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I am attempting the following problem, and was hoping if you guys could provide any feedback on whether my approach is valid. Thank you in advance for your time!

The question is as follows:

"Let $X(t) = \sum_{i=1}^n B_{i}^2(t)$ where $B_{i} = \left\{ B_{i} : t\geq 0 \right\}$ are independent one dimensional Brownian Motions. Derive an SDE satisfied by X"

I have attempted to use the multidimensional Ito's Lemma with the differential a function of $f(t, B(t))$. The partial derivatives I obtained are as follows:

$$\frac{\partial X(t)}{\partial B_{i}^2(t)} = \sum_{i=1}^{n} 2B_{i}(t)$$

$$\frac{\partial^2 X(t)}{\partial B_{i}^2(t)^2} = \sum_{i=1}^{n} 2 = 2n$$

$$\frac{\partial X(t)}{\partial t} = 0 $$ Any of the cross derivatives is zero since Wiener increments are independent.

Putting this into Ito's formula I obtain: $$dX(t) = ndt + \sum_{i=1}^{n} 2B_{i}(t) dB_{i}(t)$$

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