Regularity of monge potentials and hedging in a degenerate market

The aim of the thesis is to study the optimal transport on an abstract Wiener space and the hedging portfolio for a degenerate market model. In the first part, we discuss the Sobolev regularity of Monge potentials under the assumption that the initial measure is log-concave and the target measure has a strictly positive density on an abstract Wiener space. We prove that backward Monge potential is an element of the second-order Sobolev space. The regularity result allows us to show that backward Monge potential solves Monge- Ampére equation. We also prove that forward potential solves Monge-Ampére equation in Alexandrov sense under a weaker assumption than log-concavity. The purpose of the second part of the thesis is to derive the hedging portfolio in a financial market where the prices are governed by a stochastic equation with a singular volatility matrix. The main mathematical tools of the study are the representation theorem with respect to a minimal martingale and Malliavin calculus for the functionals of a degenerate diffusion process, which have been established in recent studies. We use those developments to prove a version of the Itô-Clark type representation formula derived for these functionals under an equivalent martingale measure. Consequently, we derive the hedging portfolio as a solution to a system of linear equations. The uniqueness of the solution is achieved by a projection idea that lies at the core of the martingale representation. We apply our result to exotic options, whose value at maturity depends on the prices over the entire time horizon.