  We first establish the existence of an unbounded solution to a backward
stochastic differential equation (BSDE) with generator $g$ allowing a general
growth in the state variable $y$ and a sub-quadratic growth in the state
variable $z$, like $|z|^\alpha$ for some $\alpha\in (1,2)$, when the terminal
condition satisfies a sub-exponential moment integrability condition like
$\exp\left(\mu L^{2/\alpha^*}\right)$ for the conjugate $\alpha^*$ of $\alpha$
and a positive parameter $\mu>\mu_0$ with a certain value $\mu_0$, which is
clearly weaker than the usual $\exp(\mu L)$ integrability and stronger than
$L^p\ (p>1)$ integrability. Then, we prove the uniqueness and comparison
theorem for the unbounded solutions of the preceding BSDEs under the additional
assumptions that the terminal conditions have sub-exponential moments of any
order and the generators are convex or concave in $(y,z)$. Afterwards, we
extend the uniqueness and comparison theorem to the non-convexity and
non-concavity case, and establish a general stability result for the unbounded
solutions of the preceding BSDEs. Finally, with these tools in hands, we derive
the nonlinear Feynman-Kac formula in this context.
