Ito Process
Overview
An Ito process is an extension of
Brownian Motion
which adds both a trend to the process and a non-constant volatility.
A Brownian motion has no drift. That is {% \mathbb{E}(B_t) = 0 %}. This can be corrected by adding a
deterministic drift term.
{% Ito(t) = B(t) + drift %}
The volatility is made non constant by using a non constant function {% \sigma(t,\omega) %} to multiply the
normal distribution term.
Discrete Time
In discrete time, the drift term is {% u(t, \omega) \Delta t %} is added to the Brownian motion
{% \Delta X(t,B(t), \omega) = u(t, \omega) \Delta t + \Delta B(t) %}
The process now has a drift, but has a constant volatility. A variable volatility can be added to the process
by the inclusion of a coefficient that multiplies the brownian motion term.
{% \Delta X(t,B(t), \omega) = u(t, \omega) \Delta t + \sigma(t, \omega) \Delta B(t) %}
Continuous Limit
The discrete time processes are then imagined in the limit as {% \Delta t \rightarrow 0 %}
and written in differential form.
{% dX(t,W(t), \omega) = u(t, \omega) dt + \sigma(t, \omega) dB(t, \omega) %}
which we usually abbreviate as :
{% dX(t,B(t)) = u(t) dt + \sigma(t) dB(t) %}
Additional Topics
