Euler-Maruyama method for stochastic differential equations with discontinuous drift
Sprache des Vortragstitels:
When solving certain stochastic optimization problems, e.g., in mathematical
finance, the optimal control policy sometimes turns out to be of threshold
type, meaning that the control depends on the state of the controlled process
in a discontinuous fashion. The stochastic
differential equations (SDEs) modeling the underlying process
then typically have discontinuous drift and degenerate diffusion parameter.
This motivates the study of a more general class of such SDEs.
We prove an existence and uniqueness result and present a numerical algorithm,
both based on certain transformations of the state space. The transform
is different from an earlier one by
Zvonkin and Verettennikov in that the drift is not removed entirely by the
transform, but is merely ``made continuous''. As a consequence the transform
becomes computable without the necessity of solving systems of partial
differential equations numerically. The resulting numerical method
is then feasible and proven to converge with strong order $1/2$. This is the first result of
this kind for this class of SDEs.
We will first present the one-dimensional case and subsequently show how the
ideas can be generalized to higher dimensions. There we find a nice
geometrical interpretation of our weakened non-degeneracy condition.