Informal Definition of Ito Process
Overview
An Ito process is one which, for a given change in time {% dt %}, the change in
the process is
{% dX(t) = \mu \times dt + \sigma \times \phi \times \sqrt{dt} %}
where {% \phi %} is a standard normal variable (mean=0, var =1)
The second term is abbreviated as
{% dW = \phi \times \sqrt{dt} %}
so that
{% dX(t) = \mu dt + \sigma dW %}
{% \mu %} is often taken to be a constant, but need not be so, it can be a function of time
{% \mu(t) %}
Moments
{% \mathbb{E}[dW] = \mathbb{E}[\phi \sqrt{dt}] = \sqrt{dt} \mathbb{E}[\phi] = 0 %}
and
{% Var[dW] = Var[\phi \sqrt{dt}] = Var[\phi] dt = dt %}
From this, we can conclude
{% \mathbb{E}[dX] = \mu dt %}
and
{% Var[dX] = \sigma dt %}
