Chapman–Kolmogorov equations
Topic of Markov chains, part of Probability · Also in Continuous-time Markov chains
Chapman–Kolmogorov equations
| Type | Topic |
| Taught in | Markov chains · Continuous-time Markov chains |
| Part | Probability · Stochastic processes |
For a time-homogeneous Markov chain the transition probabilities compose: P^{m+n}=P^mP^n in discrete time, and P(s+t)=P(s)P(t) in continuous time. To be at j after the longer interval, the chain must be somewhere at the intermediate time.
In Markov chains
From Markov chains
Each row of P is a probability distribution: the entries are non-negative and sum to 1. If the row vector \boldsymbol\pi^{(0)} is the distribution of X_0, the distribution of X_n is \boldsymbol\pi^{(0)} P^n. The Chapman–Kolmogorov equations are the matrix product P^{m+n}=P^m P^n: the way to be at j after m+n steps is to be somewhere after m steps and then travel for n more.
In Continuous-time Markov chains
Let P(t)=\bigl(p_{ij}(t)\bigr) with p_{ij}(t)=\mathbb{P}\bigl(X(t)=j\mid X(0)=i\bigr). The Chapman–Kolmogorov identity is the continuous-time product P(s+t)=P(s)P(t): to be at j at time s+t, the chain is somewhere at time s and then moves for time t. Differentiating at t=0 from the right, and using the short-time rules, gives Kolmogorov’s equations
P'(t)=P(t)Q \quad\text{(forward)}, \qquad P'(t)=QP(t) \quad\text{(backward)}, \qquad P(0)=I.
Used in
See also
Finite-state Markov chain · Limiting distribution · Markov property · Simulating a Markov chain