Well-posedness of scalar BSDEs with sub-quadratic generators and related PDEs

Abstract

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, when the terminal condition satisfies a sub-exponential moment integrability condition, which is weaker than the usual exp(μL)-integrability and stronger than Lp(p>1)-integrability. Then, we prove the uniqueness and comparison theorem for the unbounded solutions of the preceding BSDEs under some additional assumptions and establish a general stability result for the unbounded solutions. Finally, we derive the nonlinear Feynman–Kac formula in this context.