Runge–Kutta method (SDE)
Encyclopedia
In mathematics
, the Runge–Kutta method is a technique for the approximate numerical solution
of a stochastic differential equation
. It is a generalization of the Runge–Kutta method
for ordinary differential equation
s to stochastic differential equations.
Consider the Itō diffusion
X satisfying the following Itō stochastic differential equation
with initial condition X0 = x0, where Wt stands for the Wiener process
, and suppose that we wish to solve this SDE on some interval of time [0, T]. Then the Runge–Kutta approximation to the true solution X is the Markov chain
Y defined as follows:
Note that the random variables ΔWn are independent and identically distributed normal random variables with expected value
zero and variance
δ.
This scheme has strong order 1, meaning that the approximation error of the actual solution at a fixed time scales with the time step δ. 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.
The functions a and b 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. Higher-order schemes also exist, but become increasingly complex.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Runge–Kutta method is a technique for the approximate numerical solution
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
of a stochastic differential equation
Stochastic differential equation
A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....
. It is a generalization of the Runge–Kutta method
Runge–Kutta methods
In numerical analysis, the Runge–Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900 by the German mathematicians C. Runge and M.W. Kutta.See the article...
for ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s to stochastic differential equations.
Consider the Itō diffusion
Ito diffusion
In mathematics — specifically, in stochastic analysis — an Itō diffusion is a solution to a specific type of stochastic differential equation. That equation is similar to the Langevin equation, used in Physics to describe the brownian motion of a particle subjected to a potential in a...
X satisfying the following Itō stochastic differential equation
with initial condition X0 = x0, where Wt stands for the Wiener process
Wiener process
In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...
, and suppose that we wish to solve this SDE on some interval of time [0, T]. Then the Runge–Kutta approximation to the true solution X is the Markov chain
Markov chain
A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
Y defined as follows:
- partition the interval [0, T] into N equal subintervals of width δ = T ⁄ N > 0:
- set Y0 = x0;
- recursively define Yn for 1 ≤ n ≤ N by
- where
- and
Note that the random variables ΔWn are independent and identically distributed normal random variables with expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
zero and variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
δ.
This scheme has strong order 1, meaning that the approximation error of the actual solution at a fixed time scales with the time step δ. 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.
The functions a and b 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. Higher-order schemes also exist, but become increasingly complex.