In this paper, we consider an investment-consumption problem where the consumption is subject to an upper limit. This upper limit on consumption may reflect the following fact. Investors may have to finance their consumption first by using credits then pay the balance by cashing out part of their portfolio in the stock market. Credit companies set up an upper limit for the credit, thus imposing an upper bound for consumption. We also set up our model in finite horizon, which makes the problem much harder due to the loss of stationary when . We prove that the above described problem is equivalent to a free boundary problem of nonlinear parabolic equations. We aim to characterize explicitly the free boundary by applying a dual transformation technique to convert the original nonlinear parabolic equation to a linear differential equation. This trick allows us to characterize explicitly the free boundary and the optimal consumption strategy. We also prove that the regularity of the value function, which is critical for the application of Ito formula.
Accepté le :
DOI : 10.1051/cocv/2016052
Mots-clés : Optimal investment-consumption model, free boundary problem, stochastic control in finance, consumption constraint
@article{COCV_2017__23_4_1601_0,
author = {Jian, Xiongfei and Yi, Fahuai and Zhang, Jianbo},
title = {Investment and consumption problem in finite time with consumption constraint},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1601--1615},
publisher = {EDP-Sciences},
volume = {23},
number = {4},
year = {2017},
doi = {10.1051/cocv/2016052},
zbl = {1382.35347},
mrnumber = {3716934},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2016052/}
}
TY - JOUR AU - Jian, Xiongfei AU - Yi, Fahuai AU - Zhang, Jianbo TI - Investment and consumption problem in finite time with consumption constraint JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1601 EP - 1615 VL - 23 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2016052/ DO - 10.1051/cocv/2016052 LA - en ID - COCV_2017__23_4_1601_0 ER -
%0 Journal Article %A Jian, Xiongfei %A Yi, Fahuai %A Zhang, Jianbo %T Investment and consumption problem in finite time with consumption constraint %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1601-1615 %V 23 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2016052/ %R 10.1051/cocv/2016052 %G en %F COCV_2017__23_4_1601_0
Jian, Xiongfei; Yi, Fahuai; Zhang, Jianbo. Investment and consumption problem in finite time with consumption constraint. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1601-1615. doi : 10.1051/cocv/2016052. http://www.numdam.org/articles/10.1051/cocv/2016052/
and , The consumption-investment problem with subsistence consumption, bankruptcy, and random market coefficients. J. Optim. Theory Appl. 93 (1997) 243–272. | DOI | MR | Zbl
, and , Optimal consumption and investment policies allowing for consumption constraints, bankruptcy, and welfare. Inst. Oper. Research Manage. Sci. 8 (1983) 613–636. | MR | Zbl
O.A. Ladyženskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Vol. 23 of Transl. Math. Monogr. American Mathematical Society, Providence, R.I. (1967). | MR | Zbl
, Lifetime portfolio selection under uncertainty: the continuous-time case, Rev. Econ. Stat. 51 (1969) 247–257. | DOI
, Optimum consumption and portfolio rules in a continuous time model. J. Econ. Theory 3 (1971) 373-413. | DOI | MR | Zbl
, and , Asset allocation and annuity-purchase strategies to minimize probability of financial ruin. Math. Finance 16 (2006) 647–671. | DOI | MR | Zbl
H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications. Springer-Verlag, Berlin (2009). | MR | Zbl
and , Risk-aversion behavior in consumption/investment problems. Math. Finance 1 (1991) 101–124. | MR | Zbl
and , Distribution of bankruptcy time in a consumption/portfolio problem. J. Econ. Dyn. Control 20 (1996) 471–477. | DOI | MR | Zbl
, and , Explicit solution of a general consumption/portfolio problem with subsistence consumption and bankruptcy. J. Econ. Dyn. Control 16 (1992) 747–768. | DOI | MR | Zbl
and , An Optimal Consumption-Investment Model with Constraint on Consumption Rate. Math. Control Relat. Fields (MCRF) 6 (2016) 517–534. | DOI | MR | Zbl
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