On the spectral properties and convergence of the bonus-malus Markov chain model

In this paper, we study the bonus-malus model denoted by BM_k (n). It is an irreducible and aperiodic finite Markov chain but it is not reversible in general. We show that if an irreducible, aperiodic finite Markov chain has a transition matrix whose secondary part is represented by a nonnegative, irreducible and periodic matrix, then we can estimate an explicit upper bound of the coefficient of the leading-order term of the convergence bound. We then show that the BM_k (n) model has the above-mentioned periodicity property. We also determine the characteristic polynomial of its transition matrix. By combining these results with a previously studied one, we obtain essentially complete knowledge on the convergence of the BM_k (n) model in terms of its stationary distribution, the order of convergence, and an upper bound of the coefficient of the convergence bound.

کلیدواژه ها:

Bonus-malus system ، Markov chains ، convergence to stationary distribution ، the Perron-Frobenius theorem