Ito's Lemma

Ito's Lemma is a lemma used in stochastic calculus to find the differential of a function of a particular type of stochastic process. It is therefore to stochastic calculus what the chain rule is to ordinary calculus. The lemma is widely employed in mathematical finance.

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!