Studies on non-linear variational inequalities by viscosity solutions

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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.