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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 $W_3$.

Fadmad
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    Try using Ito's lemma in two variables where you consider the semimartingales $W_{1,t}$ and $\int_0^t W_{3,s} dW_{2,s}$. – Rhys Steele Dec 24 '20 at 15:28
  • @RhysSteele, I thought of that, but I am having trouble differentiating $\int W_{3,s} d W_{2,s}$ with respect to $W_{3,s}$. Can I just use the fundamental theorem of calculus? – Fadmad Dec 25 '20 at 13:15
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    $d \int_0^{t}W_{3,s}dW_{2t}=W_{3,s}dW_{2t}$. The stochastic differential is just a type of short hand for a stochastic integral. – fes Dec 25 '20 at 18:47

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Let $A_t := \int_0^t W_{3,s} dW_{2,s}$. As pointed out by @fesman in the comments, we have $dA_t = W_{3,t}dW_{2,t}$. Note that this is just notation for how we defined $A_t$, we are not actually differentiating anything. Then we have $Z_t = W_{1,t}^2 e^{W_{1,t} + A_t}$, and Ito's formula gives

\begin{align*} dZ_t &= (W_{1,t}^2 e^{W_{1,t}+A_t} + 2 W_{1,t} e^{W_{1,t}+A_t}) dW_{1,t} + W_{1,t}^2 e^{W_{1,t}+A_t} dA_t \\ & \qquad + (W_{1,t}^2 e^{W_{1,t}+A_t} + 4 W_{1,t} e^{W_{1,t}+A_t} +2e^{W_{1,t}+A_t}) dW_{1,t}dW_{1,t} + W_{1,t}^2 e^{W_{1,t}+A_t} dA_t dA_t \\ &= (W_{1,t}^2 e^{W_{1,t}+A_t} + 2 W_{1,t} e^{W_{1,t}+A_t}) dW_{1,t} + W_{1,t}^2 e^{W_{1,t}+A_t} W_{3,t}dW_{2,t} \\ & \qquad+ (W_{1,t}^2 e^{W_{1,t}+A_t} + 4 W_{1,t} e^{W_{1,t}+A_t} +2e^{W_{1,t}+A_t}) dt + W_{1,t}^2 e^{W_{1,t}+A_t} W_{3,t}^2dt. \end{align*}

Again, to emphasize, we have not differentiated $Z_t$. This is just notation for $$Z_T = \int_0^T (W_{1,t}^2 e^{W_{1,t}+A_t} + 2 W_{1,t} e^{W_{1,t}+A_t}) dW_{1,t} +\int_0^T W_{1,t}^2 e^{W_{1,t}+A_t} W_{3,t}dW_{2,t} + \int_0^T (W_{1,t}^2 e^{W_{1,t}+A_t} + 4 W_{1,t} e^{W_{1,t}+A_t} +2e^{W_{1,t}+A_t} + W_{1,t}^2 e^{W_{1,t}+A_t} W_{3,t}^2) dt.$$

user6247850
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  • This is an amazing answer. If you don't mind me asking, what is your academic background? I would guess a PhD in Probability or similar? – Jan Stuller Dec 31 '20 at 16:08
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    No problem! I'm still working on my math PhD, with a research focus in financial mathematics and stochastic processes. – user6247850 Dec 31 '20 at 19:38
  • Consider joining https://quant.stackexchange.com/ , I see you're a good contributor and the Quant Finance community on quant.stackexchange always appreciates mathematically solid contributors such as yourself. – Jan Stuller Dec 31 '20 at 19:52