Martingales

The formal study of martingales begins in Chapter 7 , adding to the Chapter 4 introduction. The first section provides the needed definitions and derives basic properties on martingales. Then Doob's regularization theory is discussed, named for Joseph Doob (1910–2004), and investigates when martingales and related stochastic processes on (S,σ(S),Σ t (S),μ) u.c. have càdlàg modifications. The next two sections develop two martingale maximal inequalities, again due to Doob. Since the notion of a martingale is fundamental to the development of a pricing theory for financial derivatives, the final section provides an introductory discussion based on the discrete pricing theory of Chapter VI.9.