Itô Integral

Integrate against Brownian wiggle: bet before the kick, and pick up a - t2 that ordinary calculus never sees.

The idea

The Itô integral $\int_{0}^{t} f\,dW$ defines integration against a Brownian path.

To see what it should measure, suppose you bet on a market that moves like a Brownian motion $W$, staking $f(s)$ at each time $s$. Over a short stretch of time your winnings are the stake times the market's move across the stretch, and your total winnings by time $t$ accumulate these products. If the path were smooth, the move across $[s,\ s + ds]$ would be $W'(s)\,ds$ and the total would be the ordinary integral of $f(s)\,W'(s)$. A Brownian path is continuous but has a derivative at no time, so the integral must be built from the increments themselves.

Partition the interval as $0 = t_{0} < t_{1} < \cdots < t_{n} = t$, write $\Delta W_{i} = W(t_{i+1}) - W(t_{i})$ for the market's move across the $i$-th stretch, and define

$\int_{0}^{t} f\, dW = \lim \sum_{i} f(t_{i})\,\Delta W_{i},$

the limit taken as the stretches shrink. The stake is sampled at $t_{i}$, the left endpoint, so that each bet is fixed before the move it multiplies arrives. Against a smooth path the sampling point would not change the limit. Against a Brownian path it does — other choices give different answers — and left-endpoint sampling is what keeps the integral a fair game.

Ways to work on it

• Walkthrough. Build the Itô integral from left-endpoint sums and establish its basic properties.
• Practice. Compute expectations and variances of Itô integrals with simple integrands.
• Hardest. Compute the variance of an Itô integral whose integrand is itself random.

Not sure where to start? Take the ten-question placement test.