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.
Most basic scheme.
Consider the Itō diffusion formula_1 satisfying the following Itō stochastic differential equation
formula_2
with initial condition formula_3, where formula_4 stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time formula_5. Then the basic Runge–Kutta approximation to the true solution formula_1 is the Markov chain formula_7 defined as follows:
The random variables formula_18 are independent and identically distributed normal random variables with expected value zero and variance formula_19.
This scheme has strong order 1, meaning that the approximation error of the actual solution at a fixed time scales with the time step formula_19. It has also weak order 1, meaning that the error on the statistics of the solution scales with the time step formula_19. See the references for complete and exact statements.
The functions formula_22 and formula_23 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.
Variation of the Improved Euler is flexible.
A newer Runge—Kutta scheme also of strong order 1 straightforwardly reduces to the improved Euler scheme for deterministic ODEs. 
Consider the vector stochastic process formula_24 that satisfies the general Ito SDE
formula_25
where drift formula_26 and volatility formula_27 are sufficiently smooth functions of their arguments.
Given time step formula_28, and given the value estimate formula_29 by formula_30 for time formula_31 via
formula_32
The above describes only one time step.
Repeat this time step formula_37 times in order to integrate the SDE from time formula_38 to formula_39.
The scheme integrates Stratonovich SDEs to formula_40 provided one sets formula_41 throughout (instead of choosing formula_42).
Higher order Runge-Kutta schemes.
Higher-order schemes also exist, but become increasingly complex.
Rößler developed many schemes for Ito SDEs,
whereas Komori developed schemes for Stratonovich SDEs. 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, along with coefficient optimization for improved stability.
References.
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