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We investigate existence, uniqueness and approximation of solutions tostochastic delay differential equations (SDDEs) under Carath\'eodory-type driftcoefficients. Moreover, we also assume that both drift $f=f(t,x,z)$ anddiffusion $g=g(t,x,z)$ coefficient are Lipschitz continuous with respect to thespace variable $x$, but only H\"older continuous with respect to the delayvariable $z$. We provide a construction of randomized Euler scheme forapproximation of solutions of Carath\'eodory SDDEs, and investigate its uppererror bound. Finally, we report results of numerical experiments that confirmour theoretical findings.

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