×

Representation theorem for generators of BSDEs driven by \(G\)-Brownian motion and its applications. (English) Zbl 1310.60082

Summary: We obtain a representation theorem for the generators of BSDEs driven by \(G\)-Brownian motions and then we use the representation theorem to get a converse comparison theorem for \(G\)-BSDEs and some equivalent results for nonlinear expectations generated by \(G\)-BSDEs.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Pardoux, É.; Peng, S. G., Adapted solution of a backward stochastic differential equation, Systems & Control Letters, 14, 1, 55-61 (1990) · Zbl 0692.93064 · doi:10.1016/0167-6911(90)90082-6
[2] Peng, S.; El Karoui, N.; Mazliak, L., Backward SDE and related \(g\)-expectation, Backward Stochastic Differential Equations. Backward Stochastic Differential Equations, Pitman Research Notes in Mathematics Series, 364, 141-159 (1997) · Zbl 0892.60066
[3] Chen, Z. J., A property of backward stochastic differential equations, Comptes Rendus de l’Académie des Sciences. Série I, 326, 4, 483-488 (1998) · Zbl 0914.60025 · doi:10.1016/S0764-4442(97)89796-0
[4] Briand, P.; Coquet, F.; Hu, Y.; Mémin, J.; Peng, S. G., A converse comparison theorem for BSDEs and related properties of \(g\)-expectation, Electronic Communications in Probability, 5, 101-117 (2000) · Zbl 0966.60054 · doi:10.1214/ECP.v5-1025
[5] Jiang, L., Representation theorems for generators of backward stochastic differential equations, Comptes Rendus Mathématique. Académie des Sciences. Paris I, 340, 2, 161-166 (2005) · Zbl 1067.60043 · doi:10.1016/j.crma.2004.10.023
[6] Jiang, L., Converse comparison theorems for backward stochastic differential equations, Statistics & Probability Letters, 71, 2, 173-183 (2005) · Zbl 1079.60054 · doi:10.1016/j.spl.2004.10.032
[7] Jiang, L., Convexity, translation invariance and subadditivity for \(g\)-expectations and related risk measures, The Annals of Applied Probability, 18, 1, 245-258 (2008) · Zbl 1145.60032 · doi:10.1214/105051607000000294
[8] Peng, S., Filtration consistent nonlinear expectations and evaluations of contingent claims, Acta Mathematicae Applicatae Sinica, 20, 2, 191-214 (2004) · Zbl 1061.60063 · doi:10.1007/s10255-004-0161-3
[9] Peng, S., Nonlinear expectations and nonlinear Markov chains, Chinese Annals of Mathematics B, 26, 2, 159-184 (2005) · Zbl 1077.60045 · doi:10.1142/S0252959905000154
[10] Peng, S., \(G\)-expectation, \(G\)-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications. Stochastic Analysis and Applications, The Abel Symposium, 2, 541-567 (2007), Berlin, Germany: Springer, Berlin, Germany · Zbl 1131.60057 · doi:10.1007/978-3-540-70847-6_25
[11] Peng, S., G-Brownian motion and dynamic risk measure under volatility uncertainty
[12] Peng, S., Multi-dimensional \(G\)-Brownian motion and related stochastic calculus under \(G\)-expectation, Stochastic Processes and Their Applications, 118, 12, 2223-2253 (2008) · Zbl 1158.60023 · doi:10.1016/j.spa.2007.10.015
[13] Peng, S., A new central limit theorem under sublinear expectations
[14] Soner, H. M.; Touzi, N.; Zhang, J., Wellposedness of second order backward SDEs, Probability Theory and Related Fields, 153, 1-2, 149-190 (2012) · Zbl 1252.60056 · doi:10.1007/s00440-011-0342-y
[15] Hu, M. S.; Ji, S. L.; Peng, S. G.; Song, Y. S., Backward stochastic differential equations driven by G-brownian motion · Zbl 1300.60074
[16] Hu, M. S.; Ji, S. L.; Peng, S. G.; Song, Y. S., Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion · Zbl 1300.60075
[17] Peng, S., G.\(G\)-Brownian motion and dynamic risk measure under volatility uncertainty
[18] Peng, S. G., Multi-dimensional \(G\)-Brownian motion and related stochastic calculus under \(G\)-expectation, Stochastic Processes and their Applications, 118, 12, 2223-2253 (2008) · Zbl 1158.60023 · doi:10.1016/j.spa.2007.10.015
[19] Peng, S. G., Nonlinear expectations and stochastic calculus under uncertainty · Zbl 1427.60004
[20] Song, Y. S., Some properties on \(G\)-evaluation and its applications to \(G\)-martingale decomposition, Science China Mathematics, 54, 2, 287-300 (2011) · Zbl 1225.60058 · doi:10.1007/s11425-010-4162-9
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.