This project advances the mathematical foundations of robust finance and decision-making under uncertain. One major focus is the development of a systematic framework to quantify divergences between stochastic models, with the goal of uncovering the underlying geometric structure of stochastic processes. These advancements will facilitate stability analysis of models used in dynamic decision-making contexts such as finance, climate science, and autonomous systems. A second theme is the exploration of optimal strategies for revealing information to capture and sustain attention --motivated by the question, “What is the most exciting game?” This problem has broad relevance to entertainment, behavioral economics, and education. Graduate students will be actively involved throughout the project. More specifically, the first direction investigates the Kullback-Leibler divergence between martingales. Using tools from convex analysis and optimal transport, the project will establish new functional inequalities, develop numerical schemes, and explore statistical applications of these divergences. The second direction frame information design as an optimization problem involving the Kullback-Leibler divergence. Building on martingale optimal transport theory, the project develops a novel control methodology for transport problems with path-dependent objectives. These results have direct applications in the pricing of financial derivatives -- such as Asian, lookback, and barrier op