Optimal stopping with \(f\)-expectations: the irregular case. (English) Zbl 1471.60055
Summary: We consider the optimal stopping problem with non-linear \(f\)-expectation (induced by a BSDE) without making any regularity assumptions on the payoff process \(\xi\) and in the case of a general filtration. We show that the value family can be aggregated by an optional process \(Y\). We characterize the process \(Y\) as the \(\mathcal{E}^f\)-Snell envelope of \(\xi \). We also establish an infinitesimal characterization of the value process \(Y\) in terms of a Reflected BSDE with \(\xi\) as the obstacle. To do this, we first establish some useful properties of irregular RBSDEs, in particular an existence and uniqueness result and a comparison theorem.
MSC:
| 60G40 | Stopping times; optimal stopping problems; gambling theory |
| 60G65 | Nonlinear processes (e.g., \(G\)-Brownian motion, \(G\)-Lévy processes) |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
Keywords:
reflected backward stochastic differential equation; non-linear optimal stopping; \( \mathcal{E}^f\)-Mertens decomposition; general filtration; American option; Tanaka-type formulaReferences:
| [1] | Bayraktar, E.; Karatzas, I.; Yao, S., Optimal stopping for dynamic convex risk measures, Illinois J. Math., 54, 3, 1025-1067 (2010) · Zbl 1259.60042 |
| [2] | Bayraktar, E.; Yao, S., Optimal stopping for non-linear expectations, Stochastic Process. Appl., 121, 2, 185-211 (2011), and 212-264 · Zbl 1221.60059 |
| [3] | Bouchard, B.; Possamaï, D.; Tan, X., A general doob-meyer-mertens decomposition for \(g\)-supermartingale system, Electron. J. Probab., 21, paper 36, 21 (2016) · Zbl 1343.60050 |
| [4] | Crépey, S.; Matoussi, A., Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison, Ann. Appl. Probab., 18, 5, 2041-2069 (2008) · Zbl 1158.60021 |
| [5] | Dellacherie, C.; Lenglart, E., (Sur Des Problèmes de Régularisation, de Recollement Et D’Interpolation En Théorie Des Processus, Sém. de Proba. XVI. Sur Des Problèmes de Régularisation, de Recollement Et D’Interpolation En Théorie Des Processus, Sém. de Proba. XVI, lect. notes in Mathematics, vol. 920 (1981), Springer-Verlag), 298-313 · Zbl 0538.60035 |
| [6] | Dellacherie, C.; Meyer, P.-A., Probabilité Et Potentiel, Chap. I-IV (1975), Nouvelle édition: Nouvelle édition Hermann |
| [7] | Dellacherie, C.; Meyer, P.-A., Probabilités Et Potentiel, ThÉorie Des Martingales, Chap. V-VIII (1980), Nouvelle édition: Nouvelle édition Hermann · Zbl 0464.60001 |
| [8] | Dumitrescu, R.; Quenez, M.-C.; Sulem, A., Generalized dynkin games and doubly reflected BSDEs with jumps, E.J.P, 21, paper 64, 32 (2016) · Zbl 1351.93170 |
| [9] | Dumitrescu, R.; Quenez, M.-C.; Sulem, A., Game options in an imperfect market with default, SIAM J. Financial Math., 8, 532-559 (2017) · Zbl 1381.93103 |
| [10] | Dumitrescu, R.; Quenez, M.-C.; Sulem, A., Mixed generalized dynkin game and stochastic control in a Markovian framework, Stochastics, 89, 1, 400-429 (2017) · Zbl 1361.60054 |
| [11] | Dumitrescu, R.; Quenez, M.-C.; Sulem, A., American options in an imperfect complete market with default, (ESAIM Proceedings & Surveys, SMAI 2017-8e Biennale Française Des Mathématiques Appliqués Et Industrielles, Vol. 64 (2018)), 93-110 · Zbl 1419.91612 |
| [12] | El Karoui, N., Les aspects probabilistes du Contrôle stochastique, (École d’été de Probabilités de Saint-Flour IX-1979 Lect. Notes in Math, Vol. 876 (1981)), 73-238, MR0637469 · Zbl 0472.60002 |
| [13] | El Karoui, N.; Kapoudjian, C.; Pardoux, E.; Peng, S.; Quenez, M.-C., Reflected solutions of backward sde’s and related obstacle problems for PDE’s, Ann. Probab., 25, 2, 702-737 (1997) · Zbl 0899.60047 |
| [14] | El Karoui, N.; Quenez, M.-C., (Runggaldier, W., Non-Linear Pricing Theory and Backward Stochastic Differential Equations. Non-Linear Pricing Theory and Backward Stochastic Differential Equations, Lect. Notes in Mathematics, 1656 (1997), Springer) · Zbl 0904.90010 |
| [15] | ElKaroui, N.; Pardoux, E.; Peng, S., Reflected backward SDEs and american options, Numer. Methods Financ., 13, 215-231 (1997) · Zbl 0898.90033 |
| [16] | Essaky, H., Reflected backward stochastic differential equation with jumps and RCLL obstacle, Bull. Sci. Math., 132, 690-710 (2008) · Zbl 1157.60057 |
| [17] | Gal’chouk, L. I., Optional martingales, Math. USSR Sb., 40, 4, 435-468 (1981) · Zbl 0472.60048 |
| [18] | Grigorova, M.; Imkeller, P.; Offen, E.; Ouknine, Y.; Quenez, M.-C., Reflected BSDEs when the obstacle is not right-continuous and optimal stopping, Ann. Appl. Probab., 27, 5, 3153-3188 (2017) · Zbl 1379.60045 |
| [19] | Grigorova, M.; Imkeller, P.; Ouknine, Y.; Quenez, M.-C., Doubly reflected BSDEs and \(\mathcal{E}^f\)-dynkin games: beyond the right-continuous case, Electron. J. Probab., 23, paper 122, 38 (2018) · Zbl 1406.60060 |
| [20] | Grigorova, M.; Quenez, M.-C., Optimal stopping and a non-zero-sum dynkin game in discrete time with risk measures induced by BSDEs, Stochastics, 89, 1, 259-279 (2017) · Zbl 1410.91133 |
| [21] | Hamadène, S., Reflected BSDE’s with discontinuous barrier and application, Stoch. Stoch. Rep., 74, 3-4, 571-596 (2002) · Zbl 1015.60057 |
| [22] | Hamadène, S.; Ouknine, Y., Reflected backward SDEs with general jumps, Teor. Veroyatn. Primen., 60, 2, 357-376 (2015) · Zbl 1341.60054 |
| [23] | Jacod, J., Calcul Stochastique et Problèmes de Martingales (1979), Springer · Zbl 0414.60053 |
| [24] | Jacod, J.; Shiryaev, A. N., (Limit Theorems for Stochastic Processes. Limit Theorems for Stochastic Processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288 (2003), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1018.60002 |
| [25] | Karatzas, I.; Shreve, S. E., Methods of Mathematical Finance, Applications of Mathematics (New York), Vol. 39 (1998), Springer: Springer New York · Zbl 0941.91032 |
| [26] | Klimsiak, T.; Rzymowski, M.; Słomiński, L., Reflected BSDEs with regulated trajectories, Stochastic Process. Appl., 129, 4, 1153-1184 (2019) · Zbl 1488.60146 |
| [27] | M. Kobylanski, M.-C. Quenez, Erratum: Optimal stopping time problem in a general framework, available at https://hal.archives-ouvertes.fr/hal-01328196, 2016. |
| [28] | Kobylanski, M.; Quenez, M.-C., Optimal stopping time problem in a general framework, Electron. J. Probab., 17, 1-28 (2012) · Zbl 1405.60055 |
| [29] | Kobylanski, M.; Quenez, M.-C.; Roger de Campagnolle, M., Dynkin games in a general framework, Stochastics, 86, 2, 304-329 (2014) · Zbl 1298.60050 |
| [30] | Lenglart, E., (Tribus de Meyer Et Théorie Des Processus, Séminaire de Probabilités de Strasbourg XIV 1978/79. Tribus de Meyer Et Théorie Des Processus, Séminaire de Probabilités de Strasbourg XIV 1978/79, Lecture Notes in Mathematics, 784 (1980)), 500-546 · Zbl 0427.60045 |
| [31] | Maingueneau, M. A., (Temps d’arrêt Optimaux et Théorie Générale, Séminaire de Probabilités, XII de Strasbourg, 1976/77, 457-467. Temps d’arrêt Optimaux et Théorie Générale, Séminaire de Probabilités, XII de Strasbourg, 1976/77, 457-467, Lecture Notes in Math, vol. 649 (1977), Springer: Springer Berlin), 457-467 · Zbl 0376.60050 |
| [32] | Protter, P. E., Stochastic Integration and Differential Equations (Stochastic Modelling and Applied Probability) (2005), Springer Verlag |
| [33] | Quenez, M.-C.; Sulem, A., SDES with jumps, optimization and applications to dynamic risk measures, Stochastic Process. Appl., 123, 0-29 (2013) |
| [34] | Quenez, M.-C.; Sulem, A., Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps, Stoch. Process. Appl., 124, 9, 3031-3054 (2014) · Zbl 1293.93783 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
