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
Keywords: Supersolutions of backward stochastic differential equations, gamma constraints, minimality under constraints, duality
Heyne, Gregor 1 ; Kupper, Michael 2 ; Mainberger, Christoph 1 ; Tangpi, Ludovic 3
@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},
year = {2016},
publisher = {EDP Sciences},
volume = {20},
doi = {10.1051/ps/2016011},
mrnumber = {3528623},
zbl = {1356.60090},
language = {en},
url = {https://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 - https://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 https://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
and , BSDE with quadratic growth and unbounded terminal value. Probab. Theory Relat. Fields 136 (2006) 604–618. | MR | Zbl | DOI
, and , Optimal Replication of Contingent Claims under Portfolio Constraints. Rev. Financ. Stud. 11 (1998) 59–79. | DOI
, and , The multi-dimensional super-replication problem under gamma constraints. Ann. Inst. Henri Poincaré, Analyse Non Linéaire 22 (2005) 633–666. | MR | Zbl | Numdam | DOI
, , and , Second Order Backward Stochastic Differential Equations and Fully Non-Linear Parabolic PDEs. Commun. Pure Appl. Math. 60 (2007) 1081–1110. | MR | Zbl | DOI
and , Hedging contingent claims with constrained portfolios. Ann. Appl. Probab. 3 (1993) 652–681. | MR | Zbl | DOI
, and , Backward SDEs with Superquadratic Growth. Probab. Theory Relat. Fields 150 (2011) 145–192. | MR | Zbl | DOI
and , A General Version of the Fundamental Theorem of Asset Pricing. Math. Annalen 300 (1994) 463–520. | MR | Zbl | DOI
F. Delbaen and W. Schachermayer, A Compactness Principle for Bounded Sequences of Martingales with Applications. Proc. of the Seminar of Stochastic Analysis, Random Fields and Applications. Progress in Probability. Birkhäuser (1996) 133–173. | MR | Zbl
, and , Minimal Supersolutions of Convex BSDEs. Ann. Probab. 41 (2013) 3973–4001. | MR | Zbl | DOI
, , and , Dual Representation of Minimal Supersolutions of Convex BSDEs. Ann. Inst. Henri Poincaré 52 (2016) 868–887. | MR | Zbl | DOI
, and , Backward Stochastic Differential Equations in Finance. Math. Finance 7 (1997) 1–71. | MR | Zbl | DOI
G. Heyne, Essays on Minimal Supersolutions of BSDEs and on Cross Hedging in Incomplete Markets. Ph.D. thesis, Humboldt-Universität zu Berlin (2012).
, and , Minimal Supersolutions of BSDEs with Lower Semicontinuous Generators. Ann. Inst. Henri Poincaré, Probabilités Statistiques 50 (2014) 524–538. | MR | Zbl | Numdam
and , Arbitrage in securities markets with short-sales constraints. Math. Finance 5 (1995) 197–232. | MR | Zbl | DOI
I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics. Springer (2004). | Zbl
N. Kazamaki, Continuous Exponential Martingales and BMO. Vol. 1579 of Lect. Notes Math.. Springer-Verlag, Berlin (1994). ISBN 3-540-58042-5. | MR | Zbl
, Backward Stochastic Differential Equations and Partial Differential Equations with Quadratic Growth. Ann. Probab. 28 (2000) 558–602. | MR | Zbl | DOI
and , Adapted Solution of a Backward Stochastic Differential Equation. System Control Lett. 14 (1990) 55–61. | MR | Zbl | DOI
, Monotonic Limit Theorem of BSDE and Nonlinear Decomposition Theorem of Doob–Meyer’s Type. Probab. Theory Relat. Fields 113 (1999) 473–499. | MR | Zbl | DOI
P.E. Protter, Stochastic Integration and Differential Equations, 2nd edition. Springer (2005). | MR
and , Superreplication under Gamma constraints. SIAM J. Control Optim. 39 (2000) 73–96. | MR | Zbl | DOI
, and , Dual Formulation of Second Order Target Problems. Ann. Appl. Probab. 23 (2013) 308–347. | MR | Zbl | DOI
Cité par Sources :
