Encyclopedia > Ito's Lemma

  Article Content

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!



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
BBC News 24

... cable television subscribers could view it. In 1999, with the advent of digital television in the UK, satellite viewers were able to view the service. The BBC were ...

 
 
 
This page was created in 26 ms