Extreme value theorems of uncertain process with application to insurance risk model. (English) Zbl 1279.60009

This is a further contribution to the author’s uncertainty theory introduced in 2004 and 2007 (see [Uncertainty theory. An introduction to its axiomatic foundations. Berlin: Springer (2004; Zbl 1072.28012); Berlin: Springer (2007; Zbl 1141.28001)]). In the present paper he proves some extreme value theorems of so-called uncertain independent increment processes and provides their uncertainty distribution of the first hitting time. The author also presents an uncertain insurance model by assuming that the claim is a renewal reward process, and proves a ruin index theorem.


60A86 Fuzzy probability
28E10 Fuzzy measure theory
91B30 Risk theory, insurance (MSC2010)
Full Text: DOI


[1] Chen XW (2011) American option pricing formula for uncertain financial market. Int J Oper Res 8(2):32-37 · Zbl 1468.91164
[2] Chen XW, Liu B (2010) Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optim Decis Making 9(1):69-81 · Zbl 1196.34005 · doi:10.1007/s10700-010-9073-2
[3] Liu B (2007) Uncertainty theory, 2nd edn. Springer, Berlin · Zbl 1141.28001
[4] Liu B (2008) Fuzzy process, hybrid process and uncertain process. J Uncertain Syst 2(1):3-16
[5] Liu B (2009a) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3-10
[6] Liu B (2009b) Theory and practice of uncertain programming, 2nd edn. Springer, Berlin · Zbl 1158.90010
[7] Liu B (2010a) Uncertain risk analysis and uncertain reliability analysis. J Uncertain Syst 4(3):163-170
[8] Liu B (2010b) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, Berlin
[9] Liu B (2012a) Why is there a need for uncertainty theory. J Uncertain Syst 6(1):3-10
[10] Liu B (2012b) Uncertainty theory, 4th edn. http://orsc.edu.cn/liu/ut.pdf
[11] Liu YH (2012c) An analytic method for solving uncertain differential equations. http://orsc.edu.cn/online/110402.pdf · Zbl 1196.34005
[12] Liu YH, Chen XW (2012) Uncertain currency model and currency option pricing. http://orsc.edu.cn/online/091010.pdf
[13] Liu YH, Ha MH (2010) Expected value of function of uncertain variables. J Uncertain Syst 4(3):181-186
[14] Peng J, Yao K (2011) A new option pricing model for stocks in uncertainty markets. Int J Oper Res 8(2):18-26 · Zbl 1468.91175
[15] Yao K, Chen XW (2012) A numerical method for solving uncertain differential equations. http://orsc.edu.cn/online/110913.pdf
[16] Yao K, Li X (2012) Uncertain alternating renewal process and its application. IEEE Trans Fuzzy Syst (to be published)
[17] Yu XC (2012) A stock model with jumps for uncertain markets. Int J Uncertain Fuzziness Knowledge-Based Syst 20(3):421-432 · Zbl 1251.91078
[18] Zhu Y (2010) Uncertain optimal control with application to a portfolio selection model. Cybern Syst 41(7):535-547 · Zbl 1225.93121
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.