Uncertain fractional differential equation (UFDE) is an useful tool for studying complex systems in uncertain environments. In this paper, we study the extreme value theorems of the solution to Caputo-Hadamard UFDEs and applications. A numerical algorithm for solving the numerical solution of a nonlinear Caputo-Hadamard UFDE is presented, the feasibility of the numerical algorithm is validated by numerical experiments. The extreme value theorems are applied to the financial markets, and the pricing formulas of the American option based on the new uncertain stock model are given. Considering the properties of the American option pricing, the algorithms for computing the expected value of the extreme values based on the Simpson’s rule are designed. Finally, the price fluctuation of the American option is illustrated by numerical experiments.

The existing researches have shown that internal degradation processes and external shocks may simultaneously interfere with the reliability of dynamic systems in uncertain random environments. Assume that failure processes are dependent, that is, shocks may accelerate degradation processes by additional degradations. Wear and additional degradations are uncertain, while shocks are considered to be random. As a natural consideration, it is necessary to maximize the reliability of multi-stage uncertain random systems. In this paper, a maximizing reliability problem is presented, and recurrence equations are provided by Bellman's principle. These are successfully applied to maximize reliability index in two special cases with linear and quadratic state equations. In addition, two effective algorithms are developed to achieve optimal solutions. Finally, a numerical example of a metallized film pulse capacitor is proposed which aims to indicate that optimization method is beneficial to maximize the reliability of multi-stage systems.