The authors investigate existence, uniqueness and continuation of solutions for stochastic functional differential equations driven by Brownian motion in which the coefficients map
is the underlying probability space and
is the maximal delay. Under suitable conditions like adaptedness and local Lipschitz conditions, they establish local existence and uniqueness of solutions. Due to the particular set-up (in
), maximal solutions are defined on a deterministic
time interval. In addition, the authors provide sufficient conditions for global existence in terms of Lyapunov functions.