the martingale transform (discrete stochastic integral)
You have a fair game and you are allowed to play it with a betting strategy: in each round you choose a stake, then collect (or pay) your stake times the round's result. What does your total fortune look like after many rounds, and can clever staking tilt the game in your favour? The bookkeeping that adds up 'stake times outcome' round by round is the martingale transform — the discrete cousin of a stochastic integral, where you integrate a betting strategy against a fair game.
Let M_0, M_1, M_2, ... be a martingale with increments DeltaM_n = M_n - M_(n-1), and let H_1, H_2, ... be a predictable process — your stakes, each chosen before the round. The transform, written (H . M)_n, is the accumulated winnings (H . M)_n = sum from k=1 to n of H_k times DeltaM_k = H_1*DeltaM_1 + ... + H_n*DeltaM_n. Think of H_k as how many units you bet on round k, and DeltaM_k as the per-unit gain or loss in that round; their product is that round's profit, and the sum is your fortune. The central theorem is short and powerful: if M is a martingale and H is predictable (and bounded, or integrability is otherwise assured), then (H . M) is again a martingale. Stake-weighting a fair game leaves it fair. If instead M is a submartingale and H is non-negative and predictable, the transform is again a submartingale — betting non-negative amounts on a favourable game keeps it favourable.
This is the exact discrete shadow of the Ito integral: H is the integrand (predictable, like betting before the outcome), M is the integrator (the fair-game noise), and (H . M) is the integral. It is the cleanest statement of why gambling systems fail — see the impossibility of beating a fair game — and it is also constructive, the device behind option pricing where the transform represents the value of a self-financing trading strategy. The caveat: the theorem requires H to be predictable and some integrability; drop predictability (let your stake see the outcome) and you can manufacture a guaranteed profit, which is exactly why the hypothesis is non-negotiable.
Play a fair +/-1 coin game with the doubling strategy: bet 1, and after each loss double the next stake. The increments DeltaM_k are +/-1; the stakes H_k are predictable. Each round (H . M) either jumps up by the current stake (a win, which ends the run with net +1) or down by it. The transform is a martingale, so its expected fortune stays 0 — there is no free lunch, only a swap of a tiny near-certain gain for a rare catastrophic loss.
(H . M) sums stake times outcome; a predictable stake on a martingale is again a martingale.
The transform stays a martingale only because H is predictable. If you could choose your stake AFTER seeing the outcome, you could force a profit — predictability is the whole point.