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) = drift(t) + B(t) %}
The volatility can be made to be non-constant by using a non-constant function {% \sigma(t,\omega) %} to multiply the
normal distribution term.
{% Ito(t) = drift(t) + \sigma(t) B(t) %}
Discrete Time
In discrete time, the process is evaluated at a discrete (countable, possibly finite) set of points in the interval over which it is defined.
Thge change in the process {% \Delta X(t,B(t), \omega) %} is defined as the change in the Brownian motion {% \Delta B(t) %} plus a drift term {% u(t, \omega) \Delta t %}
{% \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 continuous time process is imagined as the limit as {% \Delta t \rightarrow 0 %} of the discrete time process 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
- Informal Differential Ito Process: hand-waving description of Ito processes using the language of differentials
- Ornstein-Uhlenbeck Process - a simple process that includes mean reversion
- Stochastic Volatility
- Simiulating Ito Processes
- Sum of Ito Processes