Strong Converge Order of the General One-Step Method for Neutral Stochastic Delay Differential Equations under a Global Monotone Condition

We study the strong convergence of the general one-step method for neutral stochastic delay differential equations with a variable delay. First, we give the notions of C-stability and B-consistency, and then establish a fundamental theorem of strong convergence for the general one-step method solving the nonlinear neutral stochastic delay differential equations, where the corresponding diffusion coefficient with respect to the non-delay variables is highly nonlinear. Then, we construct the split-step backward Euler method which is a special implicit one-step method, and prove that it is C-stable, B-consistent, and strongly convergent of order 1/2. Finally, we give some numerical experiments to support the obtained results.