Itô calculus, named after Kiyoshi Itô, treats mathematical operations on stochastic processes. The most important is the Itô stochastic integral. Kiyoshi Itô Kiyoshi Itô (born September 7, 1915) is a Japanese mathematician, was born in Hokusei-cho, Mie Prefecture Japan. ... In the mathematics of probability, a stochastic process can be thought of as a random function. ...

Before starting, it is important to note that:

• Capitalized letters such as X denote random variables.

• 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 A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ... In the mathematics of probability, a stochastic process can be thought of as a random function. ... In mathematics, an infinitesimal, or arbitrarily small number, is a number that is greater in absolute value than zero yet smaller than any positive real number. ...

and is defined as the limit in probability of corresponding sums of the form In probability theory, there exist several different notions of convergence of random variables. ...

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. Probability theory is the mathematical study of probability. ... This article is about the mathematical concept of convolution. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...