  The chemical birth-death process, whose chemical master equation (CME) is
exactly solvable, is a paradigmatic toy problem often used to get intuition for
how stochasticity affects chemical kinetics. In a certain limit, it can be
approximated by an Ornstein-Uhlenbeck-like process which is also exactly
solvable. In this paper, we use this system to showcase eight qualitatively
different ways to exactly solve continuous stochastic systems: (i) integrating
the stochastic differential equation; (ii) computing the characteristic
function; (iii) eigenfunction expansion; (iv) using ladder operators; (v) the
Martin-Siggia-Rose-Janssen-De Dominicis path integral; (vi) the Onsager-Machlup
path integral; (vii) semiclassically approximating the Onsager-Machlup path
integral; and (viii) approximating the solution to the corresponding CME.
