Existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay in $L^p(Omega,C_h)$

Abstract

In this paper, we shall consider the existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay in $L^p(Omega,C_h)$ space: $d[x(t) − G(x_t)] = f(t,x_t)dt + g(t,x_t)dB(t),$ where we assume $f : R^+ x L^p(Omega,C_h) → L^p(Omega,R^n), g : R^+ x L^p(Omega,Ch) → L^p(Omega, L(R^m,R^n)) , G : L^p(Omega,C_h) → L^p(Omega,R^n), p > 2$, and B(t) is a given m-dimensional Brownian motion.