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Minimal Supersolutions of Convex BSDEs under Constraints

We study supersolutions of a backward stochastic differential equation, the control processes of which are constrained to be continuous semimartingales of the form $dZ = Δdt + ΓdW$. The generator may depend on the decomposition $(Δ,Γ)$ and is assumed to be positive, jointly convex and lower semicontinuous, and to satisfy a superquadratic growth condition in $Δ$ and $Γ$. We prove the existence of a supersolution that is minimal at time zero and derive stability properties of the non-linear operator that maps terminal conditions to the time zero value of this minimal supersolution such as monotone convergence, Fatou's lemma and $L^1$-lower semicontinuity. Furthermore, we provide duality results within the present framework and thereby give conditions for the existence of solutions under constraints.