Affine processes and their applications to financial mathematics

Abstract

In this work we discuss a special kind of stochastic processesX=〖{X_t}〗_(〖t∈R〗_(≥0) ) that are of exceptional interest from both the theoretical ad the applied points of view. These processes are called Affine Processes, and are characterized by the fact that their characteristic function has the form of an exponential of an affine function, i.e. by 〖∅^x〗_X (u) |_t=E^x [e^(<X_(t ),u>) ]=e^(Φ(t,u)+<x,Ψ(t,u)>), Where the exponent Φ(t,u)+ <x,Ψ(t,u)> Is an affine function of its initial state x in the state-space E=〖R^m〗_(≥0)×R^n. The above expectation E^x is the expectation which respect to the law P^x of the process started at x. In chapter 2 we introduce affine processes and discuss the main properties associated with these processes, and we give examples of such processes. In addition, we discuss in detail the semi-flow property and the Feller property for the affine processes. We also discuss the important class of regular affine processes. Chapter 3 discusses the application of affine processes to financial mathematics. In section 3.1 we introduce basic aspects of the math of finance, while in section 3.2 we discuss some applications of affine processes in financial mathematics. Finally, given that the general subjects of stochastic processes, stochastic calculus, and stochastic differential equations are highly technical subjects, and so many definitions are needed for a smooth reading of such a work, we give a quite detailed first chapter on the basics of stochastic processes and stochastic calculus, to pave the way for a clear understanding of the rest of the thesis.