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?
Best Answer
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.