A generalized reflected backward doubly SDEs with stochastic lipschitz coefficients and discontinuous barrier

In this paper, we establish existence and uniqueness results for generalized backward doubly stochastic differential equations (GRBDSDEs) with a right upper semi-continuous and left-limited (r.u.s.c.l.l.) barrier and stochastic Lipschitz coefficients. The analysis is based on Mertens' decomposition, tools from optimal stopping theory, and a Picard iteration scheme. In addition, we prove a comparison theorem and use it to establish the existence of a minimal solution when the coefficients satisfy a stochastic linear growth condition.