the Ito integral
/ EE-toh /
An ordinary integral 'integral of f dt' adds up a quantity f weighted by smooth time. But what if you want to integrate against the jagged Brownian motion B instead — to write 'integral of f dB', meaning the cumulative effect of f as the random driver B wiggles? This is the central object of stochastic calculus, and the deepest application is a betting/trading story: if B is the random price path and H(t) is how many shares you hold at time t, then 'integral of H dB' is your total trading profit. The trouble is that B has no derivative and infinite total variation, so the usual definitions of an integral simply do not apply. The Ito integral is the careful repair.
It is built as a limit of simple sums, but with one crucial rule that gives it all its character: you must evaluate the integrand H at the LEFT endpoint of each small time step, BEFORE the random increment of B happens. Chop [0, T] into small pieces; the Ito integral is the limit of sum of H(t_k) times [B(t_(k+1)) - B(t_k)] — note H is taken at t_k, the start of the step. The left-endpoint rule is not a technicality: it encodes 'no peeking into the future'. You must choose your holding H(t_k) (your bet) using only information available up to time t_k, then the market move B(t_(k+1)) - B(t_k) happens. This makes the integrand a 'predictable' process and is exactly what keeps the construction honest and gives the Ito integral its signature property.
That signature property is that the Ito integral is itself a MARTINGALE: a fair game with E['integral of H dB'] = 0. Because you always bet before the fair coin is flipped, no strategy can manufacture an expected profit — the integral inherits Brownian motion's fairness. There is also a clean variance rule, the Ito isometry: E[('integral of H dB')^2] = E['integral of H^2 dt'], which turns the variance of a stochastic integral into an ordinary integral. Two honest cautions: first, the left-endpoint choice genuinely matters — evaluating at the midpoint instead gives the different Stratonovich integral (which obeys ordinary calculus but is not a martingale). Second, this whole construction is the deep end; the rigorous limit lives in mean-square convergence and needs measure theory to make airtight.
Take the simplest case H(t) = B(t), integrating B against itself. Naive calculus would guess 'integral of B dB' = B(T)^2 / 2, but the left-endpoint rule and (dB)^2 = dt give the correct answer 'integral of B dB' = B(T)^2 / 2 - T/2. The extra -T/2 is the quadratic-variation correction that ordinary calculus misses.
Betting before each coin flip makes the Ito integral a fair game (a martingale) with mean zero.
Evaluating the integrand at the LEFT endpoint is essential — it enforces 'no peeking' and makes the integral a martingale. Using the midpoint instead gives the different Stratonovich integral, which is not a martingale.