  This paper is devoted to solving a multidimensional backward stochastic
differential equation (BSDE) with a general random terminal time, which may
take values in [0,+infinity]. The generator g satisfies a stochastic
monotonicity condition in the first unknown variable y and a stochastic
Lipschitz continuity condition in the second unknown variable z, and it can
have a more general growth with respect to y than the classical one stated in
(H5) of Briand et al. [2003]. Without imposing any restriction of finite moment
on the stochastic coefficients, we establish a general existence and uniqueness
result for the adapted solution of the previous BSDE in a proper weighted
L2-space. This result is proved via some innovative ideas and delicate
analytical techniques, and it unifies and strengthens many existing works on
BSDEs with stochastic monotonicity generators, BSDEs with stochastic Lipschitz
generators, and BSDEs with deterministic Lipschitz/monotonicity generators.
Then, a continuous dependence property and a stability theorem for the weighted
L2-solutions are given. We also derive the nonlinear Feynman-Kac formula for
both parabolic and elliptic PDEs in this context.
