Topics in Stochastic Analysis and Control

Abstract

In this dissertation, problems in stochastic analysis and control are investigated, which include mathematical finance, online learning, and mean field game. For math- ematical finance, 1) a martingale optimal transport problem with bounded volatility is studied, which allows to calibrate not only current observation (option prices) but also historical data (stock prices); see Chapter II, 2) the embedding problem in multi-dimension is solved via excursion theory in probability; see Chapter III, 3) size of most stable subgraphs of random graphs, k-core, is determined by using branching processes; see Chapter IV. For online learning, 1) an unprecedented solution to the 4-expert problem with finite stopping is provided, via an explicit construction of the solution to a nonlinear partial differential equation; see Chapter V 2) prediction prob- lems with a limited adversary are studied using partial differential equation tools; see Chapter VI and VII. For mean field game, 1) the convergence phenomenon of N + 1- player Nash equilibrium is studied by the entropy solution to scalar conservative laws; see Chapter VIII, 2) infinite horizon mean field type control and game are solved via McKean-Vlasov forward backward stochastic differential equations; see Chapter IX.