Studies on non-linear variational inequalities by viscosity solutions

本研究的目的是应用现代随机控制理论解决数学经济学和数学金融学中的优化问题，主要成果是为该问题开辟了一个非线性变分不等式的研究领域，从而使任意停止的成本函数最小化。内容如下：(1)在《联合控制与停止的变分不等式》中，给出了非线性变分不等式粘度解的新定义。发展了Bensoussan-Lions提出的线性变分不等式的惩罚方法，并证明了非线性变分不等式存在唯一的黏度解。讨论了相同条件下的博弈问题。(2)在“可离开有界速度控制的变分不等式”一文中，给出了非线性变分不等式黏性解的光滑性问题的关键。近年来，Hamilton-Jacobi-Bellman方程的黏度解对于最优策略的构造是光滑的。“用黏度解法优化再生资源开发”一文表明，类似的方法对于数学经济学中的再生资源最优消耗问题是有效的。这些结果可用于今后的各种优化问题。

The purpose of this study is to solve optimization problems in mathematical economics and mathematical finance by applications of the modern theory in stochastic control The main result in this term lies in opening up a field of research of non-linear variational inequalities for the problem so as to minimize the cost functional with discretionary stopping. The content is as follows.(1)In the paper "Variational inequalities for combined control and stopping, 2003", the new definition of viscosity solutions for non-linear variational inequalities is given. The penalty method for linear variational inequalities originated by Bensoussan-Lions is developed, and it is shown that there exists a unique viscosity solution of the non-linear variational inequality. The game problem in the same setting is discussed.(2) In the paper "Variational inequalities for leavable bounded-velocity control", the key for the smoothness problem of the viscosity solutions to non-linear variational inequalities is presented. Recently, it becomes well known when the viscosity solutions of Hamilton-Jacobi-Bellman equations are smooth for the construction of optimal policies. The paper "Optimal exploitation of renewable resources by the viscosity solution method" shows that the similar method is valid for the optimal consumption problem of renewable resources in mathematical economics. These results can be applicable for various optimization problems in future.