Past exam of the options II course of the University of Oxford / 2007 / M / M5 / a / Solution

A Poisson process of rate $\lambda$ is a counting process starting at zero with independent stationary increments, with $N(t)-N(s)$ Poisson of mean $\lambda (t-s)$ for $t\ge s$, and right-continuous paths with unit jumps. Equivalently the small-interval probabilities are $\lambda h+o(h)$ for one event and $o(h)$ for two or more.

Independence of the two processes makes their joint increments over disjoint intervals independent. On an interval of length $h$, the sum of their independent Poisson increments has probability generating functionso it is Poisson with mean $(\alpha +\beta )h$. Simultaneous arrivals have probability zero, so the sum retains unit jumps. It is therefore a Poisson process of rate $\alpha +\beta$.

The first arrival times are independent exponential variables of rates $\alpha$ and $\beta$. For positive rates,