Abstract
A finite variation process is an uncertain process whose total variation is finite over each bounded time interval. Based on the finite variation processes, a new uncertain integral is proposed in this paper. Besides, some basic properties are discussed. In the framework of the uncertain integral, uncertain differential is introduced, and the fundamental theorem of uncertain calculus is derived. Finally, the integration by parts theorem is studied.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen X, Liu B (2010) Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optim Decis Mak 9(1):69–81
Chen X (2011) American option pricing formula for uncertain financial market. Int J Oper Res 8(2):32–37
Chen X (2012) Variation analysis of uncertain stationary independent increment processes. Eur J Oper Res 222(2):312–316
Chen X, Gao J (2013) Uncertain term structure model of interest rate. Soft Comput 17(4):597–604
Chen X, Liu Y, Ralescu DA (2013) Uncertain stock model with periodic dividends. Fuzzy Optim Decis Mak 12(1):111–123
Chen X, Ralescu DA (2013) Liu process and uncertain calculus. J Uncertain Anal Appl 1(3):1–12
Doléans-Dade C, Meyer P (1970) Universté de Strasbourg Séminaire de Probabilités. Lect Notes Math 5:143–173
Gao Y (2011) Shortest path problem with uncertain arc lengths. Comput Math Appl 62(6):2591–2600
Gao Y (2012) Uncertain models for single facility location problems on networks. Appl Math Model 36(6):2592–2599
Ito K (1944) Stochastic Integral. In: Proceedings of the Japan Academy, Tokyo, Japan, pp 519–524
Jacod J (1979) Calculus stocastique et problemes de martingales. Lect Notes Math 714:66–112
Jiao D, Yao K (2014) An interest rate model in uncertain environment, Soft Comput
Kallenberg O (1997) Foundations of modern probability, 2nd edn. Springer, New York
Kunita H, Watanabe S (1967) On square integrable martingales. Nagoya Math J 30:209–245
Liu B (2007) Uncertainty theory, 2nd edn. Springer, Berlin
Liu B (2008) Fuzzy process, hybrid process and uncertain process. J Uncertain Syst 2(1):3–16
Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10
Liu B (2010) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, Berlin
Liu Y, Ha M (2010) Expected value of function of uncertain variables. J Uncertain Syst 4(3):181–186
Liu Y et al. (2014) Uncertain currency model and currency option pricing. Int J Intel Syst
Peng J, Yao K (2010) A new option pricing model for stocks in uncertainty markets. Int J Oper Res 7(4):213–224
Wiener N (1923) Differential space. J Math Phys 2:131–174
Yao K (2012) Uncertain calculus with renewal process. Fuzzy Optim Decis Mak 11(3):285–297
Yao K, Chen X (2013) A numerical method for solving uncertain differential equations. J Intell Fuzzy Syst 25(3):825–832
Zhu Y (2010) Uncertain optimal control with application to a portfolio selection model. Cybern Syst 41(7):535–547
Acknowledgments
This work was supported by National Natural Science Foundation of China Grant No. 61304182 and supported by “the Fundamental Research Funds for the Central Universities” No. NKZXB1419.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. Loia.
Rights and permissions
About this article
Cite this article
Chen, X. Uncertain calculus with finite variation processes. Soft Comput 19, 2905–2912 (2015). https://doi.org/10.1007/s00500-014-1452-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-014-1452-0