Second-order strong convergence for the bounded-domain SPDE discretisation

Let $U_T=u(T,\cdot)$ be the solution of the initial-boundary value problem at time $T$ for a given Brownian path, and let $\widehat{U}_T$ be its numerical approximation with spatial grid size $h$ and timestep $k\propto h^2$. The norm $\\|\cdot\\|$ is the $L_2$ norm. Strong error conjecture. The error at time $T$ satisfies

$$\sqrt{\mathbb{E}[\\|\widehat{U}_T-U_T\\|^2]}=O(h^2).$$

The corresponding result is known for the initial-value problem on $\mathbb{R}$; the conjecture asserts that introducing the boundary at $x=0$ does not affect the weak or strong error, as supported by the numerical experiments. A finite truncation point $x_{\max}$ would add an exponentially decaying truncation error, and reduced regularity near the boundary can increase the error constant while preserving the asymptotic order.