Computing Transition Probabilities: Kolmogorov's Forward and Backward Equations

Given jump rates, how do you compute $p_t(i,j)=\mathbb{P}(X_t=j\mid X_0=i)$ for an arbitrary $t$? This note derives the two Kolmogorov differential equations — obtained by splitting $[0,t+h]$ at either end — and shows both are solved by the matrix exponential $p_t = e^{Qt}$, the direct continuous-time analogue of $p^m$ being a matrix power in discrete time.

Learning objectives. By the end of this note you should be able to: