Markovian Dynamics Associated with d-Orthogonal Polynomials and Generalized Birth and Death Processes in Multiple-Life Insurance

Source abstract

We construct upward skip-free continuous-time Markov chains from finite-band recurrences associated with d-orthogonal polynomial systems. In the alternating-sign chamber, the recurrence yields an order-d generalized birth–death generator with one upward and at most d downward jump channels. In the non-negative chamber, an irreducible finite Hessenberg truncation admits a Perron–Doob normalization and a continuous-time uniformization. We give the stationary distribution explicitly in terms of the left and right Perron vectors and distinguish the asymptotic spectral rate from finite-time norm estimates for non-normal matrices. For Sheffer systems A(t)exB(t), we separate coefficientwise non-negativity, regular d-orthogonality, irreducibility at a fixed truncation size, and irreducibility for every truncation; this corrects the global admissibility conditions in the Sheffer and Appell chambers. For continuously observed paths, we specify the likelihood, identify the Perron-normalization nonidentifiabilities, and report a normalized Monte Carlo calibration. Finally, two recurrence-generated health generators are used directly in a product-state multiple-life valuation, and the dependence of retained-state rates on the number of living health categories is quantified. The numerical study is a simulation-based structural validation and does not claim empirical superiority on insurance data. In the actuarial interpretation, d is the largest permitted one-jump deterioration in the chosen severity ordering and m is the number of living health categories. Both are substantive modeling choices; finite-state truncation, asymmetric jump support, and the absence of an external-data calibration delimit the present practical conclusions.