I am trying to solve this problem where we're asked to compute the quadratic variation of a process. I assume that it is necessary to apply Ito's formula but not sure how to get the right solution. Furthermore, I'm also not sure about how to apply Ito's formula to a function that includes integrals like this one. I am familiar with the basic Ito Formula and know how to apply it to simpler functions.
Let N be a (P,F)-Poisson process with parameter $\lambda$ > 0 and define the process: X = $(X)_t$
$$ X_t = 2 + \int_{0}^{t} \sqrt{s} dW_s + N_t $$
Compute $[X]_t$
is this attempt correct?
$$ X_t = 2 + \int_{0}^{t} \sqrt{s} dW_s + N_t $$
which we can write in differential form as:
$$ dX_t = \sqrt{t}dW_t + dN_t $$
then the quadratic variation is given as:
$$ d[X_t] = dX_t * dX_t = (\sqrt{t}dW_t + dN_t) * (\sqrt{t}dW_t + dN_t) = tdt + dNt $$
I assumed that the cross terms cancel out. Then we can then rewrite it in integral form to get $[X_t]$
$$ [X_t] = \int_{0}^{t} t dt + \int_{0}^{t} dN_t = \frac{t^2}{2} + N_t $$
If this attempt is correct, I am not quite sure why $dN_t * dN_t = dN_t$ would be true in the quadratic covariation step.