Poisson counts with a switching rate

A Markov-modulated Poisson process, and its distribution to any order.

In a Markov-modulated Poisson process events arrive at rate $\ell_y$, with $\ell = (8, 1)$ per unit time and a symmetric chain switching at rate $\lambda$. The generating function $a_i(t) = \mathbb{E}[z^{N_t}\mid y_0 = i]$ solves $a' = \big(Q + (z-1)\operatorname{diag}\ell\big)a$, so $g_i = (z-1)\ell_i$ is constant and complex. Evaluating the expansion at $z$ on the unit circle and inverting by the discrete Fourier transform gives every probability $\Pr(N_t = k)$ at once.

The averaged model is Poisson with rate $\bar\ell = 4.5$. The corrector carries the overdispersion. The first table gives the moments of $N_1$ starting in the high-rate regime, from the numerical solution.

The second table gives the largest error over all probabilities after each order.

The same calculation applies to claim counts, defaults in a portfolio, and arrivals of any kind whose rate follows a hidden regime.