.
[3]	J. Cvitanić, Minimizing expected loss of hedging in incomplete and constrained markets, SIAM J. of Control and Optimization, 38 (2000), 1050-1066. doi: 10.1137/S036301299834185X.
[4]	J. Cvitanić and I. Karatzas, On dynamic measures of risk, Finance and Stochastics, 3 (1999), 451-482.
[5]	N. Dokuchaev, Maximin investment problems for discounted and total wealth, IMA Journal Management Mathematics, 19 (2008), 63-74. doi: 10.1093/imaman/dpm031.
[6]	N. Dokuchaev, "Mathematical Finance: Core Theory, Problems, and Statistical Algorithms," Routledge, (2007), 209 pp. doi: 10.4324/9780203964729.
[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), 257-276. doi: 10.1093/imaman/dpi041.
[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), 1239-1258. doi: 10.1137/S036301290342557X.
[9]	N. G. Dokuchaev and U. Haussmann, Optimal portfolio selection and compression in an incomplete market, Quantitative Finance, 1 (2001), 336-345. doi: 10.1088/1469-7688/1/3/305.
[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, Hong Kong Polytechnic University, Working Paper, 1998.
[11]	N. G. Dokuchaev and K. L. Teo, Optimal hedging strategy for a portfolio investment problem with additional constraints, Dynamics of Continuous, Discrete and Impulsive Systems, 7 (2000), 385-404.
[12]	I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance," Applications of Mathematics (New York), 39, Springer-Verlag, New York, 1998.
[13]	A. Khanna and M. Kulldorff, A generalization of the mutual fund theorem, Finance and Stochastics, 3 (1999), 167-185. doi: 10.1007/s007800050056.
[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), 433-442.
[15]	D. Lambertone and B. Lapeyre, "Introduction to Stochastic Calculus Applied to Finance," Chapman & Hall, London, 1996.
[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), 95-117.
[17]	M. Rutkowski, Self-financing trading strategies for sliding, rolling-horizon, and consol bonds, Mathematical Finance, 9 (1999), 361-385. doi: 10.1111/1467-9965.00074.
[18]	W. Schachermayer, M. Sîrbu and E. Taflin, In which financial markets do mutual fund theorems hold true?, Finance and Stochastics, 13 (2009), 49-77. doi: 10.1007/s00780-008-0072-x.
[19]	M. Yaari, The dual theory of choice under risk, Econometrica, 55 (1987), 95-115. doi: 10.2307/1911158.

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