Explicit stochastic Heun update for additive noise

Assume AC and the Brownian filtered-space conventions in Continuous Brownian Ito processes. Consider the scalar additive-noise It^o SDE $d{X}_t=b(X_t)dt+\sigma d{W}_t,$ where $\sigma \in \mathbb{R}$ is constant. Given a step size $h>0$, a grid value $Y_n$, and the Brownian increment $\mathrm{\Delta }{W}_n=W_{t_{n+1}}-W_{t_n}$, the explicit stochastic Heun update is the two-stage rule $Z_n=Y_n+b(Y_n)h+\sigma \mathrm{\Delta }{W}_n,Y_{n+1}=Y_n+\frac{h}{2}(b(Y_n)+b(Z_n))+\sigma \mathrm{\Delta }{W}_n\mathrm{.}$ The predictor $Z_n$ uses the same increment as the corrector. One step evaluates $b$ at $Y_n$ and $Z_n$ and uses one Gaussian increment. The weak order-two guarantee authored in this pair is only for this constant-diffusion scheme; the displayed update omits terms required for the corresponding multiplicative-noise guarantee.

In this definition and its consumers, $C_b^k(\mathbb{R};\mathbb{R})$ means that the function and all its derivatives through order $k$ are continuous and bounded.