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 with initial condition denotes the Wiener process, and suppose that we wish to solve this SDE on some interval of time. Then the Milstein approximation is the Markov chain defined as follows:

• denotes the derivative are independent and identically distributed normal random variables with expected value zero and variance will yield a better approximation.

) this method is equivalent to the Euler–Maruyama method.

which is superior to the Euler–Maruyama method.

Intuitive derivation

For this derivation, we will only look at geometric Brownian motion. Using Itō's lemma

The numerical solution is presented in the graphic for three different trajectories.

Computer implementation

The following Python

See also

• Euler–Maruyama method

References

Further reading