On the Hamilton-Jacobi-Bellman Equation for an Optimal Consumption Problem: I. Existence of Solution
On the Hamilton-Jacobi-Bellman Equation for an Optimal Consumption Problem: I. Existence of Solution
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关于最优消耗问题的 Hamilton-Jacobi-Bellman 方程：一、解的存在性
DOI：
10.1137/110794845
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
S. Sheu
中科院分区：
文献类型：
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作者：
H. Hata;S. Sheu
我们考虑了一个无限时间范围的消费问题，以优化折扣期望电力效用。资产的收益和波动性是随机的，受一些经济因素的影响，模型为扩散过程。这个问题变成了一个标准的控制问题。我们推导了Hamilton-Jacobi-Bellman(HJB)方程，并研究了它的解。在第一部分中，我们在一些弱的条件下，当我们假设存在一个有序的上下解对时，我们证明了这个HJB方程的解的存在性。为了构造一个有序的上下解对，我们还考虑了风险敏感的投资组合优化问题。在第二部分[Hata and Sheu，SIAM J.Control Opti.，50(2012)，pp.2401--2430]中，我们考虑了解的唯一性，并证明了验证定理。
We consider a consumption problem with an infinite time horizon to optimize the discounted expected power utility. The returns and volatilities of the assets are random and affected by some economic factors, modeled as diffusion process. The problem becomes a standard control problem. We derive the Hamilton--Jacobi--Bellman (HJB) equation and study its solutions. In Part I, under some weak conditions we prove the existence of a solution for this HJB equation when we assume the existence of an ordered pair of sub/supersolution. To construct an ordered pair of sub/supersolution, we also consider the risk-sensitive portfolio optimization problem. In Part II [Hata and Sheu, SIAM J. Control Optim., 50 (2012), pp. 2401--2430], we consider the uniqueness of the solution and prove the verification theorem.