Terminal perturbation method for the backward approach to continuous time mean-variance portfolio selection. (English) Zbl 1152.60051

The authors are focused on developing the backward approach to continuous time mean-variance portfolio selection with nonlinear wealth equations in a complete market model. They introduce a new method, different from the Lagrange multiplier method, which is called the terminal perturbation method, to obtain the optimal terminal wealth in the first step. The terminal perturbation method is perturbing the terminal wealth directly to obtain a necessary condition which the optimal terminal wealth satisfies. More precisely,they perturb the terminal value of backward stochastic differential equation (BSDE) with initial constraint in which the terminal condition of the BSDE is regarded as the control “variable”. the main advantage of this method is that it can deal with some state constraints of the dynamic optimization problem easily. The authors introduce Ekeland’s variational principle to derive a stochastic maximum principle which characterizes the optimal terminal wealth.


60H30 Applications of stochastic analysis (to PDEs, etc.)
93E20 Optimal stochastic control
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[1] A. Bensoussan, Lectures on stochastic control, in: Nonlinear Filtering and Stochastic Control, Proceedings, Cortona, 1983
[2] Bielecki, T.R.; Jin, H.; Pliska, S.R.; Zhou, X.Y., Continuous time Mean variance portfolio selection with bankruptcy prohibition, Math. finance, 15, 213-244, (2005) · Zbl 1153.91466
[3] Buckdahn, R.; Hu, Y., Hedging contingent claims for a large investor in an incomplete market, Adv. appl. probab., 30, 239-255, (1998) · Zbl 0904.90009
[4] Chen, S.; Li, X.; Zhou, X., Stochastic linear quadratic regulators with indefinite control weight costs, SIAM J. control optim., 36, 1685-1702, (1998) · Zbl 0916.93084
[5] Cuoco, D.; Cvitanic, J., Optimal consumption choices for a “large” investor, J. econom. dynam. control, 22, 401-436, (1998) · Zbl 0902.90031
[6] Cvitanic, J.; Ma, J., Hedging options for a large investor and forward – backward sde’s, Ann. appl. probab., 6, 370-398, (1996) · Zbl 0856.90011
[7] Ekeland, I., On the variational principle, J. math. anal. appl., 47, 324-353, (1974) · Zbl 0286.49015
[8] El Karoui, N.; Peng, S.; Quenez, M.C., Backward stochastic differential equations in finance, Math. finance, 7, 1-71, (1997) · Zbl 0884.90035
[9] El Karoui, N.; Peng, S.; Quenez, M.-C., A dynamic maximum principle for the optimization of recursive utilities under constraints, Ann. appl. probab., 11, 664-693, (2001) · Zbl 1040.91038
[10] Harrison, J.M.; Kreps, D., Martingales and multiperiod securities market, J. econom. theory, 20, 381-408, (1979) · Zbl 0431.90019
[11] S. Ji, A stochastic optimization problem with state constraints and its applications in finance, Ph.D. Thesis, 1999
[12] Kallenberg, O., Foundations of modern probability, (2002), Springer-Verlag · Zbl 0996.60001
[13] Li, D.; Ng, W.L., Optimal dynamic portfolio selection: multiperiod Mean variance formulation, Math. finance, 10, 387-406, (2000) · Zbl 0997.91027
[14] Li, X.; Zhou, X.Y.; Lim, A.E.B., Dynamic Mean variance portfolio selection with no shorting constraints, SIAM J. control optim., 40, 1540-1555, (2001) · Zbl 1027.91040
[15] Lim, A.E.B.; Zhou, X.Y., Mean variance portfolio selection with random parameters, Math. oper. res., 27, 101-120, (2002)
[16] Markowitz, H., Portfolio selection, J. finance, 7, 77-91, (1952)
[17] Pardoux, E.; Peng, S., Adapted solution of a backward stochastic differential equations, System control lett., 14, 55-61, (1990) · Zbl 0692.93064
[18] Peng, S., Backward stochastic differential equations and applications to optimal control, Appl. math. optim., 27, 125-144, (1993) · Zbl 0769.60054
[19] Pliska, S.R., A discrete time stochastic decision model, (), 290-304
[20] Pliska, S.R., Introduction to mathematical finance, (1997), Blackwell Oxford
[21] M.C. Quenez, Backward stochastic differential equation finance and optimization, Ph.D. Thesis, 1993
[22] Yong, J.; Zhou, X.Y., Stochastic controls Hamiltonian systems and HJB equations, (1999), Springer New York · Zbl 0943.93002
[23] Zhou, X.Y.; Li, D., Continuous time Mean variance portfolio selection: A stochastic LQ framework, Appl. math. optim., 42, 19-33, (2000) · Zbl 0998.91023
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