Harnack Inequality For The Solution Of A Class Of Stochastic Diferential Equations With Jumps

In this paper§we deal with the issue of the two Harnack inequalities of the semigroup associatedto the solutions of stochastic diferential equations driven by Poisson point process with jumps.Under some reasonable conditions§we establish two Harnack inequalities by using the couplingargument and Girsanov theorem§along with Ho¨lder’s inequality!Gronwall’s inequality!Young’sinequality and Ito’s formula. In addition, we give some simple examples for application.we organize the thesis as follows:In chapter1§we give the preliminary knowledge in this paper§including poisson pointprocess§definition of two kinds of random integral and main lemmasμGirsanov’s theorem andthe existence theorem of strong solutions.In chapter2§we study the problem of the Harnack inequalities and the Log-Harnackinequalities for the semigroups of the strong solution of such equation with lipschitz coefcients.We establish the relationship between two kinds of semigroupwe. Moreover, we obtain two in-equalities such as Harnack and Log-Harnack by using the coupling argument and Girsanov’stheorem§together with Ito’s formula. Finally§as an application§we solve the fast implemen-tation problems of two types of inequality.In chapter3§through the Skorohod theorem and Yamada-Watanabe theorem§we verifythe existence and uniqueness of non-explosive strong solution of stochastic diferential equationswith non-lipschitz coefcients. Furthermore, we study the problem of the Harnack inequalitiesand the Log-Harnack inequalities for the semigroups of the strong solution. First of all§undersuch conditions§the relationship between the two types of semigroup equation isn’t changed. Secondly§similar to the second chapter§we obtain the Harnack inequalities and the Log-Harnack inequalities. Finally§as application§the strong Feller property and upper bound ofheat kernel estimate are provided in terms of the corresponding semigroup.In chapter4§we give a brief summary of the thesis, and provide the direction for us to beimprove.