Viscosity solutions of nonlinear variational inequalities and their applications

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The purpose of this study is to solve the optimization problems in mathematical economics and mathematical finance by applications of the modern theory in stochastic control. The study in this period focuses on the smoothness of viscosity solutions of the nonlinear variational inequalities and elliptic equations.Due to the grant, the results will be published in the book entitled:Stochastic Control and Mathematical Modelling in Economics.The main idea to solve these optimization problems is summarized as follows:(a) We formulate the problem and define the value function.(b) We verify that the Dynamic Programming Principle (DPP) holds for the value function.(c) By the DPP, the value function becomes a unique viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation associated with the problem.(d) By the uniqueness of viscosity solutions and the existence of a unique classical solution of the boundary value problem for the HJB equation, we obtain the smoothness of the viscosity solution of the HJB equation.(e) By the solution of the HJB equation, we construct an optimal policy.