We investigate existence, uniqueness and approximation of solutions to stochastic delay differential equations (SDDEs) under Carath\'eodory-type drift coefficients. Moreover, we also assume that both drift $f=f(t,x,z)$ and diffusion $g=g(t,x,z)$ coefficient are Lipschitz continuous with respect to the space variable $x$, but only H\"older continuous with respect to the delay variable $z$. We provide a construction of randomized Euler scheme for approximation of solutions of Carath\'eodory SDDEs, and investigate its upper error bound. Finally, we report results of numerical experiments that confirm our theoretical findings.

翻译：我们研究了在Carathéodory型漂移系数下，随机延迟微分方程（SDDEs）解的存在性、唯一性及逼近问题。此外，我们还假设漂移系数$f=f(t,x,z)$和扩散系数$g=g(t,x,z)$关于空间变量$x$是Lipschitz连续的，但关于延迟变量$z$仅为Hölder连续。针对Carathéodory型随机延迟微分方程解的逼近，我们提出了一种随机化欧拉方案的构造方法，并研究了其误差上界。最后，我们报告了数值实验结果，这些结果验证了我们的理论发现。