Issue |
ESAIM: PS
Volume 20, 2016
|
|
---|---|---|
Page(s) | 178 - 195 | |
DOI | https://doi.org/10.1051/ps/2016011 | |
Published online | 14 July 2016 |
Minimal supersolutions of convex BSDEs under constraints∗
1 Humboldt-Universität zu Berlin, Unter den Linden 6, 10099
Berlin, Germany.
heyne@math.hu-berlin.de; mainberg@math.hu-berlin.de
2 University of Konstanz, Universitätsstr. 10, 78457 Konstanz,
Germany.
kupper@uni-konstanz.de
3 University of Vienna, Faculty of Mathematics,
Oskar-Morgenstern-Platz 1, 1090 Vienna .
ludovic.tangpi@univie.ac.at
Received:
6
October
2014
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
© EDP Sciences, SMAI 2016
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