The higher-dimensional sparse-grid mixed-moment conjecture for Zakai equations

Let $d\geq 3$, let $h_{x_i}=h_0\cdot 2^{-l_i}$ for $i=1,\ldots,d$, and let $\Delta=\Delta_1\otimes\cdots\otimes\Delta_d$ be the tensor product of the first-order difference operators in the coordinate directions. Write $\Delta P_{\boldsymbol l}^N$ for the corresponding mixed difference of the discrete loss functional, and let $N=T/k$. Assume the implicit finite-difference scheme is stable and

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

for an arbitrary fixed $\lambda>0$. The higher-dimensional sparse-grid mixed-moment conjecture. The first and second moments satisfy

$$\left|\mathbb{E}\left[\Delta P_{\boldsymbol l}^N\right]\right|=O(h_{x_1}^2\cdots h_{x_d}^2),\qquad \mathbb{E}\left[\left|\Delta P_{\boldsymbol l}^N\right|^2\right]=O(h_{x_1}^4\cdots h_{x_d}^4).$$

This conjecture supplies the mixed-difference estimates needed for the sparse combination method in higher dimensions; its validity depends on stability and on the corresponding higher-dimensional discretization analysis.