Ito's lemma/taylor series vs. differential of a function

  • #1

Main Question or Discussion Point

for a function f(x,t)

Ito's lemma (from Taylor series) to get df

[itex] df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial t} dt + \frac{\partial^{2} f}{\partial x^{2}} dx^{2} + ....[/itex]

higher order terms, but they cancel out in stochastics.

but this seems to contradict the standard differential of a function from multivatiable calculus

[itex] df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial t} dt [/itex]

I don't get why they are different, can anyone explain?
 

Answers and Replies

  • #2
bump....
 
  • #3
pasmith
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for a function f(x,t)

Ito's lemma (from Taylor series) to get df

[itex] df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial t} dt + \frac{\partial^{2} f}{\partial x^{2}} dx^{2} + ....[/itex]

higher order terms, but they cancel out in stochastics.

but this seems to contradict the standard differential of a function from multivatiable calculus

[itex] df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial t} dt [/itex]

I don't get why they are different, can anyone explain?
The differential also includes higher-order terms, which are routinely ignored in non-stochastic calculus. But in the stochastic calculus, you need to keep the [itex]dx^2[/itex] term because on substituting [itex]dx = a dt + b dW[/itex] you find that [itex]dx^2[/itex] includes a term of order [itex]dW^2 = dt[/itex].
 
  • #4
Thanks, but I don't get how the higher order terms can just be ignored. Like you said, in stochastic calculus any term greater than [itex]dx^{2}[/itex] is cancelled out, because [itex]dt^{2}=dWdt=0[/itex]. So it makes sense that we wouldn't include them in Ito's lemma.
 
  • #5
bump........
 
  • #6
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  • #7
Hey, anybody have any ideas?
 
  • #8
I don't know the answer, but I will at least try to brainstorm a little bit.

I would guess that the differential is defined differently in the stochastic calculus than in regular calculus. In regular calculus, a differential df is a synonym for an infinitesimal. Is the same true in the stochastic calculus? I do not know - but either way, I think what the issue comes down to is that some of the premises for developing normal calculus no longer apply, thus motivating the development of the Stochastic calculus in the first place. With a different set of postulates come different definitions and theorems, and this is probably one such.

So in other words I think the issue you mention is probably a symptom of a fundamental difference between Riemennian calculus and stochastic calculus. If this is right, your best bet to find an answer to your question is to read what postulates go into the stochastic calculus and see how it is developed from those postulates, and probably you will run into the result you are looking for fairly quickly.

Here are some links that might be good starting points.
http://en.wikipedia.org/wiki/Differential_(calculus [Broken])
http://en.wikipedia.org/wiki/Stochastic_differential
http://en.wikipedia.org/wiki/Ito_calculus
http://en.wikipedia.org/wiki/Bounded_variation
http://www.columbia.edu/~ks20/FE-Notes/4700-07-Notes-Ito.pdf

Good luck!
 
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