×

BSDEs driven by \(G\)-Brownian motion with time-varying Lipschitz condition. (English) Zbl 1471.60087

Summary: The present paper is devoted to the study of backward stochastic differential equations driven by \(G\)-Brownian motion \((G\)-BSDEs) with time-varying Lipschitz condition. With the help of nonlinear stochastic analysis technique and approximation method, we prove that the \(G\)-BSDEs admit a unique solution on finite or infinite time horizon. Furthermore, we obtain the corresponding comparison theorem.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
60G65 Nonlinear processes (e.g., \(G\)-Brownian motion, \(G\)-Lévy processes)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Briand, P.; Confortola, F., BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces, Stoch. Process. Appl., 118, 5, 818-838 (2008) · Zbl 1136.60337
[2] Briand, P.; Hu, Y., Stability of BSDEs with random terminal time and homogenization of semilinear elliptic PDEs, J. Funct. Anal., 155, 2, 455-494 (1998) · Zbl 0912.60081
[3] Chen, Z.; Wang, B., Infinite time interval BSDEs and the convergence of g-martingales, J. Aust. Math. Soc., 69, 187-211 (2000) · Zbl 0982.60052
[4] Darling, R. W.R.; Pardoux, E., Backwards SDE with random terminal time and applications to semilinear elliptic PDE, Ann. Probab., 25, 1135-1159 (1997) · Zbl 0895.60067
[5] Denis, L.; Hu, M.; Peng, S., Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths, Potential Anal., 34, 139-161 (2011) · Zbl 1225.60057
[6] El Karoui, N.; Peng, S.; Quenez, M. C., Backward stochastic differential equations in finance, Math. Finance, 7, 1-71 (1997) · Zbl 0884.90035
[7] Fan, S.; Jiang, L.; Tian, D., One-dimensional BSDEs with finite and infinite time horizons, Stoch. Process. Appl., 121, 427-440 (2011) · Zbl 1219.60059
[8] Fuhrman, M.; Hu, Y., Infinite horizon BSDEs in infinite dimensions with continuous driver and applications, J. Evol. Equ., 6, 459-484 (2006) · Zbl 1116.60032
[9] Fuhrman, M.; Hu, Y.; Tessitore, G., Ergodic BSDEs and optimal ergodic control in Banach spaces, SIAM J. Control Optim., 48, 1542-1566 (2009) · Zbl 1196.60106
[10] Hu, M.; Peng, S., On representation theorem of G-expectations and paths of G-Brownian motion, Acta Math. Appl. Sin. Engl. Ser., 25, 3, 539-546 (2009) · Zbl 1190.60043
[11] Hu, M.; Wang, F., Ergodic BSDEs driven by G-Brownian motion and applications, Stoch. Dyn., 18, 1-35 (2018) · Zbl 1417.60050
[12] Hu, M.; Ji, S.; Peng, S.; Song, Y., Backward stochastic differential equations driven by G-Brownian motion, Stoch. Process. Appl., 124, 759-784 (2014) · Zbl 1300.60074
[13] Hu, M.; Ji, S.; Peng, S.; Song, Y., Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion, Stoch. Process. Appl., 124, 1170-1195 (2014) · Zbl 1300.60075
[14] Hu, M.; Wang, F.; Zheng, G., Quasi-continuous random variables and processes under the G-expectation framework, Stoch. Process. Appl., 126, 2367-2387 (2016) · Zbl 1343.60037
[15] Hu, Y.; Lin, Y.; Soumana Hima, A., Quadratic backward stochastic differential equations driven by G-Brownian motion: discrete solutions and approximation, Stoch. Process. Appl., 128, 3724-3750 (2018) · Zbl 1401.60113
[16] Li, H.; Peng, S.; Soumana Hima, A., Reflected solutions of backward stochastic differential equations driven by G-Brownian motion, Sci. China Math., 61, 1-26 (2018) · Zbl 1390.60213
[17] Liu, G., Multi-dimensional BSDEs with diagonal generators driven by G-Brownian motion, Stochastics (2019)
[18] Pardoux, E.; Peng, S., Adapted solution of a backward stochastic differential equation, Syst. Control Lett., 14, 55-61 (1990) · Zbl 0692.93064
[19] Peng, S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stoch. Stoch. Rep., 37, 61-74 (1991) · Zbl 0739.60060
[20] Peng, S., A nonlinear Feynman-Kac formula and applications, (Control Theory, Stochastic Analysis and Applications (1991), World Sci. Publ.: World Sci. Publ. River Edge, NJ), 173-184
[21] Peng, S., Nonlinear expectation, nonlinear evaluations and risk measures, (Back, K.; Bielecki, T. R.; Hipp, C.; Peng, S.; Schachermayer, W., Stochastic Methods in Finance Lectures. Stochastic Methods in Finance Lectures, LNM, vol. 1856 (2004), Springer: Springer Heidelberg), 143-217
[22] Peng, S., G-expectation, G-Brownian motion and related stochastic calculus of Itô type, (Stochastic Analysis and Applications. Stochastic Analysis and Applications, Abel Symp., vol. 2 (2007), Springer: Springer Berlin), 541-567 · Zbl 1131.60057
[23] Peng, S., Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stoch. Process. Appl., 118, 12, 2223-2253 (2008) · Zbl 1158.60023
[24] Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty (2019), Springer-Verlag Berlin Heidelberg · Zbl 1427.60004
[25] Royer, M., BSDEs with a random terminal time driven by a monotone generator and their links with PDEs, Stoch. Stoch. Rep., 76, 4, 281-307 (2004) · Zbl 1055.60062
[26] Soner, H. M.; Touzi, N.; Zhang, J., Martingale representation theorem for the G-expectation, Stoch. Process. Appl., 121, 2, 265-287 (2011) · Zbl 1228.60070
[27] Soner, H. M.; Touzi, N.; Zhang, J., Wellposedness of second order backward SDEs, Probab. Theory Relat. Fields, 153, 1-2, 149-190 (2012) · Zbl 1252.60056
[28] Song, Y., Some properties on G-evaluation and its applications to G-martingale decomposition, Sci. China Math., 54, 287-300 (2011) · Zbl 1225.60058
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.