We study supersolutions of a backward stochastic differential equation, the control processes of which are constrained to be continuous semimartingales of the form . The generator may depend on the decomposition and is assumed to be positive, jointly convex and lower semicontinuous, and to satisfy a superquadratic growth condition in and . 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 -lower semicontinuity. Furthermore, we provide duality results within the present framework and thereby give conditions for the existence of solutions under constraints.
Accepté le :
DOI : 10.1051/ps/2016011
Mots-clés : Supersolutions of backward stochastic differential equations, gamma constraints, minimality under constraints, duality
@article{PS_2016__20__178_0,
author = {Heyne, Gregor and Kupper, Michael and Mainberger, Christoph and Tangpi, Ludovic},
title = {Minimal supersolutions of convex {BSDEs} under constraints},
journal = {ESAIM: Probability and Statistics},
pages = {178--195},
publisher = {EDP-Sciences},
volume = {20},
year = {2016},
doi = {10.1051/ps/2016011},
mrnumber = {3528623},
zbl = {1356.60090},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2016011/}
}
TY - JOUR AU - Heyne, Gregor AU - Kupper, Michael AU - Mainberger, Christoph AU - Tangpi, Ludovic TI - Minimal supersolutions of convex BSDEs under constraints JO - ESAIM: Probability and Statistics PY - 2016 SP - 178 EP - 195 VL - 20 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2016011/ DO - 10.1051/ps/2016011 LA - en ID - PS_2016__20__178_0 ER -
%0 Journal Article %A Heyne, Gregor %A Kupper, Michael %A Mainberger, Christoph %A Tangpi, Ludovic %T Minimal supersolutions of convex BSDEs under constraints %J ESAIM: Probability and Statistics %D 2016 %P 178-195 %V 20 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2016011/ %R 10.1051/ps/2016011 %G en %F PS_2016__20__178_0
Heyne, Gregor; Kupper, Michael; Mainberger, Christoph; Tangpi, Ludovic. Minimal supersolutions of convex BSDEs under constraints. ESAIM: Probability and Statistics, Tome 20 (2016), pp. 178-195. doi : 10.1051/ps/2016011. http://www.numdam.org/articles/10.1051/ps/2016011/
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