In mathematics of stochastic systems, the Runge–Kutta method is a technique for the approximate numerical solution of a stochastic differential equation.
It is a generalisation of the Runge–Kutta method for ordinary differential equations to stochastic differential equations (SDEs).
Importantly, the method does not involve knowing derivatives of the coefficient functions in the SDEs.
Consider the Itō diffusion
satisfying the following Itō stochastic differential equation
with initial condition
stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time
Then the basic Runge–Kutta approximation to the true solution
is the Markov chain
defined as follows:[1] The random variables
are independent and identically distributed normal random variables with expected value zero and variance
This scheme has strong order 1, meaning that the approximation error of the actual solution at a fixed time scales with the time step
It has also weak order 1, meaning that the error on the statistics of the solution scales with the time step
δ
See the references for complete and exact statements.
can be time-varying without any complication.
The method can be generalized to the case of several coupled equations; the principle is the same but the equations become longer.
A newer Runge—Kutta scheme also of strong order 1 straightforwardly reduces to the improved Euler scheme for deterministic ODEs.
[2] Consider the vector stochastic process
that satisfies the general Ito SDE
are sufficiently smooth functions of their arguments.
Given time step
The above describes only one time step.
Repeat this time step
times in order to integrate the SDE from time
The scheme integrates Stratonovich SDEs to
provided one sets
Higher-order schemes also exist, but become increasingly complex.
Rößler developed many schemes for Ito SDEs,[3][4] whereas Komori developed schemes for Stratonovich SDEs.
[5][6][7] Rackauckas extended these schemes to allow for adaptive-time stepping via Rejection Sampling with Memory (RSwM), resulting in orders of magnitude efficiency increases in practical biological models,[8] along with coefficient optimization for improved stability.