Minimal supersolutions of convex BSDEs under constraints∗
1 Humboldt-Universität zu Berlin, Unter den Linden 6, 10099
2 University of Konstanz, Universitätsstr. 10, 78457 Konstanz, Germany.
3 University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Vienna .
Revised: 2 March 2016
Accepted: 18 April 2016
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 L1-lower semicontinuity. Furthermore, we provide duality results within the present framework and thereby give conditions for the existence of solutions under constraints.
Mathematics Subject Classification: 60H20 / 60H30
Key words: Supersolutions of backward stochastic differential equations / gamma constraints / minimality under constraints / duality
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