# American Institute of Mathematical Sciences

December  2021, 6(4): 369-390. doi: 10.3934/puqr.2021018

## Extended conditional G-expectations and related stopping times

 1 Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, Shandong, China 2 School of Mathematics, Shandong University, Jinan 250100, Shandong, China

humingshang@sdu.edu.cn

Received  November 29, 2021 Accepted  December 05, 2021 Published  December 2021

Fund Project: Mingshang Hu is supported by National Key R&D Program of China (Grant No. 2018YFA0703900) and the National Natural Science Foundation of China (Grant No. 11671231). Shige Peng is supported by the Tian Yuan Projection of the National Natural Science Foundation of China (Grant Nos. 11526205 and 11626247) and the National Basic Research Program of China (973 Program) (Grant No. 2007CB814900 (Financial Risk)).

In this paper, we extend the definition of conditional $G{\text{-}}{\rm{expectation}}$ to a larger space on which the dynamical consistency still holds. We can consistently define, by taking the limit, the conditional $G{\text{-}}{\rm{expectation}}$ for each random variable $X$, which is the downward limit (respectively, upward limit) of a monotone sequence $\{X_{i}\}$ in $L_{G}^{1}(\Omega)$. To accomplish this procedure, some careful analysis is needed. Moreover, we present a suitable definition of stopping times and obtain the optional stopping theorem. We also provide some basic and interesting properties for the extended conditional $G{\text{-}}{\rm{expectation}}$.

Citation: Mingshang Hu, Shige Peng. Extended conditional G-expectations and related stopping times. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 369-390. doi: 10.3934/puqr.2021018
##### References:
 [1] Denis, L., Hu, M. and Peng S., Function spaces and capacity related to a sublinear expectation: Application to G-Brownian motion paths, Potential Anal., 2011, 34: 139−161. doi: 10.1007/s11118-010-9185-x. [2] Hu, M., Ji, X. and Liu, G., On the strong Markov property for stochastic differential equations driven by G-Brownian motion, Stochastic Process. Appl., 2021, 131: 417−453. doi: 10.1016/j.spa.2020.09.015. [3] Hu, M., Ji, S., Peng, S. and Song, Y., Backward stochastic differential equations driven by G-Brownian motion, Stochastic Process. Appl., 2014, 124(1): 759−784. doi: 10.1016/j.spa.2013.09.010. [4] Hu, M., Ji, S., Peng, S. and Song, Y., Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion, Stochastic Process. Appl., 2014, 124(2): 1170−1195. doi: 10.1016/j.spa.2013.10.009. [5] Hu, M. and Peng, S., On representation theorem of G-expectations and paths of G-Brownian motion, Acta Math. Appl. Sin. Engl. Ser., 2009, 25(3): 539−546. doi: 10.1007/s10255-008-8831-1. [6] Hu, M. and Peng, S., G-Lévy processes under sublinear expectations, Probab. Uncertain. Quant. Risk, 2021, 6(1): 1–22. [7] Li, X. and Peng, S., Stopping times and related Itô’s calculus with G-Brownian motion, Stochastic Process. Appl., 2011, 121(7): 1492−1508. doi: 10.1016/j.spa.2011.03.009. [8] Liu, G., Exit times for semimartingales under nonlinear expectation, Stochastic Process. Appl., 2020, 130(12): 7338−7362. doi: 10.1016/j.spa.2020.07.017. [9] Nutz, M. and Van Handel, R., Constructing sublinear expectations on path space, Stochastic Process. Appl., 2013, 123(8): 3100−3121. doi: 10.1016/j.spa.2013.03.022. [10] Nutz, M. and Zhang, J., Optimal stopping under adverse nonlinear expectation and related games, Ann. Appl. Probab., 2015, 25(5): 2503−2534. [11] Peng, S., G-expectation, G-Brownian motion and related stochastic calculus of Itô type, In: Benth, F. E., Di Nunno, G., Lindstrøm, T., Øksendal, B. and Zhang, T. (eds.), Stochastic Analysis and Applications, Abel Symp., Springer, Berlin, 2007, 2: 541–567. [12] Peng, S., Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Process. Appl., 2008, 118(12): 2223−2253. doi: 10.1016/j.spa.2007.10.015. [13] Peng, S., A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1, 2008. [14] Peng, S., Law of large numbers and central limit theorem under nonlinear expectations, Probab. Uncertain. Quant. Risk, 2019, 4: 4, doi: 10.1186/s41546-019-0038-2. [15] Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty, Springer-Verlag, Heidelberg, 2019. [16] Soner, M., Touzi, N. and Zhang, J., Martingale representation theorem under G-expectation, Stochastic Process. Appl., 2011, 121(2): 265−287. doi: 10.1016/j.spa.2010.10.006. [17] Song, Y., Some properties on G-evaluation and its applications to G-martingale decomposition, Sci. China Math., 2011, 54(2): 287−300. doi: 10.1007/s11425-010-4162-9. [18] Song, Y., Properties of hitting times for G-martingales and their applications, Stochastic Process. Appl., 2011, 121(8): 1770−1784. doi: 10.1016/j.spa.2011.04.007.

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##### References:
 [1] Denis, L., Hu, M. and Peng S., Function spaces and capacity related to a sublinear expectation: Application to G-Brownian motion paths, Potential Anal., 2011, 34: 139−161. doi: 10.1007/s11118-010-9185-x. [2] Hu, M., Ji, X. and Liu, G., On the strong Markov property for stochastic differential equations driven by G-Brownian motion, Stochastic Process. Appl., 2021, 131: 417−453. doi: 10.1016/j.spa.2020.09.015. [3] Hu, M., Ji, S., Peng, S. and Song, Y., Backward stochastic differential equations driven by G-Brownian motion, Stochastic Process. Appl., 2014, 124(1): 759−784. doi: 10.1016/j.spa.2013.09.010. [4] Hu, M., Ji, S., Peng, S. and Song, Y., Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion, Stochastic Process. Appl., 2014, 124(2): 1170−1195. doi: 10.1016/j.spa.2013.10.009. [5] Hu, M. and Peng, S., On representation theorem of G-expectations and paths of G-Brownian motion, Acta Math. Appl. Sin. Engl. Ser., 2009, 25(3): 539−546. doi: 10.1007/s10255-008-8831-1. [6] Hu, M. and Peng, S., G-Lévy processes under sublinear expectations, Probab. Uncertain. Quant. Risk, 2021, 6(1): 1–22. [7] Li, X. and Peng, S., Stopping times and related Itô’s calculus with G-Brownian motion, Stochastic Process. Appl., 2011, 121(7): 1492−1508. doi: 10.1016/j.spa.2011.03.009. [8] Liu, G., Exit times for semimartingales under nonlinear expectation, Stochastic Process. Appl., 2020, 130(12): 7338−7362. doi: 10.1016/j.spa.2020.07.017. [9] Nutz, M. and Van Handel, R., Constructing sublinear expectations on path space, Stochastic Process. Appl., 2013, 123(8): 3100−3121. doi: 10.1016/j.spa.2013.03.022. [10] Nutz, M. and Zhang, J., Optimal stopping under adverse nonlinear expectation and related games, Ann. Appl. Probab., 2015, 25(5): 2503−2534. [11] Peng, S., G-expectation, G-Brownian motion and related stochastic calculus of Itô type, In: Benth, F. E., Di Nunno, G., Lindstrøm, T., Øksendal, B. and Zhang, T. (eds.), Stochastic Analysis and Applications, Abel Symp., Springer, Berlin, 2007, 2: 541–567. [12] Peng, S., Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Process. Appl., 2008, 118(12): 2223−2253. doi: 10.1016/j.spa.2007.10.015. [13] Peng, S., A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1, 2008. [14] Peng, S., Law of large numbers and central limit theorem under nonlinear expectations, Probab. Uncertain. Quant. Risk, 2019, 4: 4, doi: 10.1186/s41546-019-0038-2. [15] Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty, Springer-Verlag, Heidelberg, 2019. [16] Soner, M., Touzi, N. and Zhang, J., Martingale representation theorem under G-expectation, Stochastic Process. Appl., 2011, 121(2): 265−287. doi: 10.1016/j.spa.2010.10.006. [17] Song, Y., Some properties on G-evaluation and its applications to G-martingale decomposition, Sci. China Math., 2011, 54(2): 287−300. doi: 10.1007/s11425-010-4162-9. [18] Song, Y., Properties of hitting times for G-martingales and their applications, Stochastic Process. Appl., 2011, 121(8): 1770−1784. doi: 10.1016/j.spa.2011.04.007.
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