Tamed Euler scheme for a superlinear drift

Assume AC and the Brownian conventions of Locally Lipschitz Itô SDE on an open state domain on $[0,T]$ with $U={\mathbb{R}}^d$. The input is an autonomous drift $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$, diffusion $\sigma :{\mathbb{R}}^d\to {\mathbb{R}}^{d\times m}$, initial state $\xi$, and integer $N\ge 1$ with $h=T\mathrm{/}N\le 1$. Define $t_n=nh$ and $b_h(x)=\frac{b(x)}{1+h\mathrm{\mid }b(x)\mathrm{\mid }}\mathrm{.}$ Return the grid values, step process, continuous interpolation, and status as follows.

1. Set $Y_0=\xi$.
2. For $n=0,\dots ,N-1$, set $\mathrm{\Delta }{W}_n=W_{t_{n+1}}-W_{t_n}$ and $Y_{n+1}=Y_n+h{b}_h(Y_n)+\sigma (Y_n)\mathrm{\Delta }{W}_n\mathrm{.}$
3. Set ${\bar{Y}}_t=Y_n$ on $[t_n,t_{n+1})$, with ${\bar{Y}}_T=Y_N$, and set $Y_t=Y_n+(t-t_n)b_h(Y_n)+\sigma (Y_n)(W_t-W_{t_n}),t\in [t_n,t_{n+1}]\mathrm{.}$ The shared endpoint formulas agree with the grid recursion.

If the dimensions, horizon, step count, or finite-valued coefficient inputs are invalid, return invalid-input without a trajectory. Otherwise return success and the sequence. In exact real arithmetic each update is a finite vector expression, and $\mathrm{\mid }h{b}_h(x)\mathrm{\mid }\le 1$, so the finite recursion has no implicit solve or branch failure. The algorithm by itself asserts no moment, stability, or convergence property.