Mathematical models for financial bubbles

Financial bubbles have been present in the history of financial markets from the early days up to the modern age. An asset is said to exhibit a bubble when its market value exceeds its fundamental valuation. Although this phenomenon has been thoroughly studied in the economic literature, a mathematical martingale theory of bubbles, based on an absence of arbitrage has only recently been developed. In this dissertation, we aim to further contribute to the developement of this theory. In the first part we construct a model that allows us to capture the birth of a financial bubble and to describe its behavior as an initial submartingale in the build-up phase, which then turns into a supermartingale in the collapse phase. To this purpose we construct a flow in the space of equivalent martingale measures and we study the shifting perception of the fundamental value of a given asset. In the second part of the dissertation, we study the formation of financial bubbles in the valuation of defaultable claims in a reduced-form setting. In our model a bubble is born due to investor heterogeneity. Furthermore, our study shows how changes in the dynamics of the defaultable claim's market price may lead to a different selection of the martingale measure used for pricing. In this way we are able to unify the classical martingale theory of bubbles with a constructive approach to the study of bubbles, based on the interactions between investors.