= Solution
Let $\xi_{1,n},\xi_{2,n}$ be <independent random variables> with the <standard normal distribution>. The unconstrained <Milstein method> step is
$$
\widetilde X_{n+1}
=X_n+a_1\Delta t+\sqrt{2\Delta t}\,\xi_{1,n},
$$
$$
\widetilde Y_{n+1}
=Y_n+a_2(X_n,Y_n)\Delta t
+\sigma(Y_n)\sqrt{\Delta t}\,\xi_{2,n}
+\frac12\sigma(Y_n)\sigma'(Y_n)\Delta t
\left(\xi_{2,n}^2-1\right).
$$
The $X$ noise is additive, so its Milstein correction vanishes. The diagonal, independent noise fields commute, so no cross iterated stochastic integral is required.
Implement each <reflecting boundary condition for a diffusion> by folding the proposed coordinate back into $[-1,1]$. One formula that also handles multiple overshoots is
$$
R(z)=1-\left|\big((z+1)\bmod4\big)-2\right|.
$$
The complete update is
$$
\boxed{X_{n+1}=R(\widetilde X_{n+1}),\qquad
Y_{n+1}=R(\widetilde Y_{n+1}).}
$$
Under the usual smoothness assumptions, the Milstein discretization has <strong order of convergence> one and <weak order of convergence> one. The reflection enforces the boundary pathwise.
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