Existence and uniqueness of solutions of stochastic functional differential equations

We provide sufficient conditions on the coefficients of a stochastic functional differential equation with bounded memory driven by Brownian motion which guarantee existence and uniqueness of a maximal local and global strong solution for each initial condition. Our results extend those of previous works. For local existence and uniqueness, we only require the coefficients to be continuous and to satisfy a one-sided local Lipschitz (or ’monotonicity’) condition. 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 – which we call Dereich-Lemma – provides tail bounds for Hölder norms of stochastic integrals. AMS 2000 Subject Classification: Primary 34K50; secondary 34K05, 60H10, 60H20.