Why I Care About Ito Calculus And How I'll Find Out More

Consider the indexed collection of random variables {X(t) | t a non-negative real}. This is a stochastic process in continuous time. A realization of it is just a time series. It makes sense to me to differentiate or integrate such a realization with respect to time. (As I remember it, though, a realization of a Wiener process is (almost always?) continuous and non-differentiable.) But what it would mean to talk about the derivative or integral of the stochastic process itself? As I understand it, this, along with Stochastic Differential Equations, is the subject of Ito or stochastic calculus.

Recently, physicists have been writing on economics. Generally, econophysicists distrust neoclassical economics and think they can do it better. Joseph L. McCauley's Dynamics of Markets: Econophysics and Finance (Cambridge University Press, 2004) is a recent textbook on econophysics. I find it difficult to read precisely where it draws on Ito calculus.

Luckily I have a resource to help me acquire some mathematical background for McCauley's text. Cosma Shalizi explains the fundamentals of Ito Calculus in Lectures 18-20 of his notes for Advanced Probability II or Almost None of Stochastic Processes.