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

G-Lévy processes under sublinear expectations

Mingshang Hu 1, and  Shige Peng 2,,

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.

Keywords: Sublinear expectation, G-normal distribution, G-Brownian motion, G-expectation, Lévy process, G-Lévy process, G-Poisson process, Lévy-Khintchine formula, Lévy-Itô decomposition.
Mathematics Subject Classification: 60H10; 60J60; 60J65.
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. Google Scholar

[2]

Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D., Thinking Coherently, RISK, 1997, 10: 68-71. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

Bertoin, J., Lévy Processes, Cambridge University Press, 1996. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[10]

Huber, P., Robust Statistics, Wiley, New York, 1981. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

Lévy, P., Théorie de l’Addition des Variables Aléatoires, GauthierVillars, Paris, 1954. Google Scholar

[14]

Peng, S., Filtration consistent nonliear expectations and evaluations of contingent claims, Acta Mathematicae Applicatae Sinica, English Series, 2004, 20(2): 1-24. Google Scholar

[15]

Peng, S., Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math., 2005, 26B(2): 159-184. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[18]

Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty, Springer, 2019. Google Scholar

[19]

Peng, S., A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1, 2008. Google Scholar

[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.  Google Scholar

[21]

Sato, K.-I., Lévy processes and infinitely divisible distributions, Cambridge University, 1999. Google Scholar

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. Google Scholar

[2]

Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D., Thinking Coherently, RISK, 1997, 10: 68-71. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

Bertoin, J., Lévy Processes, Cambridge University Press, 1996. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[10]

Huber, P., Robust Statistics, Wiley, New York, 1981. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

Lévy, P., Théorie de l’Addition des Variables Aléatoires, GauthierVillars, Paris, 1954. Google Scholar

[14]

Peng, S., Filtration consistent nonliear expectations and evaluations of contingent claims, Acta Mathematicae Applicatae Sinica, English Series, 2004, 20(2): 1-24. Google Scholar

[15]

Peng, S., Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math., 2005, 26B(2): 159-184. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[18]

Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty, Springer, 2019. Google Scholar

[19]

Peng, S., A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1, 2008. Google Scholar

[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.  Google Scholar

[21]

Sato, K.-I., Lévy processes and infinitely divisible distributions, Cambridge University, 1999. Google Scholar

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