Discrete Time Processes, Martingales

We are now interested in "time" dependent random outcomes. In this chapter we consider discrete time, that is an ordered countable index. This could be anything from a finite set \(\{0, 1, \ldots, T\}\) or at the other extreme \(\mathbb{Q}\). We will however generically consider \(\mathbb{N}_0\) as the standard index for time and denote different times by \(s\) or \(t\). The results in this chapter all holds for any countable ordered set. In the case where a specific different index shall be considered we will precise it.

Throughout we fix a probability space \((\Omega, \mathcal{F}, P)\).

• Discrete Time Processes (filtration, adapted processes, stopping times.)
• Martingales (martingale, stochastic integral, doob's optional sampling theorem)
• Martingales: Almost Sure Convergence (Doob's upcrossing lemma, martingale convergence, Borel-Cantelli)
• Martingales: \(L^p\)-convergence (\(L^p\)-convergence, law of large numbers)