Y g S,? (?) a.s. When for each n, ? n = ? a.s. , the result still holds without any assumption on ,
, Moreover, the converse statement holds
, 7) is equivalent to K ? ? ? ? Z ? a.s. Note that this constraint corresponds to the second constraint from (5.2)
, By similar arguments as those used in the proof of Theorem 7.18, it can be shown that the value process (Y t ) is a supersolution of the constrained reflected BSDE from Definition 7.15 with f replaced by g and the constraints (5.2) replaced by the constraint
,
, Y t ) is a supersolution of the above constrained reflected BSDE. Moreover, it is the minimal one
, Note that when ? = 0 and the obstacle is right-continuous, our result gives the existence of a minimal supersolution of the reflected BSDE with driver g, obstacle ? and with non positive jumps
, Moreover, our result provides a dual representation (with non linear expectation) of this minimal supersolution
On the multi-dimensional controller-and-stopper game, SIAM Journal of Control and Optimization, vol.51, pp.1263-1297, 2013. ,
A general Doob-Meyer-Mertens decomposition for g-supermartingale system, Electronic Journal of Probability, vol.21, 2016. ,
Defaultable game options in a hazard process model, International Journal of Stochastic Analysis, 2009. ,
Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers, Nonlinear Differ.Equ. Appl, vol.16, pp.381-420, 2009. ,
Reflected BSDEs with nonpositive jumps, and controller-and-stopper games, Stochastics Processes and their Applications, vol.125, pp.597-633, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-00853988
Risk under Funding Constraints-Part 1 CVA, Pricing. Mathematical Finance, vol.25, issue.1, pp.1-22, 2015. ,
Reflected and Doubly Reflected BSDEs with jumps, Annals of App. Prob, vol.18, issue.5, pp.2041-2069, 2008. ,
Hedging Contingent Claims with Constrained Portfolios, Annals of Applied Prob, 1993. ,
Hedging options for a large investor and forward-backward SDEs, Annals of applied Prob, vol.6, pp.370-398, 1996. ,
Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus, Sém. de Proba. XVI, lect. notes in Mathematics, vol.920, pp.298-313, 1981. ,
, Probabilités et Potentiel, Chap. I-IV, 1975.
BSDEs with default jump, Computation and Combinatorics in Dynamics, Stochastics and Control -The Abel Symposium, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01799335
Generalized Dynkin Games and Doubly reflected BSDEs with jumps, Electronic Journal of Probability, vol.21, issue.64, p.32, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01388022
Mixed Generalized Dynkin Games and Stochastic control in a Markovian framework, Stochastics, vol.89, pp.400-429, 2017. ,
Game options in an imperfect market with default, SIAM Journal on Financial Mathematics, vol.8, pp.532-559, 2017. ,
URL : https://hal.archives-ouvertes.fr/hal-01614758
American options in an imperfect complete market with default, ESAIM Proceedings & Surveys, 2018. ,
URL : https://hal.archives-ouvertes.fr/hal-01614741
Backward Stochastic Differential Equations in Finance, Mathematical Finance, vol.7, issue.1, pp.1-71, 1997. ,
A dynamic maximum principle for the optimization of recursive utilities under constraints, Annals of Applied Probability, vol.11, issue.3, pp.664-693, 2001. ,
Dynamic programming and pricing of contingent claims in an incomplete market, SIAM J. Control and Optimization, vol.33, pp.29-66, 1995. ,
Non-linear Pricing Theory and Backward Stochastic Differential Equations, Lectures Notes in Mathematics, vol.1656, 1996. ,
Optional decomposition and Lagrange multipliers, Finance and Stochastics, vol.2, pp.66-81, 1997. ,
Market volatility and feedback effects from dynamic hedging, Math. Finance, vol.7, pp.351-374, 1997. ,
, Optional martingales, vol.40, pp.435-468, 1981.
Reflected BSDEs when the obstacle is not right-continuous and optimal stopping, the, Annals of Applied Probability, vol.27, issue.5, pp.3153-3188, 2017. ,
URL : https://hal.archives-ouvertes.fr/hal-01141801
, Optimal stopping with f -expectations: the irregular case, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01403616
, Doubly Reflected BSDEs and E f -Dynkin games: beyond the right-continuous case, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01497914
Mixed zero-sum stochastic differential game and American game options, SIAM J. Control Optim, vol.45, issue.2, pp.496-518, 2006. ,
Martingales and arbitrage in multiperiod securities markets, J.Econom. Theory, vol.20, pp.381-408, 1979. ,
Martingale representation theorem in progressively enlarged filtrations Stochastic Processes and their Applic, vol.125, pp.4242-4271, 2015. ,
Mathematical methods for financial markets, 2009. ,
URL : https://hal.archives-ouvertes.fr/hal-00426898
Hedging American contingent claims with constrained portfolios, Finance and Stochastics, vol.2, issue.3, pp.215-258, 1998. ,
, Methods of Mathematical Finance. Applications of Mathematics, vol.39, 1998.
Dynkin games in a general framework, 2013. ,
URL : https://hal.archives-ouvertes.fr/hal-00795370
A remark on default risk models, Adv. Math. Econ, vol.1, pp.69-82, 1999. ,
Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, vol.105, pp.459-479, 1996. ,
The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Ann. Appl. Probab, vol.9, issue.3, pp.904-950, 1999. ,
Optimal Stopping under Adverse Nonlinear Expectation and Related Games, Annals of Applied Probability, vol.25, issue.5, pp.2503-2534, 2015. ,
On feedback effects from Hedging derivatives, Mathematical Finance, issue.8, 1998. ,
Nonlinear expectations, nonlinear evaluations and risk measures, Lecture Notes in Math, pp.165-253, 2004. ,
Stochastic integration and differential equations, 2004. ,
BSDEs with jumps, optimization and applications to dynamic risk measures, Stochastic Processes and Applications, vol.123, pp.3328-3357, 2013. ,
URL : https://hal.archives-ouvertes.fr/hal-00709632
Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps, Stochastic Processes and Applications, vol.124, pp.3031-3054, 2014. ,
URL : https://hal.archives-ouvertes.fr/hal-00780175
Optimal investment in incomplete financial markets, Stochastic Methods in Finance: Lectures given at the C, 2003. ,