Existence, uniqueness, almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the Euler–Maruyama approximation

In the first part of this paper the existence, uniqueness and almost sure polynomial stability of solutions for pantograph stochastic differential equations are considered, under nonlinear growth conditions. The results are obtained using the idea from (Mao and Rassias, 2005) [10], where stochastic differential equation with constant delay are considered. However, the presence of the unbounded delay in stochastic pantograph differential equations required certain modification of that idea. Moreover, the convergence in probability of the appropriate Euler–Maruyama solution is proved under the same nonlinear growth conditions. Adding the linear growth condition, we show that the almost sure polynomial stability of the Euler–Maruyama solution implies the almost sure polynomial stability of the exact solution. This part of the paper represents the extension of the idea from (Wu et al., 2010) [17]. The main novelty in this part of the paper is also related to the treatment of the unbounded delay in pantograph stochastic differential equations.