Randomized response

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 equations to stochastic differential equations.

Consider the Itō diffusion X satisfying the following Itō stochastic differential equation

$\mathrm{d}{X}_t=a(X_t)\mathrm{d}t+b(X_t)\mathrm{d}{W}_t,$

with initial condition X₀ = x₀, where W_t 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:

• partition the interval [0, T] into N equal subintervals of width δ = T ⁄ N > 0:

$0={\tau }_0<{\tau }_1<\dots <{\tau }_N=T;$

• set Y₀ = x₀;

• recursively define Y_n for 1 ≤ n ≤ N by

$Y_{n+1}=Y_n+a(Y_n)\delta +b(Y_n)\mathrm{\Delta }{W}_n+\frac{1}{2}(b({\widehat{\mathrm{{\rm Y}}}}_n)-b(Y_n))((\mathrm{\Delta }{W}_n{)}^2-\delta ){\delta }^{-1/2},$

where

$\mathrm{\Delta }{W}_n=W_{{\tau }_{n+1}}-W_{{\tau }_n}$

and

${\widehat{\mathrm{{\rm Y}}}}_n=Y_n+a(Y_n)\delta +b(Y_n){\delta }^{1/2}.$

Note that the random variables ΔW_n 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.

References

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