Ito process

An Itô process is a function of both time and a Brownian motion.

It is a stochastic process ${\left\{X_t\right\}}_{t\ge 0}$ of the form

$$X_t=x_0+{\int }_0^t{a}_{s}ds+{\int }_0^t{b}_{s}d{B}_s,$$

where ${\left\{a_t\right\}}_{t\ge 0}$ and ${\left\{b_t\right\}}_{t\ge 0}$ are adapted processes such that the integrals are defined. (An Itô process is always adapted.)

$a_t$ is the drift term, $b_t$ is the diffusion term (volatility).

An Itô process is thus the sum of an ordinary integral (w.r.t. time) and a stochastic integral (w.r.t. BM).

As an abbreviation, we can write it as an SDE:

$$d{X}_t=a_{t}dt+b_{t}d{B}_t.$$

An Itô process may be thought of as a Brownian motion with stochastic drift and volatility. (The martingale and markov property generally do not hold.)

See also: Ito’s lemma

Examples §

Brownian motion §

The standard BM ${\left\{B_t\right\}}_{t\ge 0}$ is an Itô process. (Pick $a_t\equiv 0$ and $b_t\equiv 1$.)

So is the general BM with (constant) drift (and constant volatility)

$$X_t=x_0+\mu t+\sigma B_t,$$

where $\mu \in \mathbb{R}$ and $\sigma >0$. (Pick $a_t\equiv \mu$ and $b_t\equiv \sigma$.)

The square of the Brownian motion §

$\left\{B_t^2\right\}$ is an Itô process with $a_t\equiv 1$ and $b_t=2B_t$.

$$B_t^2=2{\int }_0^t{B}_{s}d{B}_s+t={\int }_0^{t}1ds+{\int }_0^{t}2B_{s}d{B}_s.$$

If $X_t$ is an Itô process, then so is $f\left(t,X_t\right)$ where $f$ is any function which is continuously differentiable in $t$ and twice continuously differentiable in $x$. This is a special case of Itô’s lemma where $f(t,x)=x^2$.

No diffusion §

With $b_t\equiv 0$, the process

$$X_t=x_0+{\int }_0^t{a}_{s}ds$$

is an Itô process where $\left\{a_s\right\}$ is an adapted process.

This gives stochastic processes whose paths are differentiable but the derivatives are stochastic.

A concrete example is $X_t=$ $\frac{1}{t}{\int }_0^t{B}_{s}ds$, the average value of the Brownian motion up to time $t$.

No drift §

If $a_t\equiv 0$, then

$$X_t=X_0+{\int }_0^t{b}_{s}d{B}_s$$

is a martingale with an arbitrary (but fixed) starting point.