A Transformation Of Lévy Processes And Its Properties

The one-to-one correspondence between Dirichlet form and Processes providesus a convenient and applicable mathematic instrument to study some propertiesof the process.Le′vy processes are essentially stochastic processes with stationaryand independent increments.They include a number of very important processes asspecial cases,including Brownian motion,the Poisson process,stable process.Recentyears have seen a great deal of new theoretical development as well as applicationsin such diverse areas as mathematical finance and quantum field theory.Girsanovtransformation is an essential transformation,we can know more about the propertiesof Le′vy processes and some theoretical knowledge of Girsanov transformation.In this paper we mainly study the properties of some types of Le′vy pro-cesses after Girsanov transformation,such as transience property,recurrence proper-ty,translation invariant property.We mainly study Brownian motion and the Poissonprocess.In chapter1,we mainly describe the basic definitions and the notations associat-ed with Dirichlet form,Le′vy processes and Girsanov transformation.In chapter2,wefirstly give the expression of the Dirichlet form which is corresponded to complexPoisson process after Girsanov transformation.Then we prove that when u is bound-ed,the new processes which are got by Girsanov transformation of Le′vy processesin three dimension or more are transience.We also prove that the new processeswhich are got by Girsanov transformation of Brownian motion and complex Poissonprocess in one dimension do not satisfy the translation invariant property.At last,wediscuss the conditions of the new processes which are got by Girsanov transforma-tion of Le′vy processes to be recurrent.