page_title: Milstein method
section_title: Intuitive derivation
hierarchical_section_title: Milstein method / Intuitive derivation
caption_attribution_description: English: Numerical simulation of a stochastic differential equation via the Milstein method. it is plotted three different trajectories, the drift coefficient is twice the diffusion.
context_page_description: In mathematics, the Milstein method is a technique for the approximate numerical solution of a stochastic differential equation. It is named after Grigori N. Milstein who first published the method in 1974.
context_section_description: For this derivation, we will only look at geometric Brownian motion (GBM), the stochastic differential equation of which is given by:
  with real constants  and . Using Itō's lemma we get:
  Thus, the solution to the GBM SDE is:
  where
  See numerical solution is presented above for three different trajectories.
caption: Numerical solution for the stochastic differential equation just presented, the drift is twice the diffusion coefficient.
exif: {"Image Orientation": "Horizontal (normal)"}