Reading through the book "Brownian Motion & Stochastic Calculus" by Karatzas and Shreve, I found the following exercise (problem 3.9, page 15):
Let $ \ N \ $ be a poisson process with intensity $ \lambda > 0 $ (this means, in particular, that $ N_t $ is poisson-$\lambda t$-distributed, i.e. $ P(N_t = k ) = \exp(-\lambda t) \frac{(\lambda t)^k}{k!}, \ \forall \ k \geq 0$)
Use Stirling's approximation to show that $\ \lim_{t \to \infty} (1/\sqrt{\lambda t} ) \ E( N_t - \lambda t )^+ = \frac{1}{2 \pi}$.
Trying to prove the claim, I started out like this:
$ \frac{1}{\sqrt{\lambda t}} E( N_t - \lambda t )^+ = \frac{1}{\sqrt{\lambda t}} \exp(-\lambda t) \ \sum_{k \geq \lambda t} \ (k - \lambda t) \frac{(\lambda t)^k}{k!} $ $ \approx \frac{1}{\sqrt{\lambda t}} \exp(-\lambda t) \ \sum_{k \geq \lambda t} \ (k - \lambda t) \frac{(\lambda t)^k}{\sqrt{2 \pi k} \left( \frac{k}{e} \right) ^k}$
Does anybody know how to finish the proof?
Thanks a lot for your help! Regards, Si