Optimization in stochastic systems and applications to consumption problems

目的是用最新的随机控制理论研究数学经济学和数学金融学中的最优化问题。主要兴趣在于寻找非线性微分方程的解，称为Hamilton-Jacobi-Bellman方程。用黏度解法证明了这些方程具有经典解。从方程的最优性条件出发，证明了最优策略的存在性。本基金支持的研究成果可在以下三篇文章的摘要中说明。研究了具有恒定需求的随机制造系统生产计划的遍历控制问题。通过求解相应的Bellman方程给出了最优控制和最小值。我们研究具有长期时间平均效用的消费/投资问题。相关的Hamilton-Jacobi-Bellman方程可以在效用-费率函数的一些正则条件下解出，并以显式形式表示出最优投资组合和最优消费率。并给出了有限视界优化问题的一个应用。研究了可再生资源的随机优化问题，以使开发的预期贴现效用最大化。最优策略是存在的，并以反馈形式或霍特林规则的随机版本给出。

The objective is to study the optimization problems in Mathematical Economics and Mathematical Finance by the recent theory of stochastic control. The main interest lies in finding the solutions of non-linear differential equations called the Hamilton-Jacobi-Bellman equations. It is proved that these equations admit the classical solutions by using the viscosity solution method. The optimal policies are shown to exist and given from the optimality conditions of the equations. The research results supported by this grant can be stated in the following summaries of three articles below.1 : We study the ergodic control problem of production planning in stochastic manufacturing systems with constant demand. The optimal control and the minimum value are given by a solution to the corresponding Bellman equation.2 : We study consumption/investment problems with long-term time-average utilities. The associated Hamilton-Jacobi-Bellman equation can be solved under some regularity conditions of utility-rate function, and the optimal portfolio and consumption-rates are exhibited in explicit forms. An application to the optimization problem with finite horizon is also given.3 : We study the stochastic optimization problem of renewable resources to maximize the expected discounted utility of exploitation. The optimal policy is shown to exist and given in a feedback form or a stochastic version of Hotelling's rule.