On the existence of modified equations for stochastic differential equations

In this talk we describe a general framework for deriving

modified equations for stochastic differential equations with respect to

weak convergence. We will start by quickly recapping of how to derive

modified equations in the case of ODEs and describe how these ideas can

be generalized in the case of SDEs. Results will be presented for first

order methods such as the Euler-Maruyama and the Milstein method. In the

case of linear SDEs, using the Gaussianity of the underlying solutions,

we will derive a SDE that the numerical method solves exactly in the

weak sense. Applications of modified equations in the numerical study

of Langevin equations and in the calculation of effective diffusivities

will also be discussed.