Itō calculus

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Itō calculus, named after Kiyoshi Itō, treats mathematical operations on stochastic processes. Its most important concept is the Itō stochastic integral.

The Itō integral can be defined in a manner similar to the Riemann integral, that is as a limit of Riemann sums. Suppose that $W : [0, T] \times \Omega \to \mathbb{R}$ is a Wiener process and $X : [0, T] \times \Omega \to \mathbb{R}$ is a stochastic process adapted to the natural filtration of the Wiener process. Then the Itō integral

$\int_{a}^{b} X_{t} \, \mathrm{d} W_{t} : \Omega \to \mathbb{R}$

is defined as the L² limit of

$\sum_{i = 0}^{k - 1} X_{t_{i}} \left( W_{t_{i+1}} - W_{t_{i}} \right)$

as the mesh of the partition $0 = t_{0} < t_{1} < \dots < t_{k} = T$ of $[0, T]$ tends to 0 (in the style of a Riemann-Stieltjes integral).

Technically speaking, the construction is first performed on a class of "elementary processes" and then extended to the closure of this class in the L² norm. The collection of all Itō integrable processes is sometimes denoted $L 2 (W)$ .

A crucial fact about this integral is Itō'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.

[edit] Other approaches

The Stratonovich integral is another way to define stochastic integrals. Its derivation rule is simpler than Ito's lemma.

In the definition of the Stratonovich integral, the same limiting procedure is used except for choosing the average to the values of the process $X$ at the left- and right-hand endpoints of each subinterval: i.e.

$\frac{X_{t_{i+1}} + X_{t_{i}}}{2}$ in place of $X_{t_{i}}.$

Conversion between Itō and Stratonovich integrals may be performed using the formula

$\int_{0}^{T} \sigma (t, X_{t}) \circ \mathrm{d} W_{t} = \frac{1}{2} \int_{0}^{T} \sigma' (t, X_{t}) \sigma (t, X_{t}) \, \mathrm{d} t + \int_{0}^{T} \sigma (t, X_{t}) \, \mathrm{d} W_{t},$

where $X$ is some process, $\sigma' (t, x) := \frac{\partial \sigma}{\partial x} (t, x)$ , and $\int_{0}^{T} \sigma (t, X_{t}) \circ \mathrm{d} W_{t}$ denotes the Stratonovich integral.