Bounding expectation of a supremum process

Question

This is exercise 3.9(c) on page 15 of Karatzas and Shreve's Brownian Motion and Stochastic Calculus.

Let $N_t$ be a Poisson process with intensity $\lambda$. In particular, if $t$ is fixed, $N_t$ is a Poisson random variable with parameter $\lambda t$.

I would like to show $$E\left[\sup_{\sigma\le t \le \tau} \left(\frac{N_t}{t}-\lambda\right)^2 \right]\le \frac{4\tau\lambda}{\sigma^2}.$$

I'm not sure how to bound this expectation. The only theorem that seems relevant here is Doob's maximal inequality, but I don't see how to apply that.

Answer 1

Note that

$$ \left( \frac{N_t}{t}-\lambda \right)^2= \frac{1}{t^2} (N_t-t\lambda)^2 \leq \frac{1}{\sigma^2} (N_t-t\lambda)^2$$

for all $t \in [\sigma,\tau]$. Hence,

$$\mathbb{E} \left[ \sup_{t \in [\sigma,\tau]} \left( \frac{N_t}{t}-\lambda \right)^2 \right] \leq \frac{1}{\sigma^2} \mathbb{E} \left[ \sup_{t \in [\sigma,\tau]} (N_t-t\lambda)^2 \right].$$

Now apply Doob's maximal inequality to prove the assertion.