Highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler–Maruyama method

The subject of this paper is the development of discrete-time approximations for solutions of a class of highly nonlinear neutral stochastic differential equations with time-dependent delay. The main contribution is to establish the convergence in probability of the Euler–Maruyama approximate solution without the linear growth condition, that is, under Khasminskii-type conditions. The presence of the delayed argument in the equation, especially in the derivative of the state variable, requires a special treatment and some additional conditions, except the conditions that guarantee the existence and uniqueness of the exact solution. The existence and uniqueness result and the convergence in probability are directly influenced by the properties of the delay function.