Quadratic BSDEs with jumps: Related nonlinear expectations

Abstract

In this paper, we continue the study of quadratic backward SDEs with jumps, that is to say for which the generator has quadratic growth in the variables [Formula: see text], started in our accompanying paper [12]. Relying on the existence and uniqueness result of [12], we define the corresponding [Formula: see text]-expectations and study some of their properties. We obtain in particular a nonlinear Doob–Meyer decomposition for [Formula: see text]-submartingales and a downcrossing inequality which implies their regularity in time. As a consequence of these results, we also obtain a converse comparison theorem for our class of BSDEs. Finally, we provide a dual representation for the corresponding dynamic risk measures, and study the properties of their inf-convolution, giving several explicit examples.