In mathematics of stochastic systems, the Runge窶適utta method is a technique for the approximate numerical solution of a stochastic differential equation. It is a generalisation of the Runge窶適utta 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 Itナ diffusion satisfying the following Itナ stochastic differential equation
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 . See the references for complete and exact statements.
The functions and 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窶葱utta 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
The above describes only one time step. Repeat this time step times in order to integrate the SDE from time to .
The scheme integrates Stratonovich SDEs to provided one sets throughout (instead of choosing ).
Higher-order schemes also exist, but become increasingly complex. Rテカテ殕er 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. [9]
In mathematics of stochastic systems, the Runge窶適utta method is a technique for the approximate numerical solution of a stochastic differential equation. It is a generalisation of the Runge窶適utta 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 Itナ diffusion satisfying the following Itナ stochastic differential equation
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 . See the references for complete and exact statements.
The functions and 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窶葱utta 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
The above describes only one time step. Repeat this time step times in order to integrate the SDE from time to .
The scheme integrates Stratonovich SDEs to provided one sets throughout (instead of choosing ).
Higher-order schemes also exist, but become increasingly complex. Rテカテ殕er 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. [9]