Martingale
A process whose expected next value, given everything observed so far, is its current value. The formalisation of a fair game — and, less obviously, the tool that turns "fair" into a machine for proving things.
Where it sits in the subject
Two of the spoke's three stochastic-processes sources build it first, in discrete time, before anything continuous exists. lawler-stochastic-calculus opens chapter 1 with conditional expectation, then martingales, the optional sampling theorem, the martingale convergence theorem, square-integrable martingales and integrals with respect to a random walk — the entire stochastic integral, rehearsed where the sums are finite. wu-introduction-stochastic-processes gets there after the midterm (lectures 15–18) and adds uniform integrability, which is the condition under which the convergence theorem gives what one expects.
gamarnik-advanced-stochastic-processes reaches them at lecture 10, after large deviations, and pairs them with stopping times.
The two results that do the work
Optional sampling (optional stopping). A fair game stopped by a rule that cannot see the future is still fair. This is the theorem that says no stopping strategy makes a martingale profitable, and it is where the mathematics meets gambling systems directly — stirzaker-elementary-probability added both the martingale and optional stopping to its second edition, and the ruin problems in mcmullen-probability-theory are the elementary version of the same question.
Martingale convergence. A martingale with a suitable bound converges almost surely. Convergence without any assumption of a limit having been designed in.
Why finance runs on it
The connection lawler-stochastic-calculus makes explicit in chapter 5, and the reason "martingale"
is a word ../quant-trading-wiki inherits without teaching. Under a change of measure (Girsanov), the
discounted price of a traded asset becomes a martingale, and the price of a claim becomes its expected
value under that measure — the martingale approach to pricing, his section 5.6. The martingale
representation theorem (5.7) then says the hedging portfolio exists: the claim can be written as an
integral against the driving brownian-motion, which is the mathematical statement that a hedge
strategy can be found.
An arbitrage under one probability measure is an arbitrage under any equivalent one, since equivalent measures agree about which events have probability zero. That observation, which costs one line, is what lets the whole pricing argument move to a measure chosen for convenience.
Related
stochastic-processes · brownian-motion · stochastic-calculus · markov-chain · probability-theory · lawler-stochastic-calculus · wu-introduction-stochastic-processes · gamarnik-advanced-stochastic-processes · mathematics · synthesis