Optimization in stochastic systems and applications to consumption problems

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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.