Weak order two for additive-noise stochastic Heun

Assume AC and the Brownian filtration conventions in Continuous Brownian Ito processes. Consider $d{X}_t=b(X_t)dt+\sigma d{W}_t$ on $[0,T]$, where $b\in C_b^6(\mathbb{R};\mathbb{R})$, $\sigma \in \mathbb{R}$ is constant, and $X_0=x_0\in \mathbb{R}$ is deterministic. For $t\le T$ and $x\in \mathbb{R}$, let $X^{t,x}$ denote the solution of the same SDE started from $x$ at time $t$. Let $\varphi :\mathbb{R}\to \mathbb{R}$ have a bounded classical backward solution $u(t,x)=\mathbb{E}[\varphi (X_T^{t,x})]$ with $u(T,x)=\varphi (x)$, $u_t+Lu=0$, $L=b{\mathrm{\partial }}_x+\frac{1}{2}{\sigma }^2{\mathrm{\partial }}_{xx}$, and all mixed derivatives ${\mathrm{\partial }}_t^r{\mathrm{\partial }}_x^{j}u$ for $0\le r\le 3$ and $0\le j\le 6$ continuous and bounded. On a uniform grid $h=T\mathrm{/}N\le 1$, initialize $Y_0=x_0$ and apply the additive-noise stochastic Heun update in Explicit stochastic Heun update for additive noise. There is a constant $C_T$, independent of $h$, such that $\mid \mathbb{E}\varphi (Y_N)-\mathbb{E}\varphi (X_T)\mid \le C_T{h}^2\mathrm{.}$ This is weak order two, meaning the expectation error is bounded by a constant times $h^2$ for the stated payoff class. The claim is specific to constant additive diffusion; no multiplicative-noise order-two result is asserted.