# American Institute of Mathematical Sciences

March  2021, 6(1): 1-22. doi: 10.3934/puqr.2021001

## G-Lévy processes under sublinear expectations

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

Email: humingshang@sdu.edu.cn, peng@sdu.edu.cn

Received  June 22, 2020 Accepted  December 14, 2020 Published  March 2021

Fund Project: This work was supported by National Key R&D Program of China (Grant No. 2018YFA0703900), National Natural Science Foundation of China (Grant No. 11671231) , Tian Yuan Fund of the National Natural Science Foundation of China (Grant Nos. 11526205 and 11626247) and National Basic Research Program of China (973 Program) (Grant No. 2007CB814900).

We introduce G-Lévy processes which develop the theory of processes with independent and stationary increments under the framework of sublinear expectations. We then obtain the Lévy–Khintchine formula and the existence for G-Lévy processes. We also introduce G-Poisson processes.

Citation: Mingshang Hu, Shige Peng. G-Lévy processes under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 1-22. doi: 10.3934/puqr.2021001
##### References:
 [1] Alvarez, O. and Tourin, A., Viscosity solutions of nonlinear integrodifferential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1996, 13(3): 293-317. [2] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D., Thinking Coherently, RISK, 1997, 10: 68-71. [3] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D., Coherent measures of risk, Mathematical Finance, 1999, 9(3): 203-228. doi: 10.1111/1467-9965.00068. [4] Barles, G. and Imbert, C., Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited, Ann. I. H. Poincaré-AN, 2008, 25: 567-585. [5] Bertoin, J., Lévy Processes, Cambridge University Press, 1996. [6] Crandall, M., Semidifferentials, quadratic forms and fully nonlinear elliptic equations of second order, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1989, 6(6): 419-435. [7] Crandall, M. G., Ishii, H. and Lions, P.-L., User’S guide to viscosity solutions of second order partial differential equations, Bulletin of The American Mathematical Society, 1992, 27(1): 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [8] Delbaen, F., Coherent measures of risk on general probability space, In: Advances in Finance and Stochastics, Essays in Honor of Dieter Sondermann (Sandmann, K. and Schonbucher, P.J. eds.), Springer, Berlin, 2002: 1-37. [9] 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. [10] Huber, P., Robust Statistics, Wiley, New York, 1981. [11] Hu, M. and Peng, S., On representation theorem of G-expectations and paths of G-Brownian motion, Acta Mathematicae Applicatae Sinica, English Series, 2009, 25(3): 539-546. doi: 10.1007/s10255-008-8831-1. [12] Jakobsen, E.R. and Karlsen, K.H., A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations, NoDEA Nonlinear Differ. Equ. Appl., 2006, 13: 137-165. doi: 10.1007/s00030-005-0031-6. [13] Lévy, P., Théorie de l’Addition des Variables Aléatoires, GauthierVillars, Paris, 1954. [14] Peng, S., Filtration consistent nonliear expectations and evaluations of contingent claims, Acta Mathematicae Applicatae Sinica, English Series, 2004, 20(2): 1-24. [15] Peng, S., Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math., 2005, 26B(2): 159-184. [16] Peng, S., G-Expectation, G-Brownian motion and related stochastic calculus of Itô’s type, In: Stochastic Analysis and Applications, Able Symposium, Abel Symposia 2, SpringerVerlag, 2007: 541-567. [17] Peng, S., Multi-Dimensional G-Brownian motion and related stochastic calculus under G-Expectation, Stochastic Processes and their Applications, 2008, 118: 2223-2253. doi: 10.1016/j.spa.2007.10.015. [18] Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty, Springer, 2019. [19] Peng, S., A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1, 2008. [20] Peng, S., Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Science in China Series A: Mathematics, 2009, 52(7): 1391-1411. doi: 10.1007/s11425-009-0121-8. [21] Sato, K.-I., Lévy processes and infinitely divisible distributions, Cambridge University, 1999.

show all references

##### References:
 [1] Alvarez, O. and Tourin, A., Viscosity solutions of nonlinear integrodifferential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1996, 13(3): 293-317. [2] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D., Thinking Coherently, RISK, 1997, 10: 68-71. [3] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D., Coherent measures of risk, Mathematical Finance, 1999, 9(3): 203-228. doi: 10.1111/1467-9965.00068. [4] Barles, G. and Imbert, C., Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited, Ann. I. H. Poincaré-AN, 2008, 25: 567-585. [5] Bertoin, J., Lévy Processes, Cambridge University Press, 1996. [6] Crandall, M., Semidifferentials, quadratic forms and fully nonlinear elliptic equations of second order, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1989, 6(6): 419-435. [7] Crandall, M. G., Ishii, H. and Lions, P.-L., User’S guide to viscosity solutions of second order partial differential equations, Bulletin of The American Mathematical Society, 1992, 27(1): 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [8] Delbaen, F., Coherent measures of risk on general probability space, In: Advances in Finance and Stochastics, Essays in Honor of Dieter Sondermann (Sandmann, K. and Schonbucher, P.J. eds.), Springer, Berlin, 2002: 1-37. [9] 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. [10] Huber, P., Robust Statistics, Wiley, New York, 1981. [11] Hu, M. and Peng, S., On representation theorem of G-expectations and paths of G-Brownian motion, Acta Mathematicae Applicatae Sinica, English Series, 2009, 25(3): 539-546. doi: 10.1007/s10255-008-8831-1. [12] Jakobsen, E.R. and Karlsen, K.H., A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations, NoDEA Nonlinear Differ. Equ. Appl., 2006, 13: 137-165. doi: 10.1007/s00030-005-0031-6. [13] Lévy, P., Théorie de l’Addition des Variables Aléatoires, GauthierVillars, Paris, 1954. [14] Peng, S., Filtration consistent nonliear expectations and evaluations of contingent claims, Acta Mathematicae Applicatae Sinica, English Series, 2004, 20(2): 1-24. [15] Peng, S., Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math., 2005, 26B(2): 159-184. [16] Peng, S., G-Expectation, G-Brownian motion and related stochastic calculus of Itô’s type, In: Stochastic Analysis and Applications, Able Symposium, Abel Symposia 2, SpringerVerlag, 2007: 541-567. [17] Peng, S., Multi-Dimensional G-Brownian motion and related stochastic calculus under G-Expectation, Stochastic Processes and their Applications, 2008, 118: 2223-2253. doi: 10.1016/j.spa.2007.10.015. [18] Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty, Springer, 2019. [19] Peng, S., A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1, 2008. [20] Peng, S., Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Science in China Series A: Mathematics, 2009, 52(7): 1391-1411. doi: 10.1007/s11425-009-0121-8. [21] Sato, K.-I., Lévy processes and infinitely divisible distributions, Cambridge University, 1999.
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