Martingales with Integrable Variation and Square Integrable Martingales

This chapter starts to develop modern theory of martingales and stochastic integrals. It stress that all the elements of M (the collection of all uniformly integrable martingales) are supposed to be cadlag. Moreover, martingales always mean cadlag martingales. The chapter discusses integrable variation martingales, square integrable martingales, and purely discontinuous square integrable martingales. It gives theorems and lemmas for the integrable variation martingales, square integrable martingales, and purely discontinuous square integrable martingales.