Ito's Lemma

Statement of the lemma

Let <math>x(t)</math> be an Ito (or Generalized Wiener) process[?]. That is let

<math> x(t) = a(x,t)dt + b(x,t)dW_t</math>

and let f be some function with a second derivative that is continuous. Then:

<math> f(x(t)) </math> is also an Ito process.

<math> df(x(t),t) = ( a(x,t)\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t} + \frac{b(x,t)*b(x,t)* \frac{\partial^2f}{\partial x^2}}{2})dt + b(x,t)\frac{\partial f}{dx}dW_t</math>

Informal proof

A formal proof of the lemma requires us to take the limit of a sequence of random variables, which is not handled carefully here.

Expanding f(x,t) is a Taylor series in x and t we have

<math> df = \frac{\partial f}{\partial x}{dx} + \frac{\partial f}{\partial t}dt + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}dx^2+ ...</math>

and substituting in for dx from above we have

<math> df = \frac{\partial f}{\partial x}(a.dt + b.dW_t) + \frac{\partial f}{\partial t}dt + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}(a^2(dt)^2 + 2.a.b.dt.dW + b^2(dW^2))+ ...</math>

In the limit as dt tends to 0 the <math>dt^2</math> and <math>dt*dW</math> terms disappear but the <math>dW^2</math> tends to dt. Substituting this dt in, and reordering the terms so that the dt and dW terms are collected we obtain

<math> df = ( a\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t} + \frac{b*b*\frac{\partial^2f}{\partial x^2}}{2})dt + b\frac{\partial f}{dx}dW_t</math>

as required.

Formal proof

A strong-willed individual is required here!