In the book "Introduction to Stochastic Differential Equations" by Thomas C. Gard, they talk about higher-order runge-kutta type schemes. The SDE in question is a general Ito SDE of the form:
$$dX = f(t,X)dt + g(t,X)dW$$
In discrete time, they claim the analagous version is ($h = \Delta t_n$):
$$X_n = X_{n-1}+ \frac{1}{6}\left\lbrace \left[F_0 + 2F_1+2F_2+F_3 \right]h+\left[G_0 + 2G_1 + 2G_2 +G_3 \right]\Delta W_n \right\rbrace$$
where:
$F_0 = f(t_{n-1},X_{n-1})$
$F_1 = f(t_{n-1}+\frac{1}{2}h, X_{n-1}+\frac{1}{2}F_0h + \frac{1}{2}G_0\Delta W_n)$
$F_2 = f(t_{n-1}+\frac{1}{2}h, X_{n-1}+\frac{1}{2}F_1h + \frac{1}{2}G_1\Delta W_n)$
$F_3 = f(t_{n}, X_{n-1}+F_2h + \frac{1}{2}G_2\Delta W_n)$
"and the $G_i$'s are defined as $g$ evaluated at the same points".
So two questions about this:
1) Does that statement mean that:
$G_0 = g(t_{n-1},X_{n-1})$
$G_1 = g(t_{n-1}+\frac{1}{2}h, X_{n-1}+\frac{1}{2}F_0h + \frac{1}{2}G_0\Delta W_n)$
and so forth?
2) If I was interested (to save computational time) in using a 3rd order Runge-Kutta method, could I assume the analagous form for SDE is:
$$X_n = X_{n-1}+ \frac{1}{6}\left\lbrace \left[F_0 + 4F_1+F_2 \right]h+\left[G_0 + 4G_1 + G_2 \right]\Delta W_n \right\rbrace$$ with modified $F_0, F_1, F_2, G_0, G_1, G_2$ so that it matches the ODE input of the 3rd order Runge-Kutta?