| The dissertation studies modern risk management techniques for decision making in highly uncertain environments. The traditional framework of decision making under uncertainties relies on stochastic programming or simulation approaches to surpass simpler quasi-deterministic techniques, where the uncertainty is modeled by relevant statistics of stochastic parameters. In many applications, however, the mentioned methodologies in their conventional form fail to generate efficient and robust decisions. Mathematical models for such class of applications, therefore, are referred to as highly uncertain environments, with the defining features such as: large number of mutually correlated stochastic factors with dynamically changing or uncertain distributions, multiple types of risk exposure, out-of-sample application of the solution, etc. Robust decision making in such environments requires an explicit control of the risk induced by uncertainties.; The first part of the dissertation considers risk management approaches for financial applications. We present a general framework of risk/reward optimization that establishes the equivalence of different formulations of optimization problems with risk and reward functions. As an application of the general result, we consider optimization of portfolio of stocks with Conditional Value-at-Risk objective and constraints, and compare this approach with the classical Markowitz Mean-Variance methodology. An extensive study of out-of-sample performance of trading algorithms based on different risk measures is performed on the example of managing of a hedge fund portfolio. A chapter dedicated to multi-stage decision-making problems presents a new sample-path approach for multi-stage stochastic programming problems and applies it to the problem of optimal transaction implementation. The generality of the developed model allows for using it in pricing of complex derivative securities, such as exotic options.; The second part of the dissertation considers risk management techniques for military decision-making problems. The main challenges of military applications attribute to various types of risk exposure, uncertain probability measures of risk-inducing factors, and inapplicability of the "long run" convention. Different formulations for stochastic Weapon-Target Assignment problem with uncertainties in distributions are considered, and relaxation and linearization techniques for the resulting nonlinear mixed-integer programming problems are suggested. |
