Ito Lemma Heuristics
Overview
The
stochastic calculus
gives meaning to stochastic differentials such as {% dX(t) = \mu dt + \sigma dW(t) %}
by reinterpreting them as stochastic integrals.
Nevertheless it is convenient to think in terms of differentials and to re-express longer
formulas into differential notation.
(Note it is possible to formalize the notion of a stochastic differential as a mathematical object
in its own right. see
herzberg)
Ito Lemma
the content of
Ito's Lemma
can be summarized by the following heuristics on differentials.
{% dt^2 = 0 %}
{% dt \times dW = 0 %}
{% dW^2 = dt %}
in combination with the differential form of
Taylors Theorem on the function {% Z(X,t) %}
{% dZ = \frac{\partial Z}{\partial t} dt + \frac{\partial Z}{\partial X} dX + \frac{\partial ^2 Z}{\partial t ^2} dt^2 + \frac{\partial^2 Z}{\partial X^2} dX^2 +
\frac{\partial ^2 Z}{\partial t \partial X} + ...
%}
Here we assume that {% X %} is some
Ito Process
which can be specified by the differential
{% dX(t) = \mu(X,t) dt + \sigma(X,t) dW %}
Examples
Multidimensional Ito Lemma
{% Z=XY %}
{% \frac{dZ}{Z} = \frac{dX}{X} + \frac{dY}{Y} +\frac{dX}{X} \frac{dY}{Y} %}
(see Back pg 37)
{% Z=\frac{Y}{X} %}
{% \frac{dZ}{Z} = \frac{dY}{Y} - \frac{dX}{X} - \frac{dX}{X} \frac{dY}{Y} + \frac{dX}{X} ^2 %}
{% Z= e^X %}
{% \frac{dZ}{Z} = dX + \frac{dX ^ 2}{2} %}
{% Z= log(X) %}
{% dZ = \frac{dX}{X} - \frac{1}{2} (\frac{dX}{X})^2 %}
