A martingale is a special stochastic process whose increments over an interval in future have zero expectation given information of the past history of the process. Martingales play a central role in the theory of stochastic processes and stochastic calculus. In this chapter, we present the definition of martingales.

**Definition 22.1** (Martingale). Let *I* be a set. A stochastic process *X* = (*X*_{t} : *t* *I*) is called a *martingale* relative to a filtration ({_{t} : *t* *I*}, *P*) if the process satisfies the following three conditions:

(a) *X* is adapted to the filtration.

(b) *E*(|*X*_{t}|) < ∞ for all *t* *I*.

(c) *E*[*X*_{t}|_{s}] = *X*_{s} a.s. for all *s* ≤ *t*.

**Definition 22.2** (Supermartingale). Let *I* be a set. A stochastic process *X* = (*X*_{t} : *t* *I*) is called a *supermartingale* relative to a filtration ({_{t} : *t* *I*}, *P*) if the process satisfies the following three conditions:

(a)

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