# Quadratic Variation Of Mixed Brownian Motion and Poisson Process

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.

The quadratic variation for a Poisson process is: $$[N]_t=\lim_{\sup(t_{i+1}-t_i)\rightarrow0}\sum_{i:t_i\leq t}(N_{t_{i+1}}-N_{t_i})^2\tag{1}$$ for some partition $$\Pi(t)=0\leq t_0\leq\dots\leq t_n\leq t$$ of the segment $$[0,t]$$. Simply pick the partition such that the jump times $$\tau_1\leq\dots\leq\tau_k$$ of $$N$$ between $$0$$ and $$t$$ are included in $$\Pi(t)$$ then given $$N_{\tau_j}-N_{\tau_j^-}=\Delta N_{\tau_j}$$: $$[N]_t=\sum_{j:\tau_j\leq t}\Delta N_{\tau_j}^2$$ But $$\Delta N_{\tau_j}$$ is always equal to 1, and so is $$\Delta N_{\tau_j}^2$$, therefore the quadratic variation of the Poisson process jumps by $$1$$ whenever $$N$$ jumps. Consequently: $$\textrm{d}[N]_t=\textrm{d} N_t$$
Addendum. To justify picking the partition $$\Pi(t)$$ such that it contains the jump times $$\tau_1\leq\dots\leq\tau_k$$, consider the definition $$(1)$$ again. For each $$i$$, either there is no $$j\in[1,k]$$ such that $$\tau_j\in[t_i,t_{i+1})$$ in which case $$\smash{\lim_{\sup(t_{i+1}-t_i)\rightarrow0}(N_{t_{i+1}}-N_{t_i})^2=0}$$; else assuming without loss of generality that there is a single jump in the interval $$[t_i,t_{i+1})$$: \begin{align} N_{t_{i+1}}-N_{t_i} &=(N_{t_{i+1}}-N_{\tau_j})+(N_{\tau_j}-N_{t_i})\\ &=(N_{\tau_j}-N_{\tau_j})+(N_{\tau_j}-N_{\tau_{j-1}})\\ &=\Delta N_{\tau_j} \end{align} from which it entails the original argument is valid $$-$$ the argument generalizes to an arbitrary number of jumps within $$[t_i,t_{i+1})$$ at the cost of more involved notation.