# Ito's lemma and taylor series

The Wikipedia page on Ito's lemma gives a heuristic derivation using a Taylor series expansion.

I'm having trouble getting to what they give as a Taylor series expansion for $f(t,x)$ (or rather, it's differential).

$df=\frac{\partial f}{\partial t}dt+\frac{\partial f}{\partial x}dx+\frac{1}{2}\frac{\partial^2 f}{\partial x^2}dx^2 +...$

The second order Taylor series for $f(t,x)$ around the point $(a,b)$ is

$f(a,b) + f_t(a,b)(t-a) + f_x(a,b)(x-b)+\frac{1}{2}[f_{tt}(a,b)(t-a)^2+2(f_{tx}(a,b)(t-a)(x-b)+f_{xx}(a,b)(x-b)^2]+...$

If I take the derivative with respect to $t$ or $x$ and then rewrite it in differential form, I cant seem to get it to what it is in Wikipedia. So what am I missing?

• could you post the expression found on Wikipedia as part of your question so that your question is self-contained? Commented Nov 11, 2016 at 18:39
• @BenBolker done Commented Nov 11, 2016 at 18:47
• You're not missing anything--you're merely including extra irrelevant terms (because they are of higher order than needed). Commented Nov 11, 2016 at 19:32
• @whuber are you sure? If I take the derivative the 1/2 factor drops out because of the exponent 2 and the chain rule. Yet in Wikipedia, the 1/2 factor remains. Commented Nov 12, 2016 at 5:12
• @whuber I don't blame you! This question was motivated partly by a brain fart involving the differential form of the Taylor series and partly by not appreciating why some second order terms drop out and why one second order term stays. Fortunately a book I have on the topic by Hull has cleared up both issues. I will answer my own question in case anyone else has similar issues. Commented Nov 12, 2016 at 15:11

I eventually managed to figure this out but for me the problem was not appreciating in full how stochastic calculus differs from ordinary calculus in this problem.

In ordinary calculus, the following is true

$df=\frac{\partial f}{\partial t}dt+\frac{\partial f}{\partial x}dx$

for a function $f$ that depends on $t$ and $x$. This fact can be derived by starting with the Taylor series for a function of 2 variables

$f(t,x)=f(a,b)+f_t(a,b)(t-a)+f_x(a,b)(x-b)...$ as I gave in my question.

Then consider

$f(t,x)-f(a,b)=f_t(a,b)(t-a)+f_x(a,b)(x-b)...$ as a and b approach t and x respectively. In the limit, all the second order and higher terms tend to zero much faster than the first order terms and are dropped. So in the limit, this becomes

$df(t,x)=f_t(t,x)dt+f_x(t,x)dx$

But with stochastic processes, the $dx$ actually contains something on the order of $\sqrt {dt}$ so then the $dx^2$ contains something on the order of $dt$, so then in the limit earlier, it contains a term which does not tend to zero any faster than $dx$ so it is not dropped.