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An absorbing state is a state that, once entered, cannot be left. It is characterized by a transition probability of 1 to itself.

Conditions: Transition probability from the state to itself must be exactly 1.; The system eventually enters an absorbing state if reachable.

An absorbing state is a state that, once entered, cannot be left. It is characterized by a transition probability of 1 to itself.

Conditions: Transition probability from the state to itself must be exactly 1.; The system eventually enters an absorbing state if reachable.

The stationary distribution describes the long-run frequency of visiting each state in a Markov chain. It does not mean the system stops moving; rather, it represents a dynamic equilibrium where the probability flows between states balance out.

Conditions: The Markov chain has reached its long-run behavior.; The chain is ergodic (irreducible and aperiodic) for a unique stationary distribution to exist.

The stationary distribution describes the long-run frequency of visiting each state in a Markov chain. It does not mean the system stops moving; rather, it represents a dynamic equilibrium where the probability flows between states balance out.

Conditions: The Markov chain has reached its long-run behavior.; The chain is ergodic (irreducible and aperiodic) for a unique stationary distribution to exist.

Transition probabilities encode the likelihood of staying in or switching between states. In the two-state volatility example, high self-loop probabilities (0.8 for Low, 0.7 for High) mean the system tends to remain in its current state.

Conditions: The market is modeled as a two-state Markov chain.; Self-transition probabilities are high relative to switching probabilities.

Transition probabilities encode the likelihood of staying in or switching between states. In the two-state volatility example, high self-loop probabilities (0.8 for Low, 0.7 for High) mean the system tends to remain in its current state.

Conditions: The market is modeled as a two-state Markov chain.; Self-transition probabilities are high relative to switching probabilities.

In practice, market regimes or states are rarely observed directly. They are hidden and must be inferred from noisy observable data, such as returns.

Conditions: Applies to Hidden Markov Models (HMMs) or latent variable models.; Observable outputs are noisy or smoothed.

In practice, market regimes or states are rarely observed directly. They are hidden and must be inferred from noisy observable data, such as returns.

Conditions: Applies to Hidden Markov Models (HMMs) or latent variable models.; Observable outputs are noisy or smoothed.