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September & December  2018, 8(3&4): 637-651. doi: 10.3934/mcrf.2018027

Weak laws of large numbers for sublinear expectation

Zengjing Chen 1, , Qingyang Liu 1, and  Gaofeng Zong 2,,

1. 

Department of Mathematics, Shandong University, Jinan 250100, China
2. 

School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Bank of Weifang, Jinan 250014, China

* Corresponding author: Gaofeng Zong

Received  January 2018 Revised  June 2018 Published  September 2018

Fund Project: This work is supported in part by the National Science Foundation of China (Grant No.11501325, No.11231005), the China Postdoctoral Science Foundation (Grant No. 2018T110706) and the Taishan Scholars Climbing Program of Shandong.

In this paper we study the weak laws of large numbers for sublinear expectation. We prove that, without any moment condition, the weak laws of large numbers hold in the sense of convergence in capacity induced by some general sublinear expectations. For some specific sublinear expectation, for instance, mean deviation functional and one-side moment coherent risk measure, we also give weak laws of large numbers for corresponding capacity.

Keywords: Sublinear expectation, law of large numbers, coherent risk measure, capacity, independence.
Mathematics Subject Classification: Primary: 60B05, 60F05; Secondary: 60A86.
Citation: Zengjing Chen, Qingyang Liu, Gaofeng Zong. Weak laws of large numbers for sublinear expectation. Mathematical Control and Related Fields, 2018, 8 (3&4) : 637-651. doi: 10.3934/mcrf.2018027
References:
[1]

P. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.

[2]

X. Chen and Z. Chen, Weak and strong limit theorems for stochastic processes under nonadditive probability, Abstract and Applied Analysis, 2014 (2014), Art. ID 645947, 7 pp. doi: 10.1155/2014/645947.

[3]

X. Chen, Strong law of large numbers under an upper probability, Applied Mathematics, 3 (2012), 2056.

[4]

Z. Chen, Strong laws of large numbers for sub-linear expectations, Science China Mathematics, 59 (2016), 945-954.  doi: 10.1007/s11425-015-5095-0.

[5]

Z. Chen, P. Wu and B. Li, A strong law of large numbers for non-additive probabilities, International Journal of Approximate Reasoning, 54 (2013), 365-377.  doi: 10.1016/j.ijar.2012.06.002.

[6]

C. Gustave, Theory of capacities, Annales de l'institut Fourier, 5 (1953), 131-295. 

[7]

G. De Cooman and E. Miranda, Weak and strong laws of large numbers for coherent lower previsions, Journal of Statistical Planning and Inference, 138 (2008), 2409-2432.  doi: 10.1016/j.jspi.2007.10.020.

[8]

R. Durrett, Probability: Theory and Examples, Cambridge university press, 2010. doi: 10.1017/CBO9780511779398.

[9]

H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, Walter de Gruyter, 2011.

[10]

P. Ghirardato, On independence for non-additive measures with a Fubini theorem, J. Econom. Theory, 73 (1997), 261-291.  doi: 10.1006/jeth.1996.2241.

[11]

F. Maccheroni and M. Massimo, A strong law of large numbers for capacities, Annals of Probability, 33 (2005), 1171-1178.  doi: 10.1214/009117904000001062.

[12]

S. Peng, Law of large numbers and central limit theorem under nonlinear expectations, Sci. China Ser. A, 52 (2009), 1391-1411, arXiv: math/0702358. doi: 10.1007/s11425-009-0121-8.

[13]

S. Peng, Nonlinear expectations and stochastic calculus under uncertainty, preprint, arXiv: 1002.4546.

[14]

G. C. Pflug, W. Römisch, Modeling, Measuring and Managing Risk, Singapore: World Scientific, 2007. doi: 10.1142/9789812708724.

[15]

L. S. Shapley, A value for n-person games, Contributions to the Theory of Games, 2 (1953), 307-317. 

[16]

P. Terán, Laws of large numbers without additivity, Transactions of American Mathematical Society, 366 (2014), 5431-5451.  doi: 10.1090/S0002-9947-2014-06053-4.

show all references

References:
[1]

P. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.

[2]

X. Chen and Z. Chen, Weak and strong limit theorems for stochastic processes under nonadditive probability, Abstract and Applied Analysis, 2014 (2014), Art. ID 645947, 7 pp. doi: 10.1155/2014/645947.

[3]

X. Chen, Strong law of large numbers under an upper probability, Applied Mathematics, 3 (2012), 2056.

[4]

Z. Chen, Strong laws of large numbers for sub-linear expectations, Science China Mathematics, 59 (2016), 945-954.  doi: 10.1007/s11425-015-5095-0.

[5]

Z. Chen, P. Wu and B. Li, A strong law of large numbers for non-additive probabilities, International Journal of Approximate Reasoning, 54 (2013), 365-377.  doi: 10.1016/j.ijar.2012.06.002.

[6]

C. Gustave, Theory of capacities, Annales de l'institut Fourier, 5 (1953), 131-295. 

[7]

G. De Cooman and E. Miranda, Weak and strong laws of large numbers for coherent lower previsions, Journal of Statistical Planning and Inference, 138 (2008), 2409-2432.  doi: 10.1016/j.jspi.2007.10.020.

[8]

R. Durrett, Probability: Theory and Examples, Cambridge university press, 2010. doi: 10.1017/CBO9780511779398.

[9]

H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, Walter de Gruyter, 2011.

[10]

P. Ghirardato, On independence for non-additive measures with a Fubini theorem, J. Econom. Theory, 73 (1997), 261-291.  doi: 10.1006/jeth.1996.2241.

[11]

F. Maccheroni and M. Massimo, A strong law of large numbers for capacities, Annals of Probability, 33 (2005), 1171-1178.  doi: 10.1214/009117904000001062.

[12]

S. Peng, Law of large numbers and central limit theorem under nonlinear expectations, Sci. China Ser. A, 52 (2009), 1391-1411, arXiv: math/0702358. doi: 10.1007/s11425-009-0121-8.

[13]

S. Peng, Nonlinear expectations and stochastic calculus under uncertainty, preprint, arXiv: 1002.4546.

[14]

G. C. Pflug, W. Römisch, Modeling, Measuring and Managing Risk, Singapore: World Scientific, 2007. doi: 10.1142/9789812708724.

[15]

L. S. Shapley, A value for n-person games, Contributions to the Theory of Games, 2 (1953), 307-317. 

[16]

P. Terán, Laws of large numbers without additivity, Transactions of American Mathematical Society, 366 (2014), 5431-5451.  doi: 10.1090/S0002-9947-2014-06053-4.

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