Backward stochastic differential equations with applications

Abstract

In this thesis we study backward stochastic differential equations driven by a Brownian motion and by a Levy process and their applications, focusing on their applications to financial markets. We give results on the existence and uniqueness of solution of backward stochastic differential equations when the drift is Lipschitz continuous and the terminal condition is square integrable and measurable with respect to the terminal filtration. Backward stochastic differential equations associated with a forward stochastic differential equation are investigated. We use the generalisation of the Feynman-Kac formula to show the relationship between a backward stochastic differential equation associated with a forward stochastic differential equation and a partial differential equation in the Brownian motion case and a partial differential integral equation for the Levy process case. The Doob's h-transform is studied for the Brownian motion and applied to stochastic differential equations. Finally, we conclude with an application to option pricing and hedging of a European calls for both Brownian and Levy processes