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TL;DR

A stochastic process where the future state depends only on the current state, not the history: \( P(X_{n+1}|X_n, X_{n-1}, \ldots) = P(X_{n+1}|X_n) \). This is the Markov property.

By Valenke Exam Prep Team·Last updated 2026-06-03

Markov Chain

A stochastic process where the future state depends only on the current state, not the history: \( P(X_{n+1}|X_n, X_{n-1}, \ldots) = P(X_{n+1}|X_n) \). This is the Markov property.

Why it matters for interviews

Markov chains model credit rating transitions, regime-switching models, and order book dynamics. Interview questions often involve computing absorption probabilities, stationary distributions, or expected hitting times.

Definition and Mathematical Foundation

A stochastic process where the future state depends only on the current state, not the history: \( P(X_{n+1}|X_n, X_{n-1}, \ldots) = P(X_{n+1}|X_n) \). This is the Markov property.

Application in Quantitative Finance

Markov chains model credit rating transitions, regime-switching models, and order book dynamics. Interview questions often involve computing absorption probabilities, stationary distributions, or expected hitting times.

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Frequently Asked Questions

What is a stationary distribution?
A probability distribution \( \pi \) satisfying \( \pi P = \pi \) where P is the transition matrix. If the chain is irreducible and aperiodic, it converges to a unique stationary distribution.
What is the difference between ergodic and absorbing Markov chains?
Ergodic chains have a unique stationary distribution and every state is reachable from every other. Absorbing chains have states that, once entered, cannot be left. Different analysis techniques apply to each.
How are Markov chains used in credit risk?
Credit rating agencies model rating transitions (AAA to AA, AA to A, etc.) as a Markov chain. The transition matrix estimates probabilities of upgrade, downgrade, and default over fixed horizons.