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Paul
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### What is Taylor series?

And what is its connection to Ito's lemma? Thanks to greghmP

exotiq
Posts: 1888
Joined: October 13th, 2003, 3:45 pm

### What is Taylor series?

One simple definition: A Taylor series is a way of approximating an analytic function araound a single point using only the derivatives of the function at that point. Say for a function f around a point x:f(x + Dx) ~ f(x) + f'(x) (Dx) + (1/2) f''(x) (Dx)^2 + ... + (1/n!) f^(n)(x) (Dx)^n + O(Dx^(n+1))Ordinarily (x is not a stochastic process) when Dx = dx is infinitesimal, then (f(x + dx) - f(x))/dx = f'(x) + infinitesimal, which is the definition of the derivative from non-stochastic calculus.When x is a stochastic process , then dx^2 is proportional to dt and is not infinitesimal, so Ito's lemma provides for using the up to the second order term (no more, no less) to expand a function of a stochastic variable, with the remainder being infinitesimal. Just like a "total derivative" in multi-variable calculus, Ito's lemma expands the "stochastic derivative" or so to speak...

muniangel
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### What is Taylor series?

has it a connection to stochastic calculus? or Ito formula?

Witt
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Joined: February 26th, 2004, 4:20 am

### What is Taylor series?

If you write down this series for a function of two variables f(t,W) as and use a rule then you obtain the differential form of the Ito formulaSimilarly for a generic process, not just W. Or something like this, I feel I've forgotten something here

bhutes
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### What is Taylor series?

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Last edited by bhutes on July 3rd, 2005, 10:00 pm, edited 1 time in total.

rmcgowan
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### What is Taylor series?

One thing I never understood (presumably from lack of study) is: why is dWdW = dt?Considering dWdt = 0 and dtdt = 0...

Cuchulainn
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### What is Taylor series?

QuoteOriginally posted by: rmcgowanOne thing I never understood (presumably from lack of study) is: why is dWdW = dt?Considering dWdt = 0 and dtdt = 0...I had a similar question a few days ago.But what is dW and dt? Can someone give a precise definition. Book and page # is ideal. edit: It seems that dW^2 = dt is a heuristic, being based on the quadratic variation formula, lemma 2. The 'leap' from formula to heuristic is plausible I suppose? But is it true?//My hunch (disclaimer) is that dt and dW are perfectly OK if you write an SDE in integral form but the form dX = aXdt + bXdW is less precise??? (from a theoretic and computational viewpoint the integral form is also better in my opinion).
Last edited by Cuchulainn on December 29th, 2008, 11:00 pm, edited 1 time in total.
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Cuchulainn
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### What is Taylor series?

rmc,In Oksendal 1998 the Ito (dt, dW) "multiplication table" is used as a 'given' (page 44). And a series of steps (page 48) leads to dW^2 = dt (the 'striking formula') The second-last step is not clear at all. It is exactly the same issue as I raised with Lemma 2 in the previous post: how does equation (26) imply dW^2 = dt? How can these -presumably infinitesimal quantities - can be manipulated in this way?Can anyone shed a light on this patch??? Just like a "total derivative" in multi-variable calculus, Ito's lemma expands the "stochastic derivative" or so to speak... //And then when we move to the numerical solution , dW and dt becomes non-infinitesimal quantities Later.
Last edited by Cuchulainn on December 30th, 2008, 11:00 pm, edited 1 time in total.
"Compatibility means deliberately repeating other people's mistakes."
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lurchink
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Joined: May 9th, 2003, 3:56 pm

### What is Taylor series?

If I miss something in the questions raised about dW and dt, please enlighten me.But according to the part 3 of the definition of Brownian motion:W_(s+t)-W_s is normal N(0, t), and is independent of filtration up to sThis means dW is normal N(0, dt). Right?As for defining dW (or dt for that matter), you can define it as the differential form such that its integration is W (or t), bypassing all the infinitesimal nonsense. You can prove that the integral of dW*dW is t, hence dW*dW = dt.

popvivi
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Joined: November 19th, 2008, 1:28 am

### What is Taylor series?

QuoteOriginally posted by: WittIf you write down this series for a function of two variables f(t,W) as and use a rule then you obtain the differential form of the Ito formulaSimilarly for a generic process, not just W. Or something like this, I feel I've forgotten something here x should be t, right?

Cuchulainn
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### What is Taylor series?

QuoteOriginally posted by: popviviQuoteOriginally posted by: WittIf you write down this series for a function of two variables f(t,W) as and use a rule then you obtain the differential form of the Ito formulaSimilarly for a generic process, not just W. Or something like this, I feel I've forgotten something here x should be t, right?And o(..) should be O(..)? Big O
"Compatibility means deliberately repeating other people's mistakes."
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