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TL;DR

A matrix P where \( P_{ij} = P(X_{n+1} = j | X_n = i) \) gives the one-step transition probabilities of a Markov chain. Each row sums to 1. The n-step transition matrix is \( P^n \).

By Valenke Exam Prep Team·Last updated 2026-06-18

Transition Matrix

A matrix P where \( P_{ij} = P(X_{n+1} = j | X_n = i) \) gives the one-step transition probabilities of a Markov chain. Each row sums to 1. The n-step transition matrix is \( P^n \).

Why it matters for interviews

Transition matrices are used in credit rating migration analysis, regime-switching models, and to compute absorption probabilities in random walk problems. Computing powers of transition matrices is a common interview exercise.

Definition and Mathematical Foundation

A matrix P where \( P_{ij} = P(X_{n+1} = j | X_n = i) \) gives the one-step transition probabilities of a Markov chain. Each row sums to 1. The n-step transition matrix is \( P^n \).

Application in Quantitative Finance

Transition matrices are used in credit rating migration analysis, regime-switching models, and to compute absorption probabilities in random walk problems. Computing powers of transition matrices is a common interview exercise.

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Frequently Asked Questions

How do you find the stationary distribution from the transition matrix?▼
Solve \( \pi P = \pi \) with \( \sum \pi_i = 1 \). This is a left eigenvector problem for eigenvalue 1. For small matrices, solve the linear system directly.
How are transition matrices used in credit risk?▼
Rating agencies estimate transition matrices from historical data. The entry \( P_{AA, BBB} \) gives the probability of a downgrade from AA to BBB over one period. Multi-period probabilities use \( P^n \).
What are absorbing states?▼
States that, once entered, cannot be left (\( P_{ii} = 1 \)). In credit models, default is absorbing. The expected time to absorption and absorption probabilities are computed from the fundamental matrix \( (I-Q)^{-1} \).