Strong uniform Wong--Zakai approximations of L\'evy-driven Marcus SDEs

For a solution $X$ of a L\'evy-driven $d$-dimensional Marcus (canonical) stochastic differential equation, we show that the Wong--Zakai type approximation scheme $X^h$ has a strong convergence of order $\frac12$: for each $T\in [0,\infty)$ and all $x\in\mathbb R^d$ we have $$\mathbf E \sup_{kh\leq T}|X_{kh}(x)-X^h_{kh}(x)|\leq C h^{\frac{1}{2}}(1+|x|),\quad h\to 0.$$ We also determine the rate of the locally uniform strong convergence: for each $N\in(0,\infty)$ and $\varepsilon\in (0,1)$ we have $$\mathbf E\sup_{|x|\leq N}\sup_{kh\leq T}|X_{kh}(x)-X^h_{kh}(x)|\leq C h^{\frac{1-\varepsilon}{4d}},\quad h\to 0.$$