Problem: Simulating a Poisson process

Let \(N(t)\) be the number of events up to (and including) time \(t\) in a homogenuous Poisson process with intensity \(\lambda = 1\). Let \(M(t) = N(t)−\lambda t\) and \(N^* (t) = \lambda t\) so that the Doob-Meyer decomposition of \(N(t)\) is \[N(t) = N^* (t) + M(t).\] Simulate the process \(N(t)\) for \(t \in [0, 25]\) and plot the resulting sample path. Plot also \(N^* (t)\) in the same figure as \(N(t)\). In a seperate figure make a plot of the corresponding sample path of \(M(t)\). Finally, make also plots of the corresponding sample path of \(M^2 (t)\) and \(\lambda t\) (in the same plot) and (in a seperate plot) the sample path of the martingale \(M^2 (t) − \lambda t.\)

Hints: To get some hints for how to solve the problem see the link at the bottom of this page.