Drift-implicit Euler–Maruyama with explicit solve-failure status

Assume AC and the Brownian conventions of Locally Lipschitz Itô SDE on an open state domain on $[0,T]$ with $U={\mathbb{R}}^d$. An admissible mathematical input has finite dimensions $d,m\ge 1$, $T>0$, $N\ge 1$ with $h=T\mathrm{/}N\le 1$, an autonomous continuous drift $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$, a finite-valued Borel diffusion $\sigma :{\mathbb{R}}^d\to {\mathbb{R}}^{d\times m}$, and a finite ${\mathcal{F}}_0$-measurable initial state $\xi$. It also has a constant $\mu \ge 0$ such that $⟨x-y,b(x)-b(y)⟩\le -\mu \mathrm{\mid }x-y{\mathrm{\mid }}^2(x,y\in {\mathbb{R}}^d)\mathrm{.}$ This global condition is a precondition on the mathematical input, not a runtime test over all $x,y$. For $h>0$, set $H_h(x)=x-hb(x)$ and let $R_h=H_h^{-1}$, which is defined uniquely on all of ${\mathbb{R}}^d$ by Backward Euler has a globally defined nonexpansive resolvent (contractive when strongly dissipative).

First check that $d,m$ are positive integers, $T$ is finite and positive, $N$ is a positive integer, $0<h=T\mathrm{/}N\le 1$, and $\xi$ is finite. If any of these checks fails, return invalid-input. An exact execution then sets $Y_0=\xi$ and, for $n=0,\dots ,N-1$, computes $u_n=Y_n+\sigma (Y_n)(W_{t_{n+1}}-W_{t_n}),Y_{n+1}=R_h(u_n)\mathrm{.}$ It returns the grid values, the left-endpoint step process ${\bar{Y}}_t=Y_n(t_n\le t<t_{n+1}),{\bar{Y}}_T=Y_N,$ and the continuous interpolation $Y_t=Y_n+(t-t_n)b(Y_{n+1})+\sigma (Y_n)(W_t-W_{t_n}),t_n\le t\le t_{n+1}\mathrm{.}$ This is a pathwise continuous extension of the grid values. At each shared endpoint the interpolation agrees with the grid update.

A finite implementation may instead receive a root-solver interface $S$ and an integer work cap $M\ge 0$. Each call $S(H_h,u_n,M)$ must terminate after at most $M$ declared solver iterations and return either certified-root(z, proof) or no-certificate(last-iterate, residual) (with last-iterate = null if the cap permits no iterate). The supplied exact verifier must terminate and accepts a certificate precisely when it proves that $z\in {\mathbb{R}}^d$ is finite and $H_h(z)=u_n$. At the first call returning no-certificate, or at the first certificate rejected by the verifier, stop and return solve-failure with zero-based failed index $n$, the valid grid prefix through $Y_n$, the step process on $[0,t_n]$, the interpolation on $[0,t_n]$, and the solver diagnostic. For no-certificate, retain its last iterate and residual; for a rejected certificate with candidate $z$, report $z$ and residual $H_h(z)-u_n$. If the diagnostic has no finite candidate, its iterate and residual are null. Do not assign $Y_{n+1}$ or continue. A verified root equals $R_h(u_n)$ by uniqueness. If every call is verified, return success and the exact grid, step process, and interpolation above. The global dissipativity condition remains an admissibility precondition and is not tested by these finite-value checks. Thus a failed finite solve records failure of certification, not absence of a mathematical root. The algorithm makes no convergence or mean-square stability claim.