Pre-absorption dynamics in multistate run processes: exact distributions and actuarial applications

Abstract

This paper provides a complete analytical characterization of the joint distribution of state counts before run-based absorption in multistate discrete-time processes. For sequences of independent and identically distributed (i.i.d.) random variables, we derive the exact joint distribution, revealing an elegant geometric–multinomial structure with positive dependence. We then generalize this framework to Markov-dependent processes using matrix-analytic methods, providing a unified approach for both memoryless and memory-based systems. A comparative numerical study, motivated by the classification of lognormal insurance claims, demonstrates the utility of the analysis. It reveals how process memory, such as claim contagion, fundamentally alters pre-absorption statistics and risk profiles compared to the simpler i.i.d. baseline. These results provide a rigorous foundation for developing early-warning signals in dynamic risk management, allowing insurers to design adaptive reinsurance strategies and proactively manage solvency capital.