How to change SDE into a form such that I can apply Euler-Maruyama or Runge Kutta?

Question

How to get the following differential equation

$\dfrac{d^2q(t)}{dt^2} + (\Gamma_0 - \Omega_0 \eta q(t)^2)\dfrac{dq(t)}{dt} + \Omega_0^2 q(t) - \sqrt{\dfrac{2\Gamma_0 k_B T_0}{m}} \dfrac{dW(t)}{dt} = 0$

Where $W(t)$ is the Wiener function and $\Gamma_0, \Omega_0, m, k_B, T_0, \eta$ are constant with respect to time.

into the the form:

$dq(t)=a(q(t))\,{d}t+b(q(t))\,{d}W(t)$

such that I can apply Euler-Maruyama or Runge Kutta to solve this SDE using a computer to get the values of q(t) over time?

Answer 1

Do what is always done to reduce to a system of first order. Set $p=\dot q$ then \begin{align} dq&=p\,dt\\ dp&=-(Γ_0−Ω_0ηq(t)^2)p \,dt - Ω^2_0q(t)\,dt+\sqrt{\frac{2Γ_0k_BT_0}m}\,dW \end{align}

You might also try to apply the Milstein method that has half an order more in the convergence rate.

A derivative-free 2-stage Runge-Kutta variant of Milsteins method is documented in wikipedia resp. the original research in arxiv.org