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

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 both unknown variables y z) in super-linear way like |y||ln|y||?+|z||ln|z||? for some ??[0,1] ??0. Let k be maximum ?, ?+1/2 2?. For following four different ranges growth power parameter k: k=1/2, k?(1/2,1), k=1 k>1, we give reasonably weakest possible integrability conditions terminal value ensuring existence solution BSDE. In first two cases, they are stronger than LlnL-integrability weaker any Lp-integrability with p>1; third case, condition just last p>1 exp(L?)-integrability ??(0,1). We also establish three theorems, which yield immediately uniqueness, when either one generators BSDEs convex (or concave) (y,z), or satisfies one-sided Osgood variable uniform continuity second z. Finally, an application our results mathematical finance.