Faculty of Science

Objectives

To give students a solid foundation in Martingale theory, that is indispensable for studies in stochastic analysis; financial Mathematics, and statistics of stochastic processes.

Contents

Lebesgue decomposition of measures, absolute, continuous and singular measures; Radon Nokodym theorem and Radon Nikodym derivatives; Conditional expectations with respect to -algebras; Discrete Martingales: Martingale inequalities; Doob’s inequalities; Martingale convergence theorems; Uniform integrability; Doob Meyer decompositions; Continuous Martingales: Definitions and examples; Submartingale inequalities; Upcrossing inequalities and Doob’s maximal inequality; Conditions for the existence of Cadlag versions of Submartingales; Doob Meyer decompositions and natural processes; Continuous square integrable Martingales, the quadratic and cross variation processes.