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I am having trouble applying Ito's Formula to the following:

Let $Z_t = W_{1t}^2 e^{W_{1t}+ \int_0^t W_{3s}dW_{2s}}$. Find $dZ_t$. $W_1,W_2,W_3$ are independent Brownian motions.

I know the formula but I am having trouble differentiating the integral with respect to $W_2$ and $W_3$.

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The problem, I think, is that you are trying to do it in one step.

But, if you write, for example $Z_t = W_{1t}^2 e^{Y_t}$, where $Y_t = W_{1t} + \int_0^t W_{3s}dW_{2s}$, you should be able to see how to do it.

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  • $\begingroup$ Ok thanks! I think it is better to define $Y_t $without $W_{1,t}$ $\endgroup$
    – Fadmad
    Dec 25 '20 at 18:28

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