The two-dimensional <Fokker-Planck equation> is
$$
\boxed{
\partial_t p
=-a_1\partial_xp-\partial_y(a_2p)
+\partial_{xx}p+\frac12\partial_{yy}(\sigma^2p)
=-\nabla\mathbin\cdot\mathbf J,
}
$$
with <Fokker-Planck probability current>
$$
\boxed{
\mathbf J=
\left(
a_1p-\partial_xp,\,
a_2p-\frac12\partial_y(\sigma^2p)
\right).
}
$$
The point initial condition is
$$
p(x,y,0)=\delta(x)\delta(y),
$$
where $\delta$ is the <Dirac delta function>. The <reflecting boundary condition for a diffusion> is
$$
\boxed{\mathbf J\mathbin\cdot\mathbf n=0
\quad\hbox{on }\partial\Omega,}
$$
with $\mathbf n$ the outward unit normal. This zero-flux condition conserves the integral of the <probability density function> over the square.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 356 / 2 / a / 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 one and <weak order of convergence> one. The reflection enforces the boundary pathwise.

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 356 / 2 / b / Solution

A reflecting boundary confines a diffusion without losing probability. Its forward condition is zero outward <Fokker-Planck probability current>, $J\mathbin\cdot n=0$, and its backward condition is a zero conormal derivative.

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 Incoming links: Reflecting boundary condition for a diffusion