Runge–Kutta method (SDE)

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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 stochastic differential equation

$\frac{\mathrm{d} X_{t}}{\mathrm{d} t} = a(X_{t}) \, \mathrm{d} t + b(X_{t}) \, \mathrm{d} W_{t},$

with initial condition $X 0 = x 0$ , 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 $δ > 0$ :

$0 = \tau_{0} < \tau_{1} < \dots < \tau_{N} = T$ and $\delta = \frac{T}{N};$

set $Y 0 = x 0$ ;

recursively define $Y n$ for $1 \leq n \leq N$ by

$Y_{n + 1} = Y_{n} + a(Y_{n}) \delta + b(Y_{n}) \Delta W_{n} + \frac{1}{2} \left( b(\hat{\Upsilon}_{n}) - b(Y_{n}) \right) \left( (\Delta W_{n})^{2} - \delta \right) \delta^{-1/2},$