Fractional delay and anticipated backward doubly stochastic differential equations with integral-Lipschitz coefficients

This paper addresses delay and anticipated backward doubly stochastic differential equations driven by fractional Brownian motion (fractional delay and anticipated BDSDEs) with Hurst parameter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {H∈(1/2,1)} . In these equations, the generator at time t can depend not only on the past and present but also on future solutions. We establish the existence and uniqueness of solutions in the cases of both Lipschitz and integral-Lipschitz coefficients. The stochastic integral used throughout the paper is of the divergence type.