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

2019/09/22 by Fan, Shengjun, Hu, Ying
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1909.10081

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>μ₀ with a certain value μ₀, 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.

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