Existence and uniqueness of solutions of stochastic functional differential equations

Abstract

Using a variant of the Euler-Murayama scheme for stochastic functional dierential equations with bounded memory driven by Brownian motion we show that only weak one-sided local Lipschitz (or ’monotonicity’) conditions are sucient 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 stochas