In this senior thesis, we provide a stochastic analysis approach to the live sports betting context. We rigorously characterize the optimal behavior of a bookmaker and bettors of different types in a betting market for a stochastically evolving underlying sports game. We derive the optimal strategies and the dynamics satisfied by expected terminal wealth through partial differential equations of Hamilton-Jacobi type. We aimed for the thesis to be as mathematically comprehensive and rigorous as possible through techniques from functional analysis and measure theory. Moreover, we fully characterize Stackelberg equilibria for markets with bettors of both homogeneous and heterogeneous types.

This thesis investigates optimal execution strategies within a predatory trading environment through the lens of a Stackelberg mean field game. Specifically, it addresses the problem faced by a distressed institutional investor (leader) forced to liquidate a significant asset position over a finite horizon, anticipating strategic reactions from a large population of high-frequency traders (HFTs), modeled collectively as followers. Extending previous models, the framework introduced here leverages mean field approximations to capture the aggregate behavior of HFTs and the hierarchical decision-making inherent in such scenarios. Under the assumption of linear price impacts consistent with the Almgren–Chriss framework, we adopt cost functionals in the spirit of Cartea–Jaimungal. Building on the probabilistic approach of Carmona and Delarue, the equilibrium dynamics are fully determined by the fixed points of coupled forward–backward stochastic differential equations (FBSDEs), which can be solved to yield an explicit open-loop feedback control. We derive analytical solutions and present numerical results alongside a sensitivity analysis. Ultimately, this thesis proposes a realistic model that can serve as a benchmark for evaluating execution strategies, whether for a large institution or for a high-frequency trading desk.