One of the most important concepts of modern probability theory is the martingale, which formalizes the notion of a fair game. In this chapter, we first lay the foundations for the treatment of general stochastic processes (filtrations, adapted processes, stopping times). We then introduce martingales and the discrete stochastic integral as well as the martingale representation theorem and the stability theorem for discrete martingales.
We close with an application to a model from mathematical finance.
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