Euler–Maruyama for a constant-coefficient additive-noise equation

The functions $b=1$ and $\sigma =1$ are jointly Borel, globally Lipschitz in state with constant $0$, and satisfy the stated linear-growth bound with a finite $K$. Their time increments vanish, so the required half-Hölder time modulus holds. The deterministic initial value $X_0=0$ is ${\mathcal{F}}_0$-measurable, independent of future Brownian increments, and has every finite moment.

From [F1], induction gives $Y_k=kh+W_{kh}$. Substituting the coefficients into [F2] gives $Y_t=kh+(t-kh)+W_{kh}+W_t-W_{kh}=t+W_t$ on every interval. By [F4], at every fixed time the candidate path $X_t=t+W_t$ satisfies the integral equation. The continuous-version theorem [F6] and equality at all rational times identify these processes indistinguishably; hence [F5] makes this adapted continuous candidate a strong solution. Any strong solution with the same data equals $t+W_t$ directly from the constant-coefficient equation, so this is the exact path.

The initial value $X_0=0$ has every finite moment. Applying [F3] for each finite $p\ge 2$ therefore proves finite moments of the grid, step, and interpolated processes uniformly in $n$ and $t\le 1$. The explicit identity $Y_t=t+W_t$ from step 1.2 also identifies the exact path. AC in [A1] is inherited from the cited interfaces; no choice enters these computations.