Define Martingales, Sub-Martingales and Super-Martingales with examples.

A *stochastic process* is a collection of random variables indexed by time, representing the evolution of a system subject to random influences. Within the study of stochastic processes, martingales, sub-martingales, and super-martingales are important classes of processes that describe specific properties of conditional expectations over time.

A martingale is a stochastic process that models a fair game, where the expected future value, given all past information, is equal to the current value. Formally, let \(\{X_t\}_{t \geq 0}\) be a stochastic process adapted to a filtration \(\{\mathcal{F}_t\}\). Then \(\{X_t\}\) is a martingale if:

1. \(X_t\) is integrable: \(\mathbb{E}[|X_t|] < \infty\) for all \(t\),
2. \(X_t\) is adapted to the filtration \(\mathcal{F}_t\),
3. \(\mathbb{E}[X_{t+1} \mid \mathcal{F}_t] = X_t\) for all _______ ______ ____ ______ ________.
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