Topics in Stochastic Control

Brown University, Providence RI

Investigators

Abstract

This research program concerns several topics in stochastic control theory and related areas of applied probability and nonlinear partial differential equations. One topic is risk-sensitive control on an infinite time horizon, motivated by problems of robust feedback controller design for nonlinear systems. Another application of risk-sensitive control is in mathematical finance, including dynamic portfolio allocation problems on long time horizons. Yet another research topic concerns first order partial differential equations of Hamilton-Jacobi-Bellman type. While such equations are nonlinear in the usual sense, they are linear with respect to max-plus algebra operations. This allows for approximate solution via max-plus basis expansions. Finally stochastic control models for economic growth and debt which arise in international finance are being studied. Stochastic control provides a framework for modeling and analysis of dynamic decision making in the presence of uncertainty. The method of dynamic programming provides a way to obtain optimal stochastic control policies by solution of corresponding nonlinear partial differential equations. The research funded through this grant is motivated by a range of applications in engineering and financial economics, including robust feedback controller design, nonlinear estimation and filtering and optimal dynamic investment allocation. In international finance,growth/debt models are considered in which the goal is to choose national investment and consumption policies which optimize a suitably chosen criterion subject to imposed constraints. The model performance under optimal control may provide benchmarks to suggest whether actual current account deficits and levels of foreign debt are sustainable under current policies.