martingales

Martingales are a class of stochastic processes used in probability theory to model fair games or, more generally, processes with a no-advantage property given the available information. Formally, a martingale is a sequence (X_n) of random variables defined on a probability space, equipped with an increasing family of sigma-algebras (F_n) representing the information available up to time n. The process is adapted, E|X_n| < ∞ for all n, and the conditional expectation of the next value given current information equals the present value: E[X_{n+1} | F_n] = X_n almost surely. In continuous time, a martingale is a process (X_t) adapted to a filtration (F_t) with E|X_t| < ∞ and E[X_t | F_s] = X_s for all s ≤ t.

Intuition and examples: A basic example is the simple symmetric random walk S_n, where increments are ±1 with equal probability; it satisfies E[S_{n+1} | S_n] = S_n. Brownian motion (W_t) is another standard martingale with respect to its natural filtration since E[W_t | F_s] = W_s for s ≤ t. Martingales can be nonnegative, and many convergence results apply to them. In particular, nonnegative martingales often converge almost surely, and certain integrability conditions yield convergence in L1 as well.

Key results and applications: Foundational theorems include Doob’s martingale convergence theorems and the optional stopping theorem, which describe behavior at stopping times under appropriate conditions. Martingale theory has wide applications, notably in finance where discounted asset prices under a risk-neutral measure form martingales, in the analysis of gambling strategies, and in stochastic calculus and modeling of evolving systems with information flow.