Martingales

A stochastic process $(X_t)_{t \geq 0}$ with respect to a filtration $(\mathcal{F}_t)_{t \geq 0}$ is a martingale if:

• It is adapted: $X_t$ is $\mathcal{F}_t$-measurable for each $t$.

• $\mathbb{E}[|X_t|] < \infty$ for all $t$.

• For all $s < t$:

  \[ \mathbb{E}[X_t \mid \mathcal{F}_s] = X_s. \]

Intuition: A martingale is a stochastic process where the best guess for the future value is the current value. In other words, the expected change from now to the future is zero; like a ``fair game.''