Stochastic differential of a particular martingale

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SUMMARY

The discussion focuses on finding the stochastic differential of the process defined by X(t) = (W(t)^{2}-t)^{2} - 4∫(W(s))^{2}ds, where W(t) represents Brownian Motion. The user, Steve, seeks guidance on applying Itô's formula to the first term and the stochastic fundamental theorem of calculus to the integral term. The key takeaway is that Itô's formula is applicable only to the first term, while the second term requires a different approach using stochastic calculus principles.

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steve1985
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Hello everyone,
I'm studying from Oksendal's book, and I'm stuck at an exercise which asks you to find the differential form of:

X(t) = (W(t)^{2}-t)^{2} - 4\int (W(s))^{2}ds
where W(t) is a Brownian Motion.

I tried several possible functions g(t,W(t)) which could have led to a potential solution (by finding d(g(t,W(t))) with Ito), but none led me any closer to a solution.

Can you please put me in the right direction?

thanks!
Steve

PS. I posted this in the wrong place, it should have been in "Homeworks and coursework questions", but I don't know how to move it...
 
Last edited:
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Yes Ito's formula only applies to the first term in the sum. For the second term, if the integral is from 0 to t, just apply the stochastic version of the fundamental law of calculus.
 

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