Strong convergence of explicit schemes for highly nonlinear stochastic differential equations with Markovian switching

Two projected Euler type schemes are analyzed for stochastic differential equations with Markovian switching whose coefficients are super-linear. Under the polynomial growth condition and the monotone condition, we investigate the convergence in mean square sense of these numerical methods. Besides, we also discuss the convergence rates of these two schemes for highly nonlinear equations (including stochastic differential equations with and without Markovian switching) with small noise. Finally, some numerical experiments are given to verify our theoretical results.