Title: Fuzzy mortality models based on Von Neumann algebras
Authors: Agnieszka Rossa - University of Lodz (Poland) [presenting]
Andrzej Szymanski - University of Lodz (Poland)
Abstract: The generalization of the well-known Lee-Carter mortality model defined in terms of fuzzy numbers was first introduced by proposing a fuzzy representation of the log-central mortality rates and models parameters, by converting them into fuzzy numbers with triangular symmetric membership functions. In our approach we used exponential membership functions decomposed into two parts: strictly decreasing and strictly increasing functions, and transformed into the complex functions of the general form: $f(u)+ig(u)$. The pair of ordered complex numbers is called a quaternion. It can be easily seen that quaternion algebra with a proper defined definition of algebra multiplication is a $C^*$-Banach algebra, which is non-commutative. A von Neumann algebra was previously applied as a $C^*$-algebra of projections in a separable complex Hilbert space. We applied this approach to define a new mortality model based on von Neumann algebra. The research was supported by the grant from National Science Centre under contract UMO-2015/17/B/HS4/00927.