Meixner-Weighted Banded Markov Chains for Bonus–Malus Experience Rating Under Mixed-Poisson Risk Heterogeneity

Abstract

Classical bonus–malus systems are often calibrated under a homogeneous Poisson claim-count model, although motor portfolios commonly exhibit unobserved heterogeneity and overdispersion. We develop a finite Meixner-weighted d-band Markov-chain framework for experience rating. Gamma mixing of an annual Poisson intensity yields a negative-binomial count law whose probability mass function is the normalized Meixner weight. A claim-free year produces one bonus movement, whereas a claim year produces a malus movement of depth at most d. The resulting transition matrix is row-stochastic and lower Hessenberg, with one upper band and at most d lower bands. After a positive diagonal normalization, the associated finite polynomial segment satisfies a (d+2)-term recurrence; a compatible nondegenerate infinite extension is d-orthogonal by the generalized Favard theorem. We emphasize that this induced recurrence is not, without an additional identification, the specific Meixner-type d-orthogonal family studied in the special-functions literature. The count model is coupled with a severity distribution in a compound aggregate-loss framework, thereby separating stationary class relativities from monetary pure premiums. A six-class illustration calibrated from published Saudi claim-count moments compares d=1, d=3, and d=5 specifications with a Poisson benchmark. Exact stationary calculations are checked by Monte Carlo simulation, transient cohort analysis, mixing-time calculations, and a parameter-uncertainty experiment. At the selected calibration, the stationary and elasticity measures for d=3 and d=5 are close. This is a calibration-specific numerical observation rather than evidence that one value of d is universally optimal or actuarially superior.