Past exam of the options II course of the University of Oxford / 2009 / M / M4 / e

Since $\mathbb{E}{N}_t=\mathrm{Var}(N_t)=\lambda t$, use the Poisson moment-generating function to get, for $t>0$,This is finite for every real $\theta$.

For each fixed $\theta$, the Taylor expansion of the exponential yieldsThe limiting moment-generating function $e^{{\theta }^2/2}$ is that of the standard normal distribution and is finite around zero. The moment-generating function continuity theorem therefore gives