The paper studies problem of optimal portfolio selection. It is shown that, under some mild conditions, near optimal strategies for investors with different performance criteria can be constructed using a limited number of fixed processes (mutual funds), for a market with a larger number of available risky stocks. This implies dimension reduction for the optimal portfolio selection problem: all rational investors may achieve optimality using the same mutual funds plus a saving account. This result is obtained under mild restrictions for the utility functions without any assumptions on regularity of the value function. The proof is based on the method of dynamic programming applied indirectly to some convenient approximations of the original problem that ensure certain regularity of the value functions. To overcome technical difficulties, we use special time dependent and random constraints for admissible strategies such that the corresponding HJB (Hamilton–Jacobi–Bellman) equation admits “almost explicit” solutions generating near optimal admissible strategies featuring sufficient regularity and integrability.
Accepté le :
DOI : 10.1051/cocv/2017013
Mots clés : Stochastic control, near optimal strategies portfolio structure, dimension reduction, Mutual Funds Theorem
@article{COCV_2018__24_3_1043_0,
author = {Dokuchaev, Nikolai},
title = {On the structure of multifactor optimal portfolio strategies},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1043--1058},
publisher = {EDP-Sciences},
volume = {24},
number = {3},
year = {2018},
doi = {10.1051/cocv/2017013},
mrnumber = {3877192},
zbl = {1418.91464},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2017013/}
}
TY - JOUR AU - Dokuchaev, Nikolai TI - On the structure of multifactor optimal portfolio strategies JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1043 EP - 1058 VL - 24 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2017013/ DO - 10.1051/cocv/2017013 LA - en ID - COCV_2018__24_3_1043_0 ER -
%0 Journal Article %A Dokuchaev, Nikolai %T On the structure of multifactor optimal portfolio strategies %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1043-1058 %V 24 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2017013/ %R 10.1051/cocv/2017013 %G en %F COCV_2018__24_3_1043_0
Dokuchaev, Nikolai. On the structure of multifactor optimal portfolio strategies. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1043-1058. doi : 10.1051/cocv/2017013. http://www.numdam.org/articles/10.1051/cocv/2017013/
[1] and , On the convergence rate of approximation schemes for Hamilton–Jacobi–Bellman equations. Math. Model. Numer. Anal. 36 (2002) 33–54 | DOI | Numdam | MR | Zbl
[2] , The role of learning in dynamic portfolio decisions. Europ. Finance Rev. 1 (1998) 295–306 | DOI | Zbl
[3] and , Optimal portfolio selection and compression in an incomplete market. Quantitative Finance 1 (2001) 336–345 | DOI | MR | Zbl
[4] and , Optimal investment strategies with bounded risks, general utilities, and goal achieving. J. Math. Economics 35 (2001) 289–309 | DOI | MR | Zbl
[5] , Mathematical finance: core theory, problems, and statistical algorithms. New York, USA: Routledge (2007) | DOI | Zbl
[6] , Maximin investment problems for discounted and total wealth. IMA J. Manag. Math. 19 (2008) 63–74 | DOI | MR | Zbl
[7] , Mean variance and goal achieving portfolio for discrete-time market with currently observable source of correlations. ESAIM: COCV 16 2010 635–647 | Numdam | MR | Zbl
[8] , Dimension reduction and Mutual Fund Theorem in maximin setting for bond market. Discrete and Continuous Dynamical System – Series B 16 (2011) 1039–1053 | DOI | MR | Zbl
[9] , Mutual Fund Theorem for continuous time markets with random coefficients. Theory and Decision 76 (2014) 179–199 | DOI | MR | Zbl
[10] , Multifactor portfolio efficiency and multifactor asset pricing. J. Financial Quantitative Anal. 31 (1996) 441–465 | DOI
[11] , Incomplete information equilibria: Separation Theorems and Other Myths. Ann. Operat. Res. 151 (2007) 119–149 | DOI | MR | Zbl
[12] and , Deterministic and Stochastic Optimal Control. New York, USA: Springer Verlag (1975) | DOI | MR | Zbl
[13] , Theory of Financial Decision Making, Totowa, NJ, USA, Rowman and Littlefield (1987)
[14] and , Methods of Mathematical Finance. New York, USA: Springer Verlag (1998) | MR | Zbl
[15] and , A generalization of the mutual fund theorem. Finance and Stochastics 3 (1999) 167–185 | DOI | MR | Zbl
[16] , Controlled diffusion processes. New York, USA: Springer (1980) | DOI | MR | Zbl
[17] , Numerical methods for stochastic control problems in continuous time. SIAM J. Control Optimiz. 28 (1990) 999–1048 | DOI | MR | Zbl
[18] and , Optimal portfolio selection: multi-period mean-variance optimization. Math. Finance 10 (2000) 387–406 | DOI | MR
[19] , Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market. Math. Oper. Res. 29 (2004) 132–161 | DOI | MR | Zbl
[20] and , Mean-variance portfolio selection with random parameters in a complete market. Math. Oper. Res. 27 (2002) 101–120 | DOI | MR | Zbl
[21] , An intertemporal capital asset pricing model. Econometrica 41 (1973) 867–887 | DOI | MR | Zbl
[22] , , , and , Liquidity and asset pricing under the three-moment CAPM paradigm. J. Financial Res. 30 (2007) 379–398 | DOI
[23] , Optimum consumption and portfolio rules with money as an asset. J. Banking and Finance 7 (1983) 231–252 | DOI
[24] , , and In which financial markets do mutual fund theorems hold true? Finance and Stochastics 13 (2009) 49–77 | DOI | MR | Zbl
[25] , Some moment inequalities for stochastic integrals and for solutions of stochastic differential equations. Israel J. Math. 5 (1967) 170–176 | DOI | MR | Zbl
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