Riemann Integration, Stochastic Calculus, and Shift Operators on Time Scales

Abstract

AbstractThis chapter mainly introduces basic knowledge of calculus on time scales. In Sect. 1.1, concepts and fundamental properties of Riemann delta and nabla integration on time scales are introduced including some basic results of Riemann integral and fundamental theorems of calculus. In Sect. 1.2, stochastic calculus and some basic results of stochastic dynamic equations on time scales are provided. Section 1.3 is mainly devoted to introducing the concept of shift operators on time scales by which a new concept of periodicity is introduced; shift operator plays an important role in discussing shift invariance of time scales in later chapters. In Sect. 1.4, momentous hybrid derivatives called combined derivatives or diamond-α derivatives which can strictly include delta and nabla derivatives are introduced, and some basic properties of combined dynamic derivatives and integrations are established.