Convergence and almost sure polynomial stability of the backward and forward-backward Euler methods for highly nonlinear pantograph stochastic differential equations

In this paper the backward Euler and forward-backward Euler methods for a class of highly nonlinear pantograph stochastic differential equations are considered. In that sense, convergence in probability on finite time intervals is established for the continuous forward-backward Euler solution, under certain nonlinear growth conditions. Under the same conditions, convergence in probability is proved for both discrete forward-backward and backward Euler methods. Additionally, under certain more restrictive conditions, which do not include the linear growth condition on the drift coefficient of the equation, it is proved that these solutions are globally a.s. asymptotically polynomially stable. Numerical examples are provided in order to illustrate theoretical results.