When solving certain stochastic control problems in financial or
insurance mathematics, the optimal control policy sometimes turns
out to be of threshold type, meaning that the control depends on the
controlled process in a discontinuous way. The stochastic differential
equations (SDEs) modeling the underlying process then typically have
a discontinuous drift coefficient. This motivates the study of a more
general class of such SDEs.
We prove an existence and uniqueness result based on a certain
novel transformation method by which the drift is "made continuous".
As a consequence the transform becomes useful for the construction
of a numerical method. The resulting scheme is proven to converge
and its convergence speed is estimated. This is the first scheme for
which strong convergence is proven for such a general class of SDEs
with discontinuous drift.
As a next step, the transformation method is used to prove strong
convergence with positive rate of the classical Euler-Maruyama scheme
for this class of SDEs. Furthermore, we give an outlook to a convergence
result for an adaptive Euler scheme.