Doubly reflected BSDEs with quadratic growth and random terminal horizon

In this paper, we study one-dimensional backward stochastic differential equations featuring two reflecting barriers. When the terminal time is not necessarily bounded or finite and the generator $ f(t,y,z) $ exhibits quadratic growth in z, we prove existence and uniqueness of solutions. In the Markovian case, we establish the link with an obstacle problem for quadratic elliptic partial differential equation with Dirichlet boundary conditions.