The objective of the continuous time mean-variance model is to minimize the variance (risk) of an investment portfolio with a given mean at the terminal time. However, the investor can stop the investment plan at any time before the terminal time. To solve this problem, we consider to minimize the variances of the investment portfolio in the multi-time state. The advantage of this multi-time state mean-variance model is the minimization of the risk of the investment portfolio within the investment period. To obtain the optimal strategy of the model, we introduce a sequence of Riccati equations, which are connected by jump boundary conditions. In addition, we establish the relationships between the means and variances in the multi-time state mean-variance model. Furthermore, we use an example to verify that the variances of the multi-time state can affect the average of Maximum-Drawdown of the investment portfolio.
Keywords: Mean-variance, multi-time state, stochastic optimal control
@article{COCV_2021__27_1_A94_0,
author = {Yang, Shuzhen},
title = {Multi-time state mean-variance model in continuous time},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021086},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021086/}
}
TY - JOUR AU - Yang, Shuzhen TI - Multi-time state mean-variance model in continuous time JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021086/ DO - 10.1051/cocv/2021086 LA - en ID - COCV_2021__27_1_A94_0 ER -
Yang, Shuzhen. Multi-time state mean-variance model in continuous time. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 92. doi: 10.1051/cocv/2021086
[1] and , Dynamic asset allocation in a mean-variance framework. Manag. Sci. 11 (1998) 79–95.
[2] and , Dynamic mean-variance asset allocation. Rev. Financial Stud. 23 (2010) 2970–3016.
[3] , and , Linear-quadratic time-inconsistent mean field games. Dyn. Games. Appl. 3 (2013) 537–552.
[4] , , and , Linear-quadratic mean field games. J. Optim. Theory Appl. 169 (2016) 496–529.
[5] , , and , Continuous-time mean-variance portfolio selection with bankruptcy prohibition. Math. Finance 15 (2005) 213–244.
[6] , and , Mean-variance protfolio optimization with state-dependent risk aversion. Math. Finance 24 (2014) 1–24.
[7] , and , On time-inconsistent stochastic control in continuous time. Finance Stoch. 21 (2017) 331–360.
[8] , Time-consistent mean-variance portfolio selection in discrete and continuous time. Finance Stoch. 17 (2013) 227–271.
[9] , and , Continuous-time Markowitz’s model with transaction costs. SIAM J. Financial Math. 1 (2010) 96–125.
[10] , , and , A dynamic mean-variance analysis for log returns. Manag. Sci. 67 (2020) 1–16.
[11] , and , Time-inconsistent stochastic linear-quadratic control. SIAM J. Control Optim. 50 (2012) 1548–1572.
[12] , and , The Nash certainty equivalence principle and McKean-Vlasov systems: an invariance principle and entry adaptation, in Proceedings of the 46th IEEE Conference on Decision and Control (2007) 121–126.
[13] , and , Dynamic approaches for some time-inconsistent optimization problems. Ann. Appl. Probab. 27 (2017) 3435–3477.
[14] and , Time consistency of the mean-risk problem. Oper. Res. (2020) 1–37.
[15] and , Rules rather than discretion: the inconsistency of optimal plans. J. Polit. Econ. 85 (1997) 473–492.
[16] and , Optimal dynamic portfolio selection: Multi-period mean-variance formulation. Math. Finance 10 (2000) 387–406.
[17] , Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market. Math. Oper. Res. 29 (2004) 132–161.
[18] and , Quadratic hedging and mean-variance portfolio selection with random parameters in a complete market. Math. Oper. Res. 1 (2002) 101–120.
[19] , Portfolio selection. J. Finance 7 (1952) 77–91.
[20] , Portfolio Selection: Efficient Diversification of Investment. John Wiley & Sons, New York (1959).
[21] and , Static mean-variance analysis with uncertain time horizon. Manag. Sci. 52 (2006) 955–964.
[22] , An analytic derivation of the efficient frontier. J. Finance Quant. Anal. 7 (1972) 1851–1872.
[23] , Robust time-inconsistent stochastic control problems. Automatica 94 (2018) 249–257.
[24] , A minimum variance result in continuous trading portfolio optimization. Manag. Sci. 9 (1989) 1045–1055.
[25] , Mean-variance portfolio selection in a complete market with unbounded random coefficients. Automatica 55 (2015) 165–175.
[26] , and , Multi-period mean-variance portfolio selection with Markov regime switching and uncertain time horizon. J. Syst. Sci. Complex 24 (2011) 140–155.
[27] , Mean-variance portfolio choice: Quadratic partial hedging. Math. Finance 15 (2005) 533–538.
[28] and , Open-loop equilibrium strategy for mean-variance portfolio problem under stochastic volatility. Automatica 107 (2019) 211–223.
[29] , The necessary and sufficient conditions for stochastic differential systems with multi-time state cost functional. Syst. Control Lett. 114 (2018) 11–18.
[30] and , Continuous time mean-variance model with uncertain exit time. International Conference on Management and Service Science, Wuhan (2010) 1–4.
[31] , and , Multi-period portfolio selection for asset-liability management with uncertain investment horizon. J. Ind. Manag. Optim. 4 (2008) 535–552.
[32] , Continuous time mean-variance portfolio selection with random horizon. Appl. Math. Optim. 68 (2013) 333–359.
[33] and , Continuous-time mean-variance portfolio selection: A stochastic LQ framework. Appl. Math. Optim. 42 (2000) 19–33.
Cité par Sources :
This work was supported by the National Key R&D Program of China (No. 2018YFA0703900) and National Natural Science Foundation of China (No. 11701330; 11871050) and Young Scholars Program of Shandong University.
