Martingales and Local Martingales

The final Chapter 9 studies a number of results on martingales and local martingales, most originated by Joseph Doob. The first section develops statements of Doob's optional stopping theorem. For càdlàg martingales, one result generalizes the martingale property to the values of the stopped process M T and M T' and the sigma algebra Σ T (S) , where T≤ T′ are stopping times that are assumed bounded μ-a.e. The other result proves that stopped càdlàg martingales are martingales. The next sections investigate various relationships between martingales and local martingales, and then results on martingale convergence theory which address existence and properties of the limiting random variable M ∞ . For this latter investigation, Doob's upcrossings inequality is first developed. With the aid of Doob's martingale convergence results, the next section generalizes the earlier development of optional stopping. This generalization removes the μ-a.e. boundedness assumption on T and T′, and instead assumes that M t is uniformly integrable. The final section investigates alternative properties that can be used to identify a martingale.