Existence and uniqueness of solutions of stochastic functional differential equations

Using a variant of the Euler–Maruyama scheme for stochastic functional differential equations with bounded memory driven by Brownian motion we show that only weak one-sided local Lipschitz (or “monotonicity”) conditions are sufficient for local existence and uniqueness of strong solutions. In case of explosion the method yields the maximal solution up to the explosion time. We also provide a weak growth condition which prevents explosions to occur. In an appendix we formulate and prove four lemmas which may be of independent interest: three of them can be viewed as rather general stochastic versions of Gronwall's Lemma, the final one provides tail bounds for Hölder norms of stochastic integrals.