  We analyse errors of randomized explicit and implicit Euler schemes for
approximate solving of ordinary differential equations (ODEs). We consider
classes of ODEs for which the right-hand side functions satisfy Lipschitz
condition globally or only locally. Moreover, we assume that only inexact
discrete information, corrupted by some noise, about the right-hand side
function is available. Optimality and stability of explicit and implicit
randomized Euler algorithms are also investigated.
