A relation between moments of Liu process and Bernoulli numbers
- PDF / 291,241 Bytes
- 12 Pages / 439.37 x 666.142 pts Page_size
- 28 Downloads / 199 Views
A relation between moments of Liu process and Bernoulli numbers Guanzhong Ma1
· Xiangfeng Yang2 · Xiao Yao3
Accepted: 7 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper finds a relation between moments of Liu process and Bernoulli numbers. Firstly, by an exponential generating function of Bernoulli numbers, a useful integral formula is obtained. Secondly, based on this integral formula, the moments of a normal uncertain variable and Liu process are expressed via Bernoulli numbers. Keywords Uncertainty theory · Liu process · Bernoulli numbers · Inverse uncertainty distribution
1 Introduction In order to handle the belief degree of humans, uncertainty theory was founded by Liu (2007, 2009) on the foundation of normality, duality, subadditivity, and product axioms. Afterward, this theory was improved by Liu (2010) and Liu (2015). A series of fundamental concepts in uncertainty theory, such as uncertain variable, expectation, variance, and moment, were proposed by Liu (2007). For calculating the expectation of a function of uncertain variables, a compelling method was discovered by Liu and Ha (2010). Based on the inverse uncertainty distribution, Yao (2015) introduced a new approach to compute the variance. Later on, Sheng and Kar (2015) extended Yao’s result to the moments of the uncertain variable. These formulae, in Liu and Ha (2010),
B
Guanzhong Ma [email protected] Xiangfeng Yang [email protected] Xiao Yao [email protected]
1
School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China
2
School of Information Technology and Management, University of International Business and Economics, Beijing 100029, China
3
School of Mathematics Sciences, Nankai University, Tianjin 300071, China
123
X. Ma et al.
Yao (2015) and Sheng and Kar (2015), can be used to transform Lebesgue-Stieltjes integral in uncertainty theory to Lebesgue integral, which are very convenient in many applications. Inspired by the Itô integral in stochastic analysis, Liu (2009) offered a kind of stationary independent increment uncertain processes and pioneered a new integral. This uncertain process is later called Liu process, and the original integral is named Liu integral. In Liu integral, one can get an uncertain process by integrating an uncertain process with respect to the Liu process. Liu (2008) established an uncertain differential equation, which is a type of differential equations driven by Liu processes. Chen and Liu (2010) first proved that the solution of uncertain differential equation exists and is unique if linear growth condition and Lipschitz condition hold. Besides, Chen and Liu (2010) got an analytic solution of a linear uncertain differential equation. Liu (2012) presented two analytic methods for solving some nonlinear uncertain differential equations. The concept of the stability of uncertain differential equations was first proposed by Liu (2009) and investigated by numerous scholars. Yao et al. (2013) proved some stabili
Data Loading...
