Dagstuhl Seminar Proceedings. 04401 - Algorithms and Complexity for Continuous Problems
Doi
Abstract
We consider stochastic differential equations with
Markovian switching (SDEwMS). An SDEwMS is a
stochastic differential equation with drift and
diffusion coefficients depending not only on the
current state of the solution but also on the
current state of a right-continuous Markov chain
taking values in a finite state space.
Consequently, an SDEwMS can be viewed as the
result of a finite number of different scenarios
switching from one to the other according to the
movement of the Markov chain. The generator of the
Markov chain is given by transition probabilities
involving a parameter which controls the intensity
of switching from one state to another. We
construct numerical schemes for the approximation
of SDE\u27swMS and present upper error bounds for
these schemes. Our numerical schemes are based on
a time discretization with constant step-size and
on the values of a discrete Markov chain at the
discretization points. It turns out that for the
Euler scheme a similar upper bound as in the case
of stochastic ordinary differential equations can
be obtained, while for the Milstein scheme there
is a strong connection between the power of the
step-size appearing in the upper bound and the
intensity of the switching
