Existence, uniqueness and comparison theorem on unbounded solutions of scalar super-linear BSDEs

Abstract

This paper is devoted to the existence, uniqueness and comparison theorem on unbounded solutions of a scalar backward stochastic differential equation (BSDE) whose generator grows (with respect to both unknown variables y and z) in a super-linear way like |y||ln|y||δ+|z||ln|z||λ for some δ∈[0,1] and λ≥0. Let k be the maximum of δ, λ+1/2 and 2λ. For the following four different ranges of the growth power parameter k: k=1/2, k∈(1/2,1), k=1 and k>1, we give reasonably weakest possible different integrability conditions on the terminal value for ensuring existence and uniqueness of the unbounded solution to the BSDE. In the first two cases, they are stronger than the LlnL-integrability and weaker than any Lp-integrability with p>1; in the third case, the integrability condition is just some Lp-integrability for p>1; and in the last case, the integrability condition is stronger than any Lp-integrability with p>1 and weaker than any exp(Lɛ)-integrability with ɛ∈(0,1). We also establish three comparison theorems, which yield immediately the uniqueness, when either one of generators of both BSDEs is convex (or concave) in both unknown variables (y,z), or satisfies a one-sided Osgood condition in the first unknown variable y and a uniform continuity condition in the second unknown variable z. Finally, we give an application of our results in mathematical finance.