Analytical solutions to state transition probabilities in a semi-markov model

Semi-Markov models provide a more realistic framework for modeling and forecasting real-world processes than conventional Markov models. However, their practical use is severely constrained by the lack of analytical tractability. This paper proposes a viable and tractable alternative based on an expanded-state Markov model with a specific topological structure. We show that the proposed semi-Markov model remains analytically tractable when the process consists of two semi-Markovian states, each represented by at least three Markovian sub-states. Closed-form expressions for the state transition probabilities are derived for both discrete- and continuous-time settings. Empirical analysis using long-term data on U.S. business and stock market cycles demonstrates that the proposed model provides a statistically superior fit compared to conventional Markov models, underscoring its practical relevance and effectiveness.