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

In summary, Ito's lemma (from Taylor series) to get df is to take the derivative of a function with respect to x and t, and then use the chain rule to get the df. Higher order terms, but they cancel out in stochastics. But this seems to contradict the standard differential of a function from multivatiable calculus. I don't get why they are different, can anyone explain?
  • #1
saminator910
96
1
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?
 
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  • #2
bump...
 
  • #3
saminator910 said:
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 canceled 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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Do not "bump" one of your threads to the top of a forum's thread list by posting a basically empty message to it, until at least 24 hours have passed since the latest post in the thread; and then do it only once per thread.
 
  • #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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1. What is the difference between Ito's lemma and Taylor series?

Ito's lemma and Taylor series are two different mathematical tools used in stochastic calculus. Ito's lemma is used to find the derivative of a function that involves a stochastic variable, while Taylor series is used to approximate a function using a series of polynomial terms.

2. Can I use Ito's lemma to compute the derivative of any function?

No, Ito's lemma can only be used to compute the derivative of functions that involve stochastic variables. If the function does not have a stochastic component, then Ito's lemma cannot be applied.

3. How does Taylor series differ from the differential of a function?

Taylor series and the differential of a function both involve finding the derivative of a function. However, Taylor series is an approximation of the function using a series of polynomial terms, while the differential is the exact derivative of the function at a specific point.

4. When should I use Ito's lemma instead of Taylor series?

Ito's lemma is used when dealing with stochastic processes or functions that involve random variables. Taylor series is more commonly used for deterministic functions. Therefore, if your function involves a stochastic variable, it is best to use Ito's lemma to find its derivative.

5. Are there any limitations to using Taylor series or Ito's lemma?

Both Taylor series and Ito's lemma have their limitations. Taylor series can only provide an approximation of the function and may not accurately represent the function for all values. Ito's lemma can only be applied to functions that involve stochastic variables and may not be applicable in all cases. It is important to understand the assumptions and limitations of both tools before using them in your calculations.

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