Three regimes
Vasicek with a mean level and volatility that move among three regimes.
The short rate follows $dx = \kappa(\theta_{y} - x)\,dt + \sigma_y\,dW$ with $\kappa = 2$, $\theta = (0.20, 0.08, 0.02)$ and $\sigma^2 = (0.04, 0.01, 0.0025)$. The chain need not be reversible. Here its generator is $s\,Q_1$ with
$$Q_1 = \begin{pmatrix} -3 & 2 & 1 \\ 1 & -2 & 1 \\ 0.5 & 1.5 & -2 \end{pmatrix},$$whose stationary distribution is $\pi = (0.208, 0.458, 0.333)$. The averaged model is Vasicek's with the $\pi$-weighted mean level and variance. The table gives the error in the bond price $a_1(1)$, starting in the first regime, after each order, against a 30-digit numerical solution.
| scale | order 0 | order 1 | order 2 | order 3 | order 4 | order 5 | order 6 |
|---|---|---|---|---|---|---|---|
| 4 | 6.2e-3 | 1.8e-4 | 2.1e-5 | 3.7e-6 | 5.1e-7 | 1.0e-7 | 1.6e-8 |
| 8 | 3.1e-3 | 4.2e-5 | 2.4e-6 | 2.2e-7 | 1.4e-8 | 1.4e-9 | 1.1e-10 |
| 16 | 1.6e-3 | 1.0e-5 | 2.9e-7 | 1.3e-8 | 4.2e-10 | 2.2e-11 | 7.9e-13 |
Each doubling of the switching rates divides the order-$n$ error by about $2^{n+1}$. For $n$ regimes the recursion is unchanged: the two-state ratio becomes a vector $w$, and each order applies the group inverse of the generator once. See the engine.