Proposal on Optimization of Combined Cumulants in Stochastic Controls

The cumulant generating function of a cost function is the logarithm of the moment generating function of the cost function. The Taylor coefficients of the cumulant generating function are called the cumulants of the cost function. This project is concerned with the optimization of a linear combination of the cumulants of a cost function. The problem is quite general since the coefficients of the linear combination can be considered as a collection of parameters. For example, one can give heavier weights to the first several cumulants because they are usually more significant than the others. This is evident in meanvariance analysis and in certain engineering designs (minimal cost variance analysis). Of course, a very special combination of the cumulants is the Taylor expansion of the cumulant generating function, in which all coefficients depend on a single parameter. Essentially risksensitive control is concerned with the optimization of the cumulant generating function (or equivalently the moment generating function). In the past two decades, risk sensitive control has been developed rapidly into an elegant theory and a useful tool in stochastic controls. In contrast, optimization of combined cumulants has not been studied systematically. The goal of this project is to lay a rigorous mathematical foundation for the optimization of combined cumulants. The investigator plans to study the following topics. (a) characterization of cumulants in differential equations, (b) Hamilton-Jacobi-Bellman equations for value functions, (b) the conditions for existence and uniqueness of solutions, (c) efficient algorithms for approximating and computing solutions It is hoped that the outcomes of this project will enrich the theories of stochastic controls and have applications in risk analysis and engineering designs.