Resumen

This Ph.D. thesis explores stochastic optimization from a Distributionally Robust perspective, focusing on two significant themes: the innovative use of decision variable-dependent ambiguity sets in Distributionally Robust optimization (DRO), and the estimation of the mode of a random vector using the DRO perspective. Regarding the first topic, new techniques utilizing p-Wasserstein metrics in stochastic programming are proposed, where ambiguity sets are uniquely decision variable-dependent. These developments, under certain assumptions, can be reduced to finite-dimensional optimization problems, sometimes convex. They are tested within the portfolio optimization context against standard methodologies. The research also extends to stochastic programming with expected value constraints, setting feasibility criteria relative to the Wasserstein radius and constraint parameters, and benchmarking model performance using both simulated and real financial market data. Additionally, in the realm of mode estimation, an innovative strategy is devised for identifying a mode estimator in a random vector sample, even in the absence of known probability distribution or density function. This strategy employs a DRO approach and Wasserstein distance, demonstrating the resulting estimator is consistent.