Existence, uniqueness, comparison theorem and stability theorem for unbounded solutions of scalar BSDEs with sub-quadratic generators

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, like |z| α for some α ∈ (1, 2), when the terminal condition satisfies a sub-exponential moment integrability condition like exp µL 2/α * for the conjugate α * of α and a positive parameter µ > µ 0 with a certain value µ 0 , which is clearly weaker than the usual exp(µL) 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.