Chemistry - Itô calculus

Itô calculus, named after Kiyoshi Itô, treats mathematical operations on stochastic processes. The most important is the Itô stochastic integral.

Before starting, it is important to note that:

Capitalized letters with a subscript t such as B_t denote a stochastic process which is a set of random variables indexed by t.

A small letter d to the left of a random process e.g. dB_t means an infinitesimal change in the random process which is a random variable.

The stochastic integral of a process X_t with respect to a process B_t is denoted by

$\int_{a}^{b} X_t\, dB_t$

and is defined as the limit in probability of corresponding sums of the form

$\sum X_{t_i} (B_{t_{i+1}} - B_{t_i}).$

A crucial fact about this integral is Itô's lemma.

Both summation and multiplication of random variables are defined in probability theory. The summation involves a convolution of the probability density function (pdf) and multiplication is repeated summation.