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 X {\displaystyle X} satisfying the following Itō stochastic differential equation
d X t = a ( X t ) d t + b ( X t ) d W t , {\displaystyle dX_{t}=a(X_{t})\,dt+b(X_{t})\,dW_{t},}
with initial condition X 0 = x 0 {\displaystyle X_{0}=x_{0}} , where W t {\displaystyle W_{t}} stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time [ 0 , T ] {\displaystyle [0,T]} . Then the basic Runge–Kutta approximation to the true solution X {\displaystyle X} is the Markov chain Y {\displaystyle Y} defined as follows:
partition the interval [ 0 , T ] {\displaystyle [0,T]} into N {\displaystyle N} subintervals of width δ = T / N > 0 {\displaystyle \delta =T/N>0} : 0 = τ 0 < τ 1 < ⋯ < τ N = T ; {\displaystyle 0=\tau _{0}<\tau _{1}<\dots <\tau _{N}=T;}
set Y 0 := x 0 {\displaystyle Y_{0}:=x_{0}} ; recursively compute Y n {\displaystyle Y_{n}} for 1 ≤ n ≤ N {\displaystyle 1\leq n\leq N} by Y n + 1 := Y n + a ( Y n ) δ + b ( Y n ) Δ W n + 1 2 ( b ( Υ ^ n ) − b ( Y n ) ) ( ( Δ W n ) 2 − δ ) δ − 1 / 2 , {\displaystyle 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},} where Δ W n = W τ n + 1 − W τ n {\displaystyle \Delta W_{n}=W_{\tau _{n+1}}-W_{\tau _{n}}} and Υ ^ n = Y n + a ( Y n ) δ + b ( Y n ) δ 1 / 2 . {\displaystyle {\hat {\Upsilon }}_{n}=Y_{n}+a(Y_{n})\delta +b(Y_{n})\delta ^{1/2}.}
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