Dynamic Optimization in Management

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The objective of this research program is to develop mathematical models and efficient algorithms to solve practical decision-making problems that are dynamic in nature and that involve uncertainty. Two specific areas of application will be considered: investment decisions in financial markets, and the management of environmental and natural resources. Although these application areas are very different, they share a common methodological approach, that is, stochastic dynamic optimization in a multidimensional framework. In both cases, the solution of realistic problems requires the development of efficient numerical methods. Thus, in addition to addressing these specific issues, a broader objective of this research program is to develop innovative methods for the numerical solution of large-scale stochastic dynamic problems, and to contribute to the advancement of knowledge in the field of decision-making under uncertainty. ***The evaluation of innovative financial products almost always requires the use of numerical procedures. Moreover, financial institutions are increasingly concerned with managing risk and satisfying regulatory requirements. In practice, algorithms for the valuation of financial products, the evaluation of risk, and the determination of investment or hedging strategies need to be used in real time and should provide accurate results in seconds. The objectives of this first line of research are to increase the efficiency (accuracy, speed) of evaluation algorithms, and to develop new general approaches that can be used for more realistic market models. ***A growing concern in environmental and resources management is related to the so-called "tragedy of the commons". Access to a common resource (air quality, biodiversity, high-seas fishing, non-renewable resources) leads to overexploitation, and possibly, to catastrophic consequences. One of the means employed to prevent this from happening is the establishment of agreements between stakeholders. In this context, foresight is defined as the ability of stakeholders to account for the evolution over time of key features (environmental damage costs, learning, technological progress, signatories, economic development), when this evolution depends on agents' joint strategies. Clearly, dynamics and foresight are important factors in the design, stability and implementability of environmental agreements. The objectives of this second line of research is to study how agreements are affected by dynamics and uncertainty, and by the foresight and strategic outlook of participants. ***The general methodology considered is that of dynamic optimization with multidimensional state spaces. Solution approaches will include functional interpolation schemes (value functions and/or optimal strategies), and dimension reduction techniques, including aggregation of states, and learning (forward) algorithms. **