Variance of Integrated Poisson Process $U_t = \int_{0}^{t} N_u\,du$

Question

Any guidance/ hints on the following would be much appreciated!

Show that $$var(U_t) = \frac{λt^3}{3}$$ for $U_t$ is the integrated Poisson process, that is $U_t = \int_{0}^{t} N_u \,du$, $N_u$ is the Poisson process with index $u$.

From previous exercises, I have found the following. They may be useful in finding the variance of $U_t$

• $U_t = tN_t - \sum_{k = 1}^{N_t}S_k$
• $E[U_t] = \frac{λt^2}{2}$
• $E\left[ \sum_{k = 1}^{N_t}S_k\right] = \frac{λt^2}{2}$

I thought of using the definition $$var(U_t) = E({U_t}^2) - E({U_t})^2$$ but I do not know what is $E({U_t}^2)$.

Alternatively, I believe I may use \begin{align} var(U_t) & = var\left( tN_t - \sum_{k=1}^{N_t}S_k \right)\\ & = var(tN_t) + var\left(\sum_{k=1}^{N_t}S_k \right) + 2cov\left(tN_t, \sum_{k=1}^{N_t}S_k \right) \\ &=λt^3 + var\left(\sum_{k=1}^{N_t}S_k \right) + 2cov\left(tN_t, \sum_{k=1}^{N_t}S_k \right) \\ &= \cdots \end{align}

and once again I'm stuck.

Thanks!

Answer 1

Hints:

1. Compute $\mathbb{E}(N_u N_v)$ for $u \neq v$. To this end, fix say, $v>u$ and write $$\mathbb{E}(N_u N_v) = \mathbb{E}((N_v-N_u) N_u) + \mathbb{E}(N_u^2).$$ Now use that the Poisson process $(N_u)_{u \geq 0}$ has independent and stationary increment.
2. By Fubini's theorem, we have $$\left( \int_0^t N_u \, du \right)^2 = \int_0^t \int_0^t N_u \, N_v \, du \, dv$$ and $$\mathbb{E} \left( \left[ \int_0^t N_u \, du \right]^2 \right) = \int_0^t \int_0^t \mathbb{E}(N_u N_v) \, du \, dv.$$ Now plug in your results from step 1 to compute the double integral on the right-hand side. To simplify the calculations a bit, you can use symmetrization: $$\mathbb{E} \left( \left[ \int_0^t N_u \, du \right]^2 \right) = 2 \int_0^t \int_0^v \mathbb{E}(N_u N_v) \, du \, dv.$$