Topics in Stochastic Control

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This research program concerns several topics in stochastic controltheory and related areas of applied probability and nonlinear partialdifferential equations. One topic is risk-sensitive control on aninfinite time horizon, motivated by problems of robust feedbackcontroller design for nonlinear systems. Another application ofrisk-sensitive control is in mathematical finance, including dynamicportfolio allocation problems on long time horizons. Yet anotherresearch topic concerns first order partial differential equations ofHamilton-Jacobi-Bellman type. While such equations are nonlinear in theusual sense, they are linear with respect to max-plus algebraoperations. This allows for approximate solution via max-plus basisexpansions. Finally stochastic control models for economic growth anddebt which arise in international finance are being studied.Stochastic control provides a framework for modeling and analysis ofdynamic decision making in the presence of uncertainty. The method ofdynamic programming provides a way to obtain optimal stochastic controlpolicies by solution of corresponding nonlinear partial differentialequations. The research funded through this grant is motivated by arange of applications in engineering and financial economics, includingrobust feedback controller design, nonlinear estimation and filteringand optimal dynamic investment allocation. In internationalfinance,growth/debt models are considered in which the goal is to choosenational investment and consumption policies which optimize a suitablychosen criterion subject to imposed constraints. The model performanceunder optimal control may provide benchmarks to suggest whether actualcurrent account deficits and levels of foreign debt are sustainableunder current policies.