# Runge–Kutta method (SDE)

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 generalization of the Runge–Kutta method for ordinary differential equations to stochastic differential equations (SDEs). Importantly, the method does not involve knowing derivatives the coefficient functions in the SDEs.

## Most basic scheme

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

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

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 basic Runge–Kutta approximation to the true solution $X$ is the Markov chain $Y$ defined as follows:[1]

• partition the interval $[0,T]$ into $N$ subintervals of width $\delta=T/N>0$:
$0 = \tau_{0} < \tau_{1} < \dots < \tau_{N} = T;$
• set $Y_0:=x_0$;
• recursively compute $Y_n$ for $1\leq n\leq N$ by
$Y_{n + 1} := Y_{n} + a(Y_{n}) \delta + b(Y_{n}) \Delta W_{n} + \frac{1}{2} \left( b(\hat{\Upsilon}_{n}) - b(Y_{n}) \right) \left( (\Delta W_{n})^{2} - \delta \right) \delta^{-1/2},$

where $\Delta W_{n} = W_{\tau_{n + 1}} - W_{\tau_{n}}$ and $\hat{\Upsilon}_{n} = Y_{n} + a(Y_n) \delta + b(Y_{n}) \delta^{1/2}.$ The random variables $\Delta W_{n}$ are independent and identically distributed normal random variables with expected value zero and variance $\delta$.

This scheme has strong order 1, meaning that the approximation error of the actual solution at a fixed time scales with the time step $\delta$. It has also weak order 1, meaning that the error on the statistics of the solution scales with the time step $\delta$. 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.

## Variation of the Improved Euler is flexible

A newer Runge—Kutta scheme also of strong order 1 straightforwardly reduces to the Improved Euler scheme for deterministic ODEs. [2] Consider the vector stochastic process $\vec X(t)\in \mathbb R^n$ that satisfies the general Ito SDE

$d\vec X=\vec a(t,\vec X)\,dt+\vec b(t,\vec X)\,dW,$

where drift $\vec a$ and volatility $\vec b$ are sufficiently smooth functions of their arguments. Given time step $h$, and given the value $\vec X(t_k)=\vec X_k$, estimate $\vec X(t_{k+1})$ by $\vec X_{k+1}$ for time $t_{k+1}=t_k+h$ via

$\begin{array}{rl} &\vec K_1=h\vec a(t_k,\vec X_k)+(\Delta W_k-S_k\sqrt h)\vec b(t_k,\vec X_k), \\&\vec K_2=h\vec a(t_{k+1},\vec X_k+\vec K_1)+(\Delta W_k+S_k\sqrt h)\vec b(t_{k+1},\vec X_k+\vec K_1), \\&\vec X_{k+1}=\vec X_k+\frac12(\vec K_1+\vec K_2), \end{array}$
• where $\Delta W_k=\sqrt hZ_k$ for normal random $Z_k\sim N(0,1)$;
• and where $S_k=\pm1$, each alternative chosen with probability $1/2$.

The above describes only one time step. Repeat this time step $(t_m-t_0)/h$ times in order to integrate the SDE from time $t=t_0$ to $t=t_m$.

The scheme integrates Stratonovich SDEs to $O(h)$ provided one sets $S_k=0$ throughout (instead of choosing $\pm 1$).

## Higher order Runge-Kutta schemes

Higher-order schemes also exist, but become increasingly complex. Rossler developed many schemes for Ito SDEs. [3] [4] Whereas Komori developed schemes for Stratonovich SDEs. [5] [6] [7]

## References

1. ^ P. E. Kloeden and E. Platen. Numerical solution of stochastic differential equations, volume 23 of Applications of Mathematics. Springer--Verlag, 1992.
2. ^ A. J. Roberts. Modify the improved Euler scheme to integrate stochastic differential equations. [1], Oct 2012.
3. ^ Rößler, A. (2009). "Second Order Runge–Kutta Methods for Itô Stochastic Differential Equations". SIAM Journal on Numerical Analysis 47 (3): 1713. doi:10.1137/060673308. edit
4. ^ Rößler, A. (2010). "Runge–Kutta Methods for the Strong Approximation of Solutions of Stochastic Differential Equations". SIAM Journal on Numerical Analysis 48 (3): 922. doi:10.1137/09076636X. edit
5. ^ Komori, Y. (2007). "Multi-colored rooted tree analysis of the weak order conditions of a stochastic Runge–Kutta family". Applied Numerical Mathematics 57 (2): 147. doi:10.1016/j.apnum.2006.02.002. edit
6. ^ Komori, Y. (2007). "Weak order stochastic Runge–Kutta methods for commutative stochastic differential equations". Journal of Computational and Applied Mathematics 203: 57. doi:10.1016/j.cam.2006.03.010. edit
7. ^ Komori, Y. (2007). "Weak second-order stochastic Runge–Kutta methods for non-commutative stochastic differential equations". Journal of Computational and Applied Mathematics 206: 158. doi:10.1016/j.cam.2006.06.006. edit