1 Milstein Scheme
References: [[higham-kloeden]] §17.4.
The Milstein Scheme is a methodology for simulating continuous time stochastic processes with dynamics given by some SDE, improving upon the Euler-Maruyama Method.
improves upon the Euler-Maruyama Method approach by considering this Taylor expansion, grouping terms based on powers of \(\Delta t\) and removing all terms with powers \(>1\). The general form of the Milstein method is given by \[
\begin{align}
X_{n+1} = X_{n}+\Delta t f(t_{n}, X_{n}) + \Delta W_{n}g(t_{n},X_{n})+L^1g(t_{n}, X(t_{n}))\int_{t_{n}}^{t_{n+1}}{\int_{t_{n}}^{s}{}~d{W(u)}}~d{W(s)},
\end{align}
\] where \(L_{1}=g \frac{\partial}{\partial x}\) and \(\int_{t_{n}}^{t_{n+1}}{\int_{t_{n}}^{t}{}~d{W(s)}}~d{W(t)}= \frac{1}{2}((\Delta W_{n})^2 - \Delta t)\). Again returning to our simple BSM model example we can write our approximation as \[
S_{t+\Delta t}=S_{t}\left\{ 1+\sigma(\Delta t)^{\frac{1}{2}}\xi+\Delta t \left[ r+ \frac{1}{2}\left( \sigma^2(\xi^2-1) \right) \right] \right\}.\tag{11}
\] This is again very easy to simulate with a for loop:
## Milstein's Algorithm
S <- S0
for(i in 2:length(t)){
xi <- rnorm(1,0,1)
S[i] <- S[i-1]*(1+sigma*h^(1/2)*xi +
h*(r+(1/2)*(sigma^2*(xi^2 - 1))))
}For both the Euler-Maruyama method and the Milstein method it seems natural to suggest that the approximation improves as \(N \to \infty\) and to test this presumption we must consider the two standard approached for measuring the error \(X_{n}\to X(t_{n})\), leading to the concepts of weak and strong convergence. This will also allow us to more concretely quantify the improvement over the Euler-Maruyama method that the Milstein method provides.