In this paper, we study the martingale inequalities under $G$-expectation and their applications. To this end, we introduce a new kind of random time, called $G$-stopping time, and then investigate the properties of a $G$-martingale (supermartingale) such as the optional sampling theorem and upcrossing inequalities. With the help of these properties, we can show the martingale convergence property under $G$-expectation.
Hanwu LI
. MARTINGALE INEQUALITIES UNDER G-EXPECTATION AND THEIR APPLICATIONS[J]. Acta mathematica scientia, Series B, 2021
, 41(2)
: 349
-360
.
DOI: 10.1007/s10473-021-0201-6
[1] Chung K L. A Course in Probability Theory. New York:Academic Press, 1974
[2] Denis L, Hu M, Peng S. Function spaces and capacity related to a sublinear expectation:application to G-Brownian motion pathes. Potential Anal, 2011, 34:139-161
[3] Hu M, Ji S, Peng S, et al. Backward stochastic differential equations driven by G-Brownian motion. Stochastic Processes and their Applications, 2014, 124:759-784
[4] Hu M, Peng S. Extended conditional G-expectations and related stopping times. arXiv:1309.3829v1, 2013
[5] Karatzas I, Shreve S E. Brownian Motion and Stochastic Calculus. New York:Springer, 1991
[6] Li H, Peng S, Song Y. Supermartingale decomposition theorem under G-expectation. Electron J Probab, 2018, 23:1-20
[7] Li H. Optimal stopping under G-expectation. arXiv:1812.08626, 2018
[8] Peng S. G-expectation, G-Brownian Motion and Related Stochastic Calculus of Itô type. Stochastic analysis and applications, 2007, Abel Symp, 2:541-567
[9] Peng S. Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Processes and their Applications, 2008, 118(12):2223-2253
[10] Peng S. Nonlinear expectations and stochastic calculus under uncertainty. arXiv:1002.4546v1, 2010
[11] Song Y. Some properties on G-evaluation and its applications to G-martingale decomposition. Science China Mathematics, 2011, 54:287-300
[12] Song Y. Properties of hitting times for G-martingales and their applications. Stochastic Processes and their Applications, 2011, 121:1770-2784
