Two Path Properties On The Volterra Processes

Volterra Processes are mathematical models proposed by Volterra,a famous Italian mathematician and physicist.An integral of kernel function F(t,r)on semi-martingale X(r)is given by This plays an important role in physical kinematics,such as fractional motion,fractional Levy process.For the study of Volterra Processes,one is to make proper assumptions for semimartingales X(r),the other is to make proper assumptions for kernel function F(t,r).If the kernel function F(t,r)is a smooth variation function,L.Mydtnik and E.Neuman(2012)gave two important sample properties of Volterra Processes.and where △x(t)= X(t)-X(t-).Because L.Mydtnik and E.Neuman’s(2012)assumption of semimartingale X(r)is very ordinary,but the requirement of kernel function is very high.So the pur-pose is to generalize the kernel function F(t,r)in order to get the same conclu-sion.In the first chapter,we introduce the background,significance,history,current situation and primarily conclusion about Volterra Processes.The second chapter is mainly about the basic definition and properties need-ed later.The third chapter is the core one.We redefine the kernel function of Volterra Processes.It is proved that the kernel function in reference[l]is included in the kernel function defined in this paper and the sample of orbital properties remain valid.The forth chapter conclude the paper,including all the conclusions.