In the foundation of uncertainty theory, uncertain stock model has been put forward to portray the price fluctuation of stocks in a market with uncertain information. In this paper, the model is depicted by uncertain differential equations involved by a Liu process that is a sequence of uncertain variables varying with time. According to this model, we mainly investigate the formulas to price the American barrier option for rights of buying or selling the stock with a set price in the uncertain financial market. Then, four new types of concepts are introduced that are, respectively, American call options, including both up-and-in and down-and-out, and American put options, including both down-and-in and up-and-out. Moreover, several formulas are derived for giving the price of the corresponding four types of options. At the same time, some examples are given.

Wave equation is a commonly used tool for describing various kinds of wave phenomena in nature such as sound wave, water wave, electromagnetic wave and string vibration. It is a second-order partial differential equation and describe the wave propagation without noises. However, real world is filled with noises everywhere. So deterministic wave equation is not enough to model some problems with additive noises. As a remedy method, stochastic wave equation driven by Wiener process was presented where the noise is considered random and modeled by using Wiener process. Except for randomness, uncertainty associated belief degrees is another different type of indeterministic phenomenon. For modeling the wave phenomena with uncertain noises, this paper aims at deriving an uncertain wave equation driven by Liu process, which is a type of partial differential equation. Here, Liu process is a Lipschitz continuous uncertain process with stationary and independent increments. Then, we prove the existence and uniqueness of the solution of an uncertain wave equation. Additionally, we give the inverse uncertainty distribution of a solution of an uncertain wave equation.