Homogenization and Calibration Problems through the Lens of Mean Field Games and Mean Field Control

In this thesis, we apply techniques from mean field game theory and mean field control theory to establish homogenization results for various types of slow-fast stochastic differential equations (SDEs) and to address calibration problems for stochastic volatility models. The first part is devoted to proving a homogenization result for conditional slow-fast McKean–Vlasov SDEs. In the second part, we extend this analysis to the controlled setting, establishing a homogenization result for Hamilton–Jacobi equations posed in the Wasserstein space. The final part focuses on the use of martingale optimal transport and martingale Schrödinger bridges as tools for calibrating stochastic volatility models, either to SPX option data or jointly to SPX and VIX options.