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

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.