Optimization in stochastic systems and applications to consumption problems

本文的目的是利用随机控制理论研究数理经济学和数理金融学中的最优化问题。主要的兴趣在于寻找非线性微分方程的解，称为Hamilton-Jacobi-Bellman方程。利用粘性解方法证明了这些方程存在经典解。证明了最优策略的存在性，并由方程的最优性条件给出。本研究的成果可归纳为以下三篇文章：1.研究需求为常数的随机制造系统中生产计划的遍历控制问题。最优控制和最小值是由相应的Bellman方程的解决方案。2：我们研究消费/投资问题的长期时间平均效用。在效用率函数满足一定的正则性条件下，相应的Hamilton-Jacobi-Bellman方程可解，最优投资组合和最优消费率以显式形式给出。最后给出了有限时域优化问题的一个应用。3.研究了可再生资源的随机优化问题，以最大化开发的期望贴现效用。最优策略的存在，并给出了反馈形式或随机版本的霍特林规则。

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.