Topics in McKean-Vlasov equations: rank-based dynamics and Markovian projection with applications in finance and stochastic control.

Abstract

In this thesis, we consider McKean-Vlasov stochastic differential equations (SDEs), arising from various fields, such as the large-system limit of mean field games, particle systems with mean field interactions, financial mathematics, optimal control, game theory and mathematical physics. We study three aspects of the equations: as limits of interacting particle systems, the existence and uniqueness for them and the connection between the time-marginal distribution and the law of the process.

Firstly, in the setting of rank-based models, we use the mean field limit and the Gaussian fluctuations to characterize the dynamics of observables which capture the diversity of a financial market. The results can be used to study the performance of functionally generated portfolios over short-term and medium-term horizons.

Secondly, we study the McKean-Vlasov SDE arising from the calibration of local stochastic volatility models in finance. Despite the limited theoretical understanding, we give the strong existence result of stationary solutions for these SDEs, as well as their strong uniqueness in an important special case.

Thirdly, we consider conditional McKean-Vlasov stochastic differential equations where the conditional time-marginals of the solutions satisfy non-linear stochastic partial differential equations (SPDEs) of the second order and the laws of the conditional time-marginals follow Fokker-Planck equations (FPEs) on the space of probability measures. We establish connections between the SDEs, SPDEs and the FPEs. This provides a useful tool to obtain Markovian controls in the context of controlled McKean-Vlasov dynamics.