Optimal Control of Nonlinear Dynamic Stochastic Models

The overall aim of the project is to develop two new decision-support instruments to be used by policy makers when choosing their optimal strategies for designing economic and financial policy. Optimal control theory has a great number of applications not only in economics or finance, but also in many other areas of science from engineering to medicine. One of the main questions in economics is which strategy should be used in order to influence certain economic variables such as unemployment, inflation or GDP, and to bring about the desired values. The suitable method allows policy makers to design economic policies so as to maximize some measure of social welfare or to minimize costs in terms of policy objectives deviating from desired or ideal time paths. The aim of this project is to create two sophisticated instruments for calculating approximately optimal policies for nonlinear discrete- time stochastic dynamic systems with a quadratic objective function under system and parameter uncertainties. The first instrument is based on classical optimization methods and provides a solution to an optimal control problem in a system with rational expectation using an active learning strategy. The new aspects allow getting more reliable and realistic optimal solution. The second proposed instrument uses a heuristic optimization method and allows handling different limitations, which are very difficult or impossible to consider in classical solution algorithms. In this project two constraints are considered, namely inequality constraints and different frequencies of the control and state variables. The idea is to give decision-makers a more flexible instrument for finding optimal policies. Along theoretical research, an important focus is put on real-world applications of the new frameworks. Each of the aims consists of two parts, namely methodological development of the algorithms combined with implementation in MATLAB and empirical work. First, the new algorithms have to be applied to some existing models and subsequently a new large (or, rather, more sophisticated) macroeconomic model of the Austrian economy has to be developed for the purpose to application. The new approaches are of use not only in the area of economic, for example to analyze current situation in countries and to test different alternative ways of applying fiscal and monetary policies, but also other research areas like medicine or pollution control can take advantage of these innovations.

The overall aim of the project was to develop two new optimal control algorithms to be used by policy makers when choosing their optimal strategies for designing economic and financial policies. One of the main questions in macroeconomics is which strategy should be used in order to influence certain economic variables, such as unemployment, inflation or GDP, and to bring about the desired values. To answer such questions, optimal control theory is usually applied and a suitable decision-support optimal control instrument is required. Thus, the goal of the project was to create two sophisticated instruments for calculating approximately optimal policies for nonlinear discrete-time stochastic dynamic systems with a quadratic objective function under system and parameter uncertainties. The first instrument is based on classical optimization methods and provides a solution to an optimal control problem in a system with rational expectation (some variables are forward-looking) using an active learning strategy (the dual problem of choosing the best strategy and minimizing the uncertainty about the system). To summarise the research process, we developed the mathematical framework (OPTCON3), implemented the algorithm in MATLAB and tested it on some existing, simplified models. Initial evaluations show that the OPTCON3 approach may be promising to enhance our understanding of the adaptive economic policy problem under uncertainty. In addition, a model of Austrian economy was created in order to apply the OPTCON3 algorithm to a bigger economic system. The resulting evaluations show that the new algorithm allows getting more reliable and realistic optimal solution. The second instrument uses a heuristic optimization method and allows handling different limitations, which are very difficult or impossible to consider in "classical" solution algorithms. In this project, a special case with inequality constraints in the dynamic system is considered. The new instrument uses a heuristic optimization method, Differential Evolution, and adapts it to passive learning strategy (the idea is to observe the outcomes of the system at the end of each time period and to use this information in order to adjust the system). The new approach is implemented in MATLAB and firstly tested on some existing models and then on a novel nonlinear dynamic model of the Austrian economy targeting on the output - public debt trade-off. Optimal control of stochastic processes is a topic, which occurs in many contexts of applied mathematics such as engineering, biology, chemistry, economics, and management science. Thus, the new algorithms are of a multidisciplinary nature and can be applied not only to economic models but also to models in other fields of science.