Upper Error Bounds for Approximations of Stochastic Differential Equations with Markovian Switching

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'swMS 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.