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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.

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1 Answer 1

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Your attempt is correct.

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.

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  • $\begingroup$ thank you so much, this clears things up $\endgroup$
    – Niko
    May 19, 2023 at 1:48

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