uncertainty inherent in the financial market was usually consid- ered to be random. however, randomness is only one special type of uncer- tainty and appropriate when describing objective information. for describing subjective information it is preferred to assume that uncertainty is fuzzy. this paper defines the expected payoof trading strategies in a fuzzy financial market within the framewor...

In this paper, multi-dimensional Wiener-Liu process is proposed. Wiener-Liu process is a type of hybrid process, it corresponds to Brownian motion (Wiener process) in stochastic process and Liu process in fuzzy process. In classical analysis, the basic operations are differential and integral. Correspondingly, Ito-Liu formula plays the role of Ito formula in stochastic process and Liu formula i...

Credibility measure was created by Baoding Liu [5, 7] and his Chinese co workers [4] around 2004. Then, Baoding Liu designed a grand view of fuzzy process, hybrid process and uncertain process [6]. In response to his work, Dai [1] sketched how to create standard normal fuzzy process under credibility measure, which he called standard Liu process. Independently Kageyama and Iwamura [3] construct...

Liu process is a type of fuzzy process. It is a fuzzy counterpart of Brownian motion. In this paper, the continuity property of Liu process is studied. It is proved that almost all Liu paths are Lipschitz continuous.

A complex stochastic process is a fundamental connection between two dimensional stochastic processes. Similarly, a complex fuzzy process is proposed and the complex differential and complex Liu integral of complex Liu process are studied in this paper. Especially, some properties of analytic functions of complex Liu process are proved. Finally, complex differential equations are defined.

In this paper, we consider fractional Liu process. First, the membership functions, expected values and variances of arithmetic and geometric fractional Liu process are given. Then we suppose that stock price follows geometric fractional Liu process and formulate fractional Liu’s stock model. Based on this model, European option pricing formulas are obtained.

Uncertain calculus deals with the integral and differential of some uncertain processes. So far, uncertain integrals have been defined with respect to Liu process, renewal process, finite variation process, and multiple Liu processes. This paper presents an uncertain integral of a matrix of uncertain processes with respect to multi-dimensional Liu process, and verifies its linearity property. T...