On the Hamilton-Jacobi-Bellman Equation for an Optimal Consumption Problem: II. Verification Theorem
On the Hamilton-Jacobi-Bellman Equation for an Optimal Consumption Problem: II. Verification Theorem
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关于最优消耗问题的 Hamilton-Jacobi-Bellman 方程：II。
DOI：
10.1137/110794857
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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：
--
作者：
H. Hata;S. Sheu
我们考虑了一个最优消费问题，其中投资者试图最大化无限时域期望的贴现双曲绝对风险厌恶效用。我们把一个随机因子模型，使风险资产的平均收益取决于潜在的经济因素制定为一个随机微分方程的解决方案。利用动态规划原理，导出了汉密尔顿-雅可比-贝尔曼（HJB）方程，并研究了它的解.在第一部分中，我们在适当的条件下证明了HJB方程古典解的存在性。在第二部分中，我们考虑了证明由HJB方程的解导出的最优策略的候选确实是最优的验证定理。我们证明了两组条件下的结果。一个是线性模型，一个是非线性模型。在这两种情况下，HJB方程没有解析解。
We consider an optimal consumption problem where an investor tries to maximize the infinite horizon expected discounted hyperbolic absolute risk aversion utility of consumption. We treat a stochastic factor model such that the mean returns of risky assets depend on underlying economic factors formulated as the solution of a stochastic differential equation. Using a dynamic programming principle, we derive the Hamilton--Jacobi--Bellman (HJB) equation and study its solutions. In part I, we prove the existence of a classical solution for HJB equation under suitable conditions. In part II, we consider the verification theorem stating a candidate of optimal strategy derived from a solution of HJB equation is indeed optimal. We prove the result under two sets of conditions. One is a linear model, and one is a nonlinear model. In both cases, the HJB equations do not have analytical solutions.