×

On the strict value of the non-linear optimal stopping problem. (English) Zbl 1471.60056

Summary: We address the non-linear strict value problem in the case of a general filtration and a completely irregular pay-off process \((\xi_t)\). While the value process \((V_t)\) of the non-linear problem is only right-uppersemicontinuous, we show that the strict value process \((V^+_t)\) is necessarily right-continuous. Moreover, the strict value process \((V_t^+)\) coincides with the process of right-limits \((V_{t+})\) of the value process. As an auxiliary result, we obtain that a strong non-linear \(f\)-supermartingale is right-continuous if and only if it is right-continuous along stopping times in conditional \(f\)-expectation.

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60G07 General theory of stochastic processes
91G80 Financial applications of other theories
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] Bayraktar, E., Karatzas, I. and Yao, S.: Optimal Stopping for Dynamic Convex Risk Measures, Illinois Journal of Mathematics, 54(3), (2010), 1025-1067. · Zbl 1259.60042
[2] Bayraktar, E. and Yao, S.: Optimal stopping for Non-linear Expectations, Stochastic Processes and Their Applications 121(2), (2011), 185-211 and 212-264. · Zbl 1221.60059
[3] Bouchard, B., Possamaï, D. and Tan, X.: A general Doob-Meyer-Mertens decomposition for \(g\)-supermartingale system, Electronic Journal of Probability 21, paper no. 36, (2016), 21 pages. · Zbl 1343.60050
[4] Dellacherie, C. and Lenglart, E.: Sur des problèmes de régularisation, de recollement et d’interpolation en théorie des processus, Sém. de Proba. XVI, lect. notes in Mathematics, 920, (1981), 298-313, Springer-Verlag. · Zbl 0455.60032
[5] Dellacherie, C. and Meyer, P.-A.: Probabilités et Potentiel, Théorie des Martingales, Chap. V-VIII. Nouvelle édition, Hermann, 1980.
[6] El Karoui, N.: Les aspects probabilistes du contrôle stochastique. École d’été de Probabilités de Saint-Flour IX-1979 Lect. Notes in Math. 876, (1981), 73-238.
[7] El Karoui, N. and Quenez, M.-C.: Non-linear Pricing Theory and Backward Stochastic Differential Equations, Lect. Notes in Mathematics 1656, Ed. W. Runggaldier, Springer, 1996.
[8] Grigorova, M., Imkeller, P., Offen, E., Ouknine, Y. and Quenez, M.-C.: Reflected BSDEs when the obstacle is not right-continuous and optimal stopping, The Annals of Applied Probability, 27(5), (2017), 3153-3188. · Zbl 1379.60045
[9] Grigorova M., Imkeller, P., Ouknine, Y. and Quenez, M.-C.: Optimal stopping with \(f\)-expectations: The irregular case, Stochastic Processes and Their Applications, 130(3), (2020), 1258-1288, https://doi.org/10.1016/j.spa.2019.05.001. · Zbl 1471.60055
[10] Grigorova, M. and Quenez, M.-C.: Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEs, Stochastics, 89(1), (2017), 259-279. · Zbl 1410.91133
[11] Grigorova, M., Quenez, M.-C. and Sulem, A.: American options in a non-linear incomplete market with default, preprint, https://hal.archives-ouvertes.fr/hal-02025835/document. · Zbl 1452.91308
[12] Karatzas, I. and Shreve, S. E.: Methods of Mathematical Finance, Springer, 1998. · Zbl 0941.91032
[13] Kim, E., Nie, T., and Rutkowski, M.: Arbitrage-Free Pricing of American Options in Nonlinear Markets, preprint, arXiv:1804.10753.
[14] Kobylanski, M. and Quenez, M.-C.: Optimal stopping time problem in a general framework, Electronic Journal of Probability, 17, paper no. 72, (2012), 28 pp. Corrected In Erratum: Optimal stopping time problem in a general framework, available at https://hal.archives-ouvertes.fr/hal-01328196. · Zbl 1405.60055
[15] Kobylanski, M., Quenez, M.-C., and Rouy-Mironescu E.: Optimal multiple stopping time problem, Ann. Appl. Probab. 21(4), (2011), 1365-1399. · Zbl 1235.60040
[16] Peng, S.: Nonlinear expectations, nonlinear evaluations and risk measures, Lecture Notes in Math., 1856, Springer, Berlin, 2004. pp. 165-253. · Zbl 1127.91032
[17] Quenez, M.-C. and Sulem, A.: Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps, Stochastic Processes and their Applications 124(9), (2014), 3031-3054. · Zbl 1293.93783
[18] Royer, M.: Backward stochastic differential equations with jumps and related non-linear expectations, Stochastic Processes and Their Applications 116, (2006), 1358-1376. · Zbl 1110.60062
[19] Touzi, N. · Zbl 1322.60047
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.