# Numerical integration of SDE: choice of $dt$ and algorithm

I am working on the following Stochastic Differential Equation (SDE) in the Quantum Mechanics context:

$$dX_{t} = a X_{t} dt + b X_{t} dW$$

where $$X_{t}$$ is my stochastic varible, $$dt$$ is my integration time step, and $$dW = N[0,1] \sqrt{dt}$$ is the Wiener differential.

Due to the "quantumness" of this SDE, it's not possible to obtain an analytical solution, in contrast to the classical case (Geometric Brownian Motion) for which it is available.

So, to solve this equation numerically I have implemented Euler-Maruyama and Milstein methods. However, the choice of the time step $$dt$$ for the integration seems to be arbitrary since I have no comparison with an exact solution. Is there any general rule for the choice of $$dt$$?

I was thinking about study the limitations of the choice of $$dt$$ in the framework of ODEs Euler scheme for the equation without the noise term (for which I have an analytical solution) and use that to choose $$dt$$ for the SDE. Is it feasible?

Moreover I have to change the magnitude of the noise term to be from a perturbative term to the predominant one. I read in another thread that it is not possible to adapt regular ODEs solver like 4th-order Runge-Kutta method to add the noise term $$dW$$ in an Euler/Milstein fashion, but in the case of perturbative noise the deterministic part is predominant over the noise one, and using a better approximation for the deterministic part can allow choosing bigger $$dt$$, saving computational time. Is still wrong to adapt 4th-order Runge-Kutta to the SDE? Is there any other method suitable for small noise?

• For any case of interest there is no exact solution to compare against; otherwise no need for numerical solution. The solution should converge (at a certain rate, given by the order accuracy of the numerical scheme) as dt is reduced; that's the metric for choosing the time step. – Maxim Umansky Sep 10 at 18:43