On approximation of solutions of stochastic delay differential equations via randomized Euler scheme

We investigate existence, uniqueness and approximation of solutions to stochastic delay differential equations (SDDEs) under Carathéodory-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 Hölder continuous with respect to the delay variable $z$. We provide a construction of randomized Euler scheme for approximation of solutions of Carathéodory SDDEs, and investigate its upper error bound. Finally, we report results of numerical experiments that confirm our theoretical findings.