This paper deals with numerical solutions of nonlinear stiff stochastic differential equations with jump-diffusion and piecewise continuous arguments.By combining compensated split-step methods and balanced methods,a class of compensated split-step balanced(CSSB)methods are suggested for solving the equations.Based on the one-sided Lipschitz condition and local Lipschitz condition,a strong convergence criterion of CSSB methods is derived.It is proved under some suitable conditions that the numerical solutions produced by CSSB methods can preserve the mean-square exponential stability of the corresponding analytical solutions.Several numerical examples are presented to illustrate the obtained theoretical results and the effectiveness of CSSB methods.Moreover,in order to show the computational advantage of CSSB methods,we also give a numerical comparison with the adapted split-step backward Euler methods with or without compensation and tamed explicit methods.