Numerical Integration of stochastic di erential equations 1 Basic algorithm and relation to Fokker - Planck equations Dipartimento

The aim of this lecture is to study numerical algorithms for the integration of stochastic di erential equations. I will derive an algorithmwhich exactly integrates the SDE, using a generalization of a Taylor series in the presence of stochastic forces. Given the complexity, we will nd that this algorithm, although \exact", it is not particularly fast or, in general, the most convenient in a digital simulation, although it is a useful benchmark to test other algorithms, and I will try to improve it. I will discuss the features of di erent algorithms, both in terms of accuracy in a deterministic sense, and also in statistical terms, i.e. how well the algorithm is able to reproduce, for instance, the correct equilibrium distribution. I will then brie y introduce algorithms to integrate stochastic di erential equations which are driven by correlated noise. The lecture will close with the discussion of a few algorithms for the special case of a two dimensional system in a potential, subject to damping and noise. The interest for this particular case is due to the importance and the interest in algorithms which can integrate, for instance, particles in the liquid state.