Ito Lemma
Overview
Itos lemma is one of the fundamental theorems of the stochastic calculus. It gives a method for finding analytic solutions of
stochastic integrals.
Ito Lemma
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{% df(t, W(t)) = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial W}dW + \frac{1}{2} \frac{\partial f^2}{\partial ^2 W} dt%}
Ocasionally youll see a more general form of the equation, stated as
{% df(t, W(t)) = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial W}dW + \frac{1}{2} \frac{\partial f^2}{\partial ^2 W} d[W] %}
Here, the last term is stated in terms of the quadratic variation of W.
However, for Brownian motion, the quadratic variation is equation to t.
{% [W] = t %}
Topics
Heuristic Justification
{% f(x_0 + \Delta x) - f(x_0) = f'(x_0)\Delta x + \frac{1}{2} f''(x_0) \Delta x ^2 + Remainder %}
Then, if we take multiple such steps, such that {% x-x_0 = \sum \Delta x %}
{% f(x) - f(x_0) = \sum [f'(x_0)\Delta x + \frac{1}{2} f''(x_0) \Delta x ^2 + Remainder] %}
Then as {% \Delta x \rightarrow 0 %}, this becomes
{% f(x) - f(x_0) = \int f'(x_0)\Delta x %}
This is true because both {% \sum \Delta x^2 + Remainder \rightarrow 0 %}
However, in the case that x is an
Borwnian Motion
{% \sum \Delta x^2 \rightarrow t %}, the quadratic variation.
