Singular Control of Diffusion Processes and its Applications to the Models of Economic Dynamics

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

ABSTRACT SINGULAR CONTROL OF DIFFUSION PROCESSES AND ITS APPLICATIONS TO THE MODELS OF ECONOMIC DYNAMICS The proposed research lies within the area of optimal stochastic control. It deals with optimal control of diffusion processes by means of singular with respect to time functionals. This type of control naturally appears in the problems in which there is no natural restriction on the rates of control and as a result the optimal control rate is infinite. The optimal policy in these types of problems is to reflect the process from an a priori unknown boundary. Such type of action also arises in problems with additive control, as an approximation for a "bang-bang" optimal policy. The proposed research consists of developing the theory of the related Partial Differential Equations with gradient constraints, studying optimal reflecting barriers and optimal policies in one dimensional and multidimensional cases. Applications include stochastic control models of flexible manufacturing systems as well as dividend optimization and multidimensional portfolio optimization models. It is also intended to consider application of the singular stochastic control theory to optimization problems in insurance. We will also study stochastic control models of large economies and analysis of the relationship between the general framework of the no arbitrage asset pricing in mathematical finance and the equilibrium paths in those models. In addition to developing new stochastic control and stochastic processes theory, the applications of this research would include devising better models for optimization of the manufacturing processes and outputs. In addition our research would also result in developing new optimization models in mathematical finance and insurance. While perceived by many nonspecialists as a tool for making a fast profit, mathematical finance in fact is a "technology" for risk reduction. In this regard its merge with insurance is only too natural. The issues of controlling the risk as well as insurance aspects of the financial risk has loomed large recently in both financial markets as well as in the insurance industry. The importance of optimization in devise of the policies employed by insurance and financial institutions is hard to overestimate. Exposure to unnecessary economic and financial risks and failure to employ the optimal procedures may have serious economic and social impacts. Our research deals with development of mathematical models of optimal risk control techniques for financial and insurance corporations. This will also enable one to get a better insight into the nature of the optimal risk reduction techniques a publicly traded financial corporation should adhere to, as well as the optimal dividend distribution policy it should follow.