### Citations

5386 | Convex Analysis - Rockafellar - 1970 |

468 |
Adapted solution of a backward stochastic differential equation
- Pardoux, Peng
- 1990
(Show Context)
Citation Context ...hu-berlin.de ∗MATHEON project E.11 † Funding: Berlin Mathematical School, phase II scholarship PAPER INFO AMS CLASSIFICATION: 60H20; 60H30 1. Introduction Since their introduction by Pardoux and Peng =-=[15]-=-, nonlinear Backward Stochastic Differential Equations (BSDEs) have found numerous applications in mathematical finance. For instance, they are used to constructively describe the optimal solution of ... |

349 |
Stochastic Finance: An Introduction in Discrete Time
- Föllmer, Schied
- 2004
(Show Context)
Citation Context ...revious section can be seen as a risk measure. In fact, when the generator does not depend on y, the functional ρ defined by ρ(X) := E0(−X) is a convex risk measure in the sense of Föllmer and Schied =-=[10]-=-, and u(X) := −E0(−X) defines a monetary utility function. If the generator g does depend on y and satisfies (DEC), then ρ is instead a cash-subadditive risk measure as defined in [8]. In particular, ... |

250 | Brownian motion and stochastic calculus, volume 113 - Karatzas, Shreve - 1991 |

202 |
Dynamic programming and pricing of contingent claims in an incomplete market
- Karoui, Quenez
- 1995
(Show Context)
Citation Context ...e that this representation is obtained in the static case. Our representation results can be seen as extensions of the dual representation of the minimal superreplicating cost of El Karoui and Quenez =-=[7]-=- to the case where we allow for a nonlinear cost function in the dynamics of the wealth process. The second theme of this work is to give conditions based on convex duality under which a dynamic cash-... |

175 |
Backward stochastic differential equations and partial differential equations with quadratic growth
- Kobylanski
(Show Context)
Citation Context ...du ≤ C1 T∫ 0 |Yu| du+ C2 T∫ 0 ‖Zu‖ du. (3.9) It is known that if X is bounded and f is Lipschitz, then the solution (Y, Z) of the BSDE is such that Y is bounded and ∫ Z dW is in BMO, see for instance =-=[14]-=- and [1, Proposition 7.3]1. Equation (3.9) and BMO ⊆ Hp for all 1 ≤ p <∞, see [13], together with Hölder’s inequality imply E dQq̄n dP T∫ 0 ∣∣f∗u(β̄nu , q̄nu)∣∣ du 2 ≤ C̃1E [( dQq̄ n dP )2... |

92 |
Utility maximization in incomplete markets.
- Hu, Imkeller, et al.
- 2005
(Show Context)
Citation Context ...ions (BSDEs) have found numerous applications in mathematical finance. For instance, they are used to constructively describe the optimal solution of some utility maximization problems, see Hu et al. =-=[11]-=-. Through the g-expectations of Peng [16], BSDEs offer a framework to study nonlinear expectations and time consistent dynamic risk measures as described by Rosazza Gianin [19] and Delbaen et al. [4].... |

75 |
Continuous Exponential Martingales and BMO.
- Kazamaki
- 1994
(Show Context)
Citation Context ...f is Lipschitz, then the solution (Y, Z) of the BSDE is such that Y is bounded and ∫ Z dW is in BMO, see for instance [14] and [1, Proposition 7.3]1. Equation (3.9) and BMO ⊆ Hp for all 1 ≤ p <∞, see =-=[13]-=-, together with Hölder’s inequality imply E dQq̄n dP T∫ 0 ∣∣f∗u(β̄nu , q̄nu)∣∣ du 2 ≤ C̃1E [( dQq̄ n dP )2] + C̃2E [( dQq̄ n dP )4]1/2 E T∫ 0 ‖Zu‖2 du 2 1/2 ≤ C, where C is a ... |

75 | Integral functionals, normal integrands and measurable selections. In: Nonlinear Operators and the Calculus of Variations - Rockafellar - 1976 |

56 | Pricing, hedging and optimally designing derivatives via minimization of risk measures,
- Barrieu, Karoui
- 2005
(Show Context)
Citation Context ...f solutions of BSDE with quadratic growth in the control variable, linear growth in the value process and bounded terminal condition are by now well understood, see for instance Barrieu and El Karoui =-=[1]-=- and El Karoui and Ravanelli [8]. In this work we give the dual representation of the minimal supersolution functional of a BSDE in the framework of Drapeau et al. [6]. The H1-L∞ duality turns out to ... |

37 |
Backward SDE and related g-expectation, Backward stochastic differential equations
- Peng
- 1997
(Show Context)
Citation Context ...ions in mathematical finance. For instance, they are used to constructively describe the optimal solution of some utility maximization problems, see Hu et al. [11]. Through the g-expectations of Peng =-=[16]-=-, BSDEs offer a framework to study nonlinear expectations and time consistent dynamic risk measures as described by Rosazza Gianin [19] and Delbaen et al. [4]. Mainly driven by its financial applicati... |

32 | Representation of the penalty term of dynamic concave utilities.
- Delbaen, Peng, et al.
- 2010
(Show Context)
Citation Context ...[11]. Through the g-expectations of Peng [16], BSDEs offer a framework to study nonlinear expectations and time consistent dynamic risk measures as described by Rosazza Gianin [19] and Delbaen et al. =-=[4]-=-. Mainly driven by its financial applications, the study of BSDEs has been extended in various ways beyond the question of existence and uniqueness of solutions. Many authors have been interested in q... |

32 |
Risk measures via g-expectations
- E
(Show Context)
Citation Context ...problems, see Hu et al. [11]. Through the g-expectations of Peng [16], BSDEs offer a framework to study nonlinear expectations and time consistent dynamic risk measures as described by Rosazza Gianin =-=[19]-=- and Delbaen et al. [4]. Mainly driven by its financial applications, the study of BSDEs has been extended in various ways beyond the question of existence and uniqueness of solutions. Many authors ha... |

28 |
Dynamic Risk Measures: Time Consistency and Risk Measures from
- Bion-Nadal
- 2008
(Show Context)
Citation Context ...on technique appears already in the work of Delbaen et al. [4] where it is used to construct a sequence of µ-dominated risk measures. Furthermore, prior to us Barrieu and El Karoui [1] and Bion-Nadal =-=[2]-=- already used the BMO-martingale theory in the study of financial risk measures, but in different settings from ours. Using standard convex duality arguments such as the Fenchel-Moreau theorem and the... |

27 | A Compactness Principle for Bounded Sequences of Martingales with Applications - Delbaen, Schachermayer - 1996 |

25 | Backward SDEs with superquadratic growth . Probab. Theory and Related Fields,
- Delbaen, Hu, et al.
- 2011
(Show Context)
Citation Context ...hich a dynamic cash-subadditive risk measure with a given representation can be seen as the solution, or the minimal supersolution of a BSDE. The cash-additive case has been studied by Delbaen et al. =-=[5]-=-. Their results are based on m-stability of the dual space, some supermartingale property and Dood-Meyer decomposition of the risk measure. We shall show that in the cash-subadditive case, discounting... |

21 | Cash subadditive risk measures and interest rate ambiguity.
- Karoui, Ravanelli
- 2009
(Show Context)
Citation Context ...au et al. [6] show existence of the minimal supersolution of a BSDE. They study the properties of minimal supersolutions and give the link to cash-subadditive risk measures of El Karoui and Ravanelli =-=[8]-=-. Our main objectives are, on the one hand, to derive a dual representation of minimal supersolutions of BSDEs, and, on the other hand, to study conditions under which an operator satisfying such a re... |

3 | Minimal Supersolutions of Convex BSDEs
- Drapeau, Heyne, et al.
(Show Context)
Citation Context ... for an overview. The subject of this paper is to study BSDEs by convex duality theory. Deviating from the usual quadratic growth or Lipschitz assumptions on the generator of the BSDE, Drapeau et al. =-=[6]-=- show existence of the minimal supersolution of a BSDE. They study the properties of minimal supersolutions and give the link to cash-subadditive risk measures of El Karoui and Ravanelli [8]. Our main... |