Euler–Maruyama Scheme and Coefficient Conditions — Examples

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.

The diffusion $\sigma =1$ is globally Lipschitz and has linear growth. For the drift, $\mathrm{\mid }b(x)\mathrm{\mid }\mathrm{/}(1+\mathrm{\mid }x\mathrm{\mid })=\mathrm{\mid }x{\mathrm{\mid }}^3\mathrm{/}(1+\mathrm{\mid }x\mathrm{\mid })\to \mathrm{\infty }$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$, so no finite linear-growth constant exists. Also $\mathrm{\mid }b(x)-b(0)\mathrm{\mid }\mathrm{/}\mathrm{\mid }x\mathrm{\mid }=\mathrm{\mid }x{\mathrm{\mid }}^2\to \mathrm{\infty }$, so no finite global Lipschitz constant exists.

Since both required drift conditions fail, the uniform-moment theorem [F2] cannot be invoked for this coefficient pair. The conclusion is only that its stated hypotheses are absent; no contrary convergence or moment claim follows. AC in [A1] is inherited and no choice is used.