Optimal Consumption in the Stochastic Ramsey Problem without Boundedness Constraints
Optimal Consumption in the Stochastic Ramsey Problem without Boundedness Constraints
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DOI：
10.1137/18m1188410
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发表时间：
2018-05
期刊：
影响因子：
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通讯作者：
Yu‐Jui Huang;S. Khalili
中科院分区：
文献类型：
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作者：
Yu‐Jui Huang;S. Khalili
本文研究了具有Cobb-Douglas生产函数的随机Ramsey问题的最优消费问题。与以前的研究相反，我们考虑了一般的消费过程，没有任何先验的有界性约束。一个非标准的随机微分方程，既不具有Lipschitz连续性，也不具有线性增长性，它描述了受控状态过程的动力学。使用混合的概率论证来构造状态过程，并建立其非爆炸性和严格正性。这导致了反馈消费过程的最优化，该反馈消费过程定义为价值函数和状态过程。基于附加粘性解技巧，我们将值函数刻画为一类适当函数中非线性椭圆型方程的唯一经典解。这种刻画涉及到一个关于价值函数在原点的极限行为的条件，这是处理无界消耗的关键。最后，放松有界性约束可以严格地提高所有财富水平的预期效用。
This paper investigates optimal consumption in the stochastic Ramsey problem with the Cobb-Douglas production function. Contrary to prior studies, we allow for general consumption processes, without any a priori boundedness constraint. A non-standard stochastic differential equation, with neither Lipschitz continuity nor linear growth, specifies the dynamics of the controlled state process. A mixture of probabilistic arguments are used to construct the state process, and establish its non-explosiveness and strict positivity. This leads to the optimality of a feedback consumption process, defined in terms of the value function and the state process. Based on additional viscosity solutions techniques, we characterize the value function as the unique classical solution to a nonlinear elliptic equation, among an appropriate class of functions. This characterization involves a condition on the limiting behavior of the value function at the origin, which is the key to dealing with unbounded consumptions. Finally, relaxing the boundedness constraint is shown to increase, strictly, the expected utility at all wealth levels.