What is Itô Calculus?

Calculus is all well and good. Some people think it’s a pretty neat idea, and certainly plenty of decent maths and real-world applications rest on it.

But it’s a bit… [dons sunglasses]… limited ((Look. I don’t get paid enough to edit out bad puns, ok?)).

In particular, when you’re dealing with stochastic processes – for example, random walks or Brownian motion – the usual rules of calculus break down. Limits aren’t properly defined, and your calculations end up shooting off to infinity.

The Itô Calculus addresses some of those problems: under certain circumstances ((namely: time is continuous, the distribution of the next step on the path depends only on the current state (the Markov property), the expected value of the next step is equal to the current state (the Martingale property), quadratic variation (which I don’t want to explain) and normality (that the difference between any two steps is normally distributed)), it’s possible to integrate a function stochastically over a random walk.

It’s still a limiting process, and you might naively expect it to follow all the same rules as regular calculus. But no, of course it doesn’t.

In particular, the chain rule becomes $\int \d F = \int \diff F X \d X + \int \diffn{2}{F}{X} \d t$ ((this is Itô’s Lemma)) – the integral has a dependency both on the path and time.

Why is it important?

I mean, it sort of brought down the global economy once upon a time.

Wait. That’s not entirely fair: it’s like blaming calculus for the time your car crashed into the sea. It’s more reasonable to blame user error in both cases.

In any event: a notorious application of Itô calculus is to how stock prices are expected to behave over time. In particular, it can be used to derive the value distribution of a stock at some point in the future – which means you can place a value on stock options. This leads to the Black-Scholes-Merton equation.

Black-Scholes-Merton is good, actually. It’s an excellent model of stock behaviour under normal circumstances. Unfortunately, as soon as the assumptions it’s based on stop holding, it stops being useful, just like any other model. It seems that some banks weren’t quite au fait with that basic tenet of applied maths, and here we are more than a decade later still with dramatically reduced public services as if it were our fault. Ahem.

Who was Kiyosi Itô?

Kiyosi Itô was born in Hokusei (now part of Inabe), Japan in 1915. He studied at the University of Tokyo, earning his PhD in 1945 while working at the Japanese National Statistical Bureau.

In 1952, he became a professor at the University of Kyoto, and retired in 1979.

He died in Kyoto in 2008, aged 93.