Dynamic Systems Involving Human Uncertainty and Probability Distribution With Ambiguity: Part ⅰ. Well-Posedness, Stability and Numerical Method

The operation of a system in the real world is inevitably influenced by various indeterminate factors. In many cases, a hybrid system always exists both human uncertainty and probability distribution with ambiguity. To model this complex phenomenon, uncertain-sublinear (U-S) chance theory offers an effective mathematical method by introducing the concept of uncertain random variable under U-S chance spaces, and it has proved significant effectiveness in handling issues related to static systems. To model the evolution of this type of uncertain stochastic dynamic system, this paper introduces an uncertain stochastic differential equation driven by canonical Liu process and G-Brownian motion (LG-USDE). Firstly, we construct the method of uncertain stochastic integral (named as G-Itô-Liu integral) and present a G-Itô-Liu formula under U-S chance framework. Next, the well-posedness (existence and uniqueness of solution) of the uncertain stochastic dynamic system is proved under the linear growth condition and standard Lipschitz condition. Additionally, we propose three different concepts of stability related to this dynamic system. Finally, we apply the Euler-Maruyama numerical method to solve the dynamic system and prove that the numerical approximate solutions produced by this method converge to the exact solution.