What is the quadratic variation of compensated poisson process?

Question

I want to know what is the quadratic variation of a compensated poisson process.

$$[N-\lambda t, N - \lambda t]_t = \sum_{0 \leq s \leq t} (\Delta (N_s - \lambda s))^2 = ? $$

This is as far as I can go. I am not sure how to proceed further.

I am more interested in the steps to derive the answer rather than the answer itself.

Answer 1

Hints:

1. Recall: $\Delta(f(s)-g(s)) = \Delta(f(s))-\Delta(g(s))$ for any two (deterministic) functions $f,g$.
2. Recall: $\Delta(f(s))=0$ if $f$ is continuous at $s$.
3. Using step 1,2 show that $$(\Delta (N_s-\lambda s))^2 = (\Delta N_s)^2.$$
4. Deduce from the fact that $(N_t)_{t \geq 0}$ is a Poisson process that $$(\Delta N_s)^2 = \Delta N_s.$$
5. Conclude.

Solution: $$[N_t-\lambda_t] = N_t.$$