Existence results for doubly reflected backward doubly stochastic differential equations with continuous drivers

We study a class of doubly reflected backward doubly stochastic differential equations with continuous reflecting barriers. The first component of the solution is constrained to remain between a lower obstacle $ L $ and an upper obstacle $ U $. The coefficient $ g $ is assumed to be Lipschitz continuous with respect to $ (y, z) $, with a contraction condition in the $ z $ variable, whereas the driver $ f $ is only assumed to be continuous and of linear growth. Under a semimartingale approximation condition on the upper barrier, we prove the existence of both minimal and maximal adapted solutions. The proof combines the theory of one-barrier reflected BDSDEs with an upper-obstacle penalization procedure. A central point of the construction is that the Lipschitz approximation of the continuous driver and the penalization of the upper barrier are handled by two separate parameters. This separation makes it possible to use the comparison theorem in a rigorous way and avoids the monotonicity difficulty that arises when both parameters are varied simultaneously. The minimal solution is obtained through an increasing sequence of Lipschitz approximations of $ f $, while the maximal solution is obtained through a decreasing sequence. The proof relies on uniform a priori estimates, a uniform control of the upper penalization term, convergence of the martingale integrands, and verification of the two Skorokhod conditions. The examples show that the assumptions are non-empty and include genuinely non-Lipschitz drivers, as well as deterministic and bounded random barriers.