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In this paper pantograph stochastic differential equations are considered under nonlinear growth conditions. The existence, uniqueness and almost sure polynomial stability of solution is established. The whole consideration is affected by the presence of the unbounded delay in the arguments of coefficients of the equation of that type. 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.

Pantograph stochastic differential equations; nonlinear growth conditions; Euler-Maruyama approximation; almost sure polynomial stability; convergence in probability