Autonomous Euler–Maruyama scheme and interpolation

Assume AC. Fix finite integers $d,m\ge 1$, $T>0$, and $N\ge 1$ such that $h=T\mathrm{/}N\le 1$. Work on a filtered probability space whose filtration is right-continuous and contains every $P$-null subset of $\mathcal{F}$ in ${\mathcal{F}}_0$. Let $W$ be an adapted standard $m$-dimensional Brownian motion whose increments after time $s$ are independent of ${\mathcal{F}}_s$. Let $X_0$ be finite, ${\mathcal{F}}_0$-measurable, and independent of $\sigma (W_t-W_0:0\le t\le T)$. Let $f:{\mathbb{R}}^d\to {\mathbb{R}}^d$ and $g:{\mathbb{R}}^d\to {\mathbb{R}}^{d\times m}$ be $C^1$. Assume finite constants $\mu ,c,D\ge 0$ and an integer $q\ge 1$ satisfy, for all $x,y\in {\mathbb{R}}^d$, $⟨x-y,f(x)-f(y)⟩\le \mu \mathrm{\mid }x-y{\mathrm{\mid }}^2,\mathrm{\parallel }g(x)-g(y){\mathrm{\parallel }}_F^2\le c\mathrm{\mid }x-y{\mathrm{\mid }}^2,$ $\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^2\le D(1+\mathrm{\mid }x{\mathrm{\mid }}^q+\mathrm{\mid }y{\mathrm{\mid }}^q)\mathrm{\mid }x-y{\mathrm{\mid }}^2\mathrm{.}$ Here $\mathrm{\mid }\cdot \mathrm{\mid }$ is the Euclidean norm and $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_F$ is the Frobenius norm.

Set $t_k=kh$ and construct the grid approximation recursively by $Y_0=X_0,Y_{k+1}=Y_k+f(Y_k)h+g(Y_k)(W_{t_{k+1}}-W_{t_k}),0\le k<N\mathrm{.}$ The output also includes the left-endpoint step process ${\bar{Y}}_t=Y_k(t_k\le t<t_{k+1}),{\bar{Y}}_T=Y_N,$ and the continuous interpolation $Y_t=Y_k+f(Y_k)(t-t_k)+g(Y_k)(W_t-W_{t_k})(t_k\le t\le t_{k+1})\mathrm{.}$ The recursion makes the two interpolation formulas agree at every shared endpoint. The finite recursion is defined pathwise: $C^1$ coefficients are finite-valued, each $Y_k$ is finite almost surely by induction, and Brownian paths are continuous. This algorithm defines no moment or convergence claim.