In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in
the state variable. We deal with the case of a fixed terminal time, as well
as the case of random terminal time. The need for this type of extension
of the classical existence and uniqueness results comes from the desire to
provide a probabilistic representation of the solutions of semilinear partial
differential equations in the spirit of a nonlinear Feynman-Kac formula.
Indeed, in many applications of interest, the nonlinearity is polynomial,
e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.