The Study Of Quadratic Harness

In this paper,we investigate the general properties of the class of Markov pro-cesses,which is Markov process with polynomial regression(MPR process).We give the definition of a family of orthogonal polynomials martingales {Mn(Xt,t)} satisfy the following 3-term recurrence:XtMn(t)=αn+1(t)=Mn+1(t)+βn(t)Mn(t)+γn-1(t)Mn-1(t)Under the condition,we get some general properties of {Mn(Xt,t)}.Regarding the research of harness,R.Mansuy and M.Yor analyze harnesses on R+,Therefore,we strictly give the definition of the harness and the quadratic har-ness,and then,we use the orthogonal polynomial martingale Mn(Xt,t)to study MPR process.We get the MPR process not only a harness,but also a quadrat-ic harness.Then,we show the specific expression of its parameters to guarantee that the MPR process are harness and quadratic harness.And then we also study the necessary and sufficient conditions under the MPR process are harness and quadratic harness..we show that {Mi(Xt,t)/m(t)},i=1,2 are reversed martin-gales.Finally,we also constructed several typical examples to verify the accuracy of the conclusions which we get.