The higher-dimensional pointwise error expansion conjecture for Zakai equations

Let $d\geq 3$, and let $v$ solve the $d$-dimensional Zakai SPDE. For a spatial grid with spacings $h_{x_i}>0$, let $V_{i_1,\ldots,i_d}^N$ denote the implicit finite-difference approximation at time $T=Nk$ and spatial point $(i_1h_{x_1},\ldots,i_dh_{x_d})$. Assume the implicit finite-difference scheme is stable, and let the timestep and mesh sizes satisfy

$$k\leq \lambda\min\{h_{x_1}^2,h_{x_2}^2,\ldots,h_{x_d}^2\},$$

for an arbitrary fixed $\lambda>0$. The higher-dimensional pointwise error expansion conjecture. Under these assumptions,

$$\sqrt{\mathbb{E}\left[\left|V_{i_1,\ldots,i_d}^N-v(T,x_1,\ldots,x_d)\right|^2\right]}=O(h_{x_1}^2)+O(h_{x_2}^2)+\cdots+O(h_{x_d}^2),$$

where $N=T/k$ is the number of time steps. The result would extend the corresponding two-dimensional mean-square error expansion to dimensions $d\geq 3$; stability for sufficiently small correlations is expected, but the conditions for specific dimensions greater than two are not established in the source.