Inequalities For Two Classes Of Convex Stochastic Processes

Stochastic process is an important research field of probability theory.The characterization of some random phenomena needs to be studied by stochastic process.With the development of science and technology,stochastic process theory is widely used in physics,biology,economy,management,engineering technology and many other fields.At the same time,the application requirements of these fields also promote the development of stochastic process theory.In recent years,many scholars have studied the properties,inequalities and applications of stochastic processes whose orbits are convex functions,and obtained a series of results.This paper mainly studies two types of convex stochastic processes and their inequalities.In the sense of mean square continuous,mean square differentiable and mean square integrable,by analogy with convex function,we improve and generalize the existing results of convex stochastic process and strongly convex stochastic process.This paper is divided into three parts.The first part introduces the development background,research status of convex stochastic processes,and some basic knowledge,including the related conceptual properties and inequalities of convex stochastic process and strongly convex stochastic process.In the second part,we mainly study the inequality and application of convex stochastic process.The classical Hermite-Hadamard inequality of convex stochastic process is improved.At the same time,we extend a version of convex stochastic process on mean square stochastic pseudo-integral by combining stochastic integral with pseudo-integral.In the third part,strongly convex stochastic process is mainly studied.Firstly,the strong convexity characterization of stochastic process under Hermite-Hadamard condition is proved by using Fubini’s theorem.Then we obtain the Jensen type inequality for strongly convex stochastic process and the generalized form of Hermite-Hadamard inequality for strongly Jensen convex stochastic process.Finally,some Ostrowski type inequalities are established and analyzed by using H(?)lder and Power-mean inequalities.