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AN OPTIMAL INVESTMENT／CONSUMPTIONPROBLEM WITH HIGHER BORROWING RATE

作者姓名：

作者单位：

[1]DepartmentofAppliedMathematics,ZhejiangUniversity,Hangzhou3]000~7. [2]DepartmentotAppliedMathematics,ZhejiangUniversity,Hangzhou310027／／PaymentandScience~.TechnologyDepartment,People＇sBankotChina,BeijlngI0{）800. [3]DepartmentofAppliedMathematics,ZhejiangUniversity,Hangzhou3]027.

摘要：

In this paper, optimal investment and consumption decisions for an optimal choiceproblem in infinite borizon are considered, for an investor who has available a bank account anda stock whose price is a log normal diffusion. The bank pays at an interest rate r for any de-posit, and takes at a larger rate / for any loan. As in the paper of Xu Wensheng and ChenShuping in JAMS(B), where an analogous problem in finite horizon is studied, optimal strategies are obtained via Hamilton-Jacobi-Bellman (ladE) equation which is derived from dynamic c1-programming principle. For the specific HARA case, i.e. U(t,c)=e^-βtc^1-R/1-R, this paper getsthe optimal consumption and optimal investment in the form of c^‘1 =β -^-g/Rwi and π^‘1= b -- γ / Rσ^2wr, with γ1,=max{γ,min{γ‘,b--Rσ^2‘} },^-g=(1--R)γ (b-γ)^2/2Rσ^2]. This result coincides with the classical one under condition γ‘ ≡γ.

关键词：

投资消费最优化贷款率法向扩散 HJB

An optimal investment/consumption problem with higher borrowing rate

WUXIONGWEI XUWENSHENG CHENSHUPING.An optimal investment/consumption problem with higher borrowing rate[J].Applied Mathematics A Journal of Chinese Universities,1998,13(1):68-76.

Authors:

Wu Xiongwei Xu Wensheng Chen Shuping

Institution:

WUXIONGWEI XU WENSHENG AND CHEN SHUPING

Abstract:

In this paper, optimal investment and consumption decisions for an optimal choice problem in infinite horizon are considered. for an investor who has available a bank account and a stock whose price is a log normal diffusion. The bank pays at an interest rate r for any deposit, and takes at a larger rate r′ for any loan. As in the paper of Xu Wensheng and Chen Shuping in JAMS(B), where an analogous problem in finite horizon is studied, optimal strategies are obtained via Hamilton-Jacobi-Bellman (HJB) equation which is derived from dynamic programming principle. For the specific HARA case, i.e. $U(t,c) = e^{\_\beta t} \frac{{c^{1 - R} }}{{1 - R}}$ this paper gets the optimal consumption and optimal investment in the form of

$c_t^* = \frac{{\beta - \bar g}}{R}w_t and \pi _t^* = \frac{{b - \gamma }}{{R\sigma ^2 }}w_t$

with $\gamma : = \max \{ r,\min \{ r',b - R\sigma ^2 \} \} ,\bar g = (1 - R)\left {\gamma + \frac{{(b - \gamma )^2 }}{{2R\sigma ^2 }}} \right]$ . This result coincides with the classical one under condition r′=r. This work was supported by the National Natural Science Foundation of China.

Keywords:

Investment consumption interest rate borrowing rate stock market

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