Kolmogorov forward and backward equations
Section of Continuous-time Markov chains, part of Stochastic processes
Kolmogorov forward and backward equations
| Type | Section |
| Taught in | Continuous-time Markov chains |
| Part | Stochastic processes |
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.
The forward equation conditions on the state just before time t. The backward equation conditions on the first jump after time 0. For a finite-state homogeneous chain the unique solution is the matrix exponential
P(t)=e^{tQ}=\sum_{n=0}^{\infty}\frac{(tQ)^n}{n!}.
The forward and backward products agree because this series is a power series in the single matrix Q.
Used in
- Taught in Continuous-time Markov chains
- Also used in Poisson processes
- All 3 questions on Continuous-time Markov chains
See also
Generator · Two-state continuous-time chain · Uniformisation · Jump chain · Holding time