Representation of solutions to quadratic 2BSDEs with unbounded terminal values

Abstract

Second order backward stochastic differential equations (2BSDEs, for short) are one of useful tools in solving stochastic control problems with model uncertainty. In this paper, we prove a representation formula for quadratic 2BSDEs with an unbounded terminal value under a convex assumption on the generator. Because of the unboundedness of the terminal value, we are unable to use some fine properties of BMO martingales, which are often employed in the literature to deal with bounded solutions to quadratic backward stochastic differential equations. Instead, we utilize the $\theta$-technique. We also prove an existence result under an additional assumption that the terminal value is of uniformly continuous.