Milstein

class brainpy.integrators.sde.Milstein(f, g, dt=None, name=None, show_code=False, var_type=None, intg_type=None, wiener_type=None, state_delays=None)

Milstein method for Ito or Stratonovich integrals.

The Milstein scheme represents the order 1.0 strong Taylor scheme. For the Ito integral,

\[\begin{split}\begin{aligned} &Y_{n+1}=Y_{n}+f_{n} h+g_{n} \Delta W_{n}+\frac{1}{2} g_{n} g_{n}^{\prime}\left[\left(\Delta W_{n}\right)^{2}-h\right] \\ &\Delta W_{n}=\left[W_{t+h}-W_{t}\right] \sim \sqrt{h} \mathcal{N}(0,1) \end{aligned}\end{split}\]

where \(g_{n}^{\prime}=\frac{d g\left(Y_{n}\right)}{d Y_{n}}\) is the first derivative of \(g_n\).

For the Stratonovich integral, the Milstein method is given by

\[\begin{split}\begin{aligned} &Y_{n+1}=Y_{n}+f_{n} h+g_{n} \Delta W_{n}+\frac{1}{2} g_{n} g_{n}^{\prime}\left(\Delta W_{n}\right)^{2} \\ &\Delta W_{n}=\left[W_{t+h}-W_{t}\right] \sim \sqrt{h} \mathcal{N}(0,1) \end{aligned}\end{split}\]