Problems in Mathematical Finance Related to Transaction Costs and Model Uncertainty.

Abstract

This thesis is devoted to the study of three problems in mathematical finance which involve either transaction costs or model uncertainty or both.

In Chapter II, we investigate the Fundamental Theorem of Asset Pricing (FTAP) under both transaction costs and model uncertainty, where model uncertainty is described by a family of probability measures, possibly non-dominated. We first show that the recent results on the FTAP and the super-hedging theorem in the context of model uncertainty can be extended to the case where only options available for static hedging (hedging options) are quoted with bid-ask spreads. In this set-up, we need to work with the notion of robust no-arbitrage which turns out to be equivalent to no-arbitrage under the additional assumption that hedging options with non-zero spread are non-redundant. Next, we look at the more difficult case where the market consists of a money market and a dynamically traded stock with bid-ask spread. Under a continuity assumption, we prove using a backward-forward scheme that no-arbitrage is equivalent to the existence of a suitable family of consistent price systems.

In Chapter III, we study the problem where an individual targets at a given consumption rate, invests in a risky financial market, and seeks to minimize the probability of lifetime ruin under drift uncertainty. Using stochastic control, we characterize the value function as the unique classical solution of an associated Hamilton-Jacobi-Bellman (HJB) equation, obtain feedback forms for the optimal investment and drift distortion, and discuss their dependence on various model parameters. In analyzing the HJB equation, we establish the existence and uniqueness of viscosity solution using Perron's method, and then upgrade regularity by working with an equivalent convex problem obtained via the Cole-Hopf transformation.

In Chapter IV, we adapt stochastic Perron's method to the lifetime ruin problem under proportional transaction costs which can be formulated as a singular stochastic control problem. Without relying on the Dynamic Programming Principle, we characterize the value function as the unique viscosity solution of an associated variational inequality. We also provide a complete proof of the comparison principle which is the main assumption of stochastic Perron's method.