Rough Differential Equations in Rough Function Model

Abstract

To describe rough differential equations in rough function model, a series of new concepts of rough differential equations are proposed by generalizing theory of difference equations and ordinary differential equations into rough function model. Based on these basic concepts, fundamental properties of linear rough differential equations solutions are discussed and four principles of rough superposition are put forward, which lay foundations for solving methods of rough differential equations. Several insufficiencies in original solving methods of rough differential equations are pointed out and improved respectively. Three types of typical rough differential equations are given. According to characteristics of different types of rough differential equations, corresponding solving methods and solving steps as well as different forms of solutions are discussed. Finally, some general remarks are made on other types of rough differential equations.