Markov Chain Models for Multi-State Insurance

Continuous-Time Markov Chains

Multi-state models generalize traditional life contingencies by allowing multiple living states (e.g., healthy, disabled, critically ill) plus death. The model is defined by transition intensities mu_x^{ij} giving the instantaneous rate of moving from state i to state j at age x. The transition probability t_p_x^{ij} is the probability of being in state j at age x+t given the individual is in state i at age x. For a Markov model, transition probabilities depend only on the current state, not on history.

Insurance and Annuity Valuation

Multi-state annuity values sum discounted sojourn probabilities: a-bar_x^{ii} = integral from 0 to infinity of v^t * t_p_x^{ii} dt gives the annuity payable while in state i. Insurance values for transitions use: A-bar_x^{ij} = integral from 0 to infinity of v^t * sum over k of t_p_x^{ik} * mu_{x+t}^{kj} dt. Thiele's equations extend to multi-state models with one equation per state. Kolmogorov's forward and backward equations govern the transition probabilities. Exam LTAM tests two-state, three-state, and disability models extensively.