In mathematics, the Milstein method is a technique for the approximate numerical solution of a stochastic differential equation. It is named after Grigori Milstein who first published it in 1974.
Description Consider the autonomous Itō stochastic differential equation:
d X t = a ( X t ) d t + b ( X t ) d W t {\displaystyle \mathrm {d} X_{t}=a(X_{t})\,\mathrm {d} t+b(X_{t})\,\mathrm {d} W_{t}}
with initial condition X 0 = x 0 {\displaystyle X_{0}=x_{0}} , where W t {\displaystyle W_{t}} denotes the Wiener process, and suppose that we wish to solve this SDE on some interval of time [ 0 , T ] {\displaystyle [0,T]} . Then the Milstein approximation to the true solution X {\displaystyle X} is the Markov chain Y {\displaystyle Y} defined as follows:
Partition the interval [ 0 , T ] {\displaystyle [0,T]} into N {\displaystyle N} equal subintervals of width Δ t > 0 {\displaystyle \Delta t>0} : 0 = τ 0 < τ 1 < ⋯ < τ N = T with τ n := n Δ t and Δ t = T N {\displaystyle 0=\tau _{0}<\tau _{1}<\dots <\tau _{N}=T{\text{ with }}\tau _{n}:=n\Delta t{\text{ and }}\Delta t={\frac {T}{N}}}
Set Y 0 = x 0 ; {\displaystyle Y_{0}=x_{0};}
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