Overview
Statement
Given a process {% Z(t) = g(t, X(t), Y(t)) %} where both {% X(t) %} and {% Y(t) %} are Ito Processes
{% dZ = \frac{\partial g}{\partial t} dt + \frac{\partial g}{\partial x} dX + \frac{\partial g}{\partial y} dY
+ \frac{1}{2} \frac{\partial ^2 g}{\partial x ^2} (dX)^2 + \frac{1}{2} \frac{\partial ^2 g}{\partial y ^2} (dY)^2
+ \frac{\partial ^2 g}{\partial x \partial y} (dX)(dY)
%}
Multidimensional Ito Lemma Heuristics
-
{% Z=XY %}
{% \frac{dZ}{Z} = \frac{dX}{X} + \frac{dY}{Y} +\frac{dX}{X} \frac{dY}{Y} %}
-
{% 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 %}