Segment convergence for super-linear stochastic functional differential equations by the truncated Euler-Maruyama method

Most existing literature focuses on pointwise convergence (i.e., convergence at a fixed time point) of numerical solutions for Stochastic functional differential equations (SFDEs). In contrast, this paper investigates the strong segment convergence (i.e., the strong order of convergence of the numerical segment process). For SFDEs with super-linear drift and diffusion coefficients, we employ the explicit truncated Euler-Maruyama (EM) scheme. First, we establish the uniform moment boundedness of the truncated EM solution over a finite time interval. Second, we derive the $L^2$-error estimate between the continuous numerical segment and the step numerical segment. Finally, we prove the strong convergence order of the numerical segment generated by the truncated EM. The results can be used to analyze invariant measures and ergodicity of numerical segment, and have important applications in practical problems such as path-dependent financial options. We also provide a numerical example to support the theoretical results.