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Martingales and Fair Games

The martingale property
E[Xt+1∣Ft]=XtE[X_{t+1} \mid \mathcal{F}_t] = X_t

The best forecast of tomorrow is today. A statement about fairness, not about staying still.

Given everything known up to now, the expected future value is the present value. A fair game: no expected gain, no expected loss.

Ft\mathcal{F}_t is the filtration, the information available at time tt. Martingale-ness is always relative to an information set, which matters: a process can be a martingale to someone who knows less and not to someone who knows more.

The three cases

Martingale: E[Xt+1∣Ft]=XtE[X_{t+1}\mid\mathcal{F}_t] = X_t, fair.

Submartingale: ≥\geq, drifts up, favourable.

Supermartingale: ≤\leq, drifts down. Your position in a casino.

Optional stopping

The theorem with the most practical bite. Under suitable conditions, for any stopping time τ\tau (a rule for quitting that uses only information available at the time):

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