The martingale problem for generators of variable order

我们研究了马尔可夫工艺过渡概率的平滑密度的存在，用跳跃来解决D-dimensional stochastic integro-differential equation的问题：dx D2t D2 = a D20 D2（x ― D2 t ― D2）dt +模糊D 6 m（±）k=1 ― D 6a ― D2 k ― D2（xイイD2tイエD2）·dイイD1kイエD1（t） +·b（xイイD2t-イエD2，\）J（dtd\），where\（t）\（イイD1kイエD1（t）） is an m-dimensional Brownian motion and J（dtd\） is a Poisson random measure with E[J（dtd\）] =\（d\）dt。平滑密度的存在是相等于平行伪伪差运算符的hypoellipticity of the parabolic pseudo-differential operator（/）（A型D20型D2 +型D 71（±）2型D 7型D 6 m（±）k=1型D 6（A-D2k-D2）-D12-D1 +-（BイイD2·イエD2 - I）π（dθ），where AイイD2k·イD2 = aイD2k·イD2（x）·イD2 and BイD2 are operators defined by BイD2·イD2（x） = φ（x + b（x，θ））.在Malliavin计算课程中研究了一个类似的问题，而Hormander的条件众所周知，被认为是一个有效的条件。我们通过Girzanov的Levy过程转换了跳跃型马尔可夫过程的变体，并通过部分集成在Cad-lag空间中提出了一种特殊必要的公式。我们还面临着一个问题，以展示与Malliavin covariance相关的特殊功能相关的拉普拉斯变换的显著变化。到目前为止，一个显著的衰退是由长时间复杂的参数提出的。我们用一种新方法展示了这一点，其中的关键lemma是一般半语的估计。我们提供了马尔可夫工艺的平滑过渡密度，在某些条件下跳跃，其中在先前的意义上比霍曼德类型条件更弱。这一方法也为普通的抛物线式差异运算符提供了一个简单的证明。

We studied on the existence of smooth densities of transition probabilities of Markov processes with jumps which are solutions to d-dimensional stochastic integro-differential equations:dxィイD2tィエD2 = aィイD20ィエD2(xィイD2tィエD2)dt + ΣィイD6m(/)k=1ィエD6aィイD2kィエD2(xィイD2tィエD2)・dβィイD1kィエD1(t) + ∫b(xィイD2t-ィエD2,θ)J(dtdθ),where β(t) = (βィイD1kィエD1(t)) is an m-dimensional Brownian motion and J(dtdθ) is a Poisson random measure with E[J(dtdθ)] = π(dθ)dt. The existence of smooth densities is equivalent to the hypoellipticity of the parabolic pseudo-differential operatorィイD7∂(/)∂tィエD7 + (AィイD20ィエD2 + ィイD71(/)2ィエD7ΣィイD6m(/)k=1ィエD6(AィイD2kィエD2)ィイD12ィエD1 + ∫(BィイD2θィエD2 - I)π(dθ)),where AィイD2kィエD2 = aィイD2kィエD2(x)・∂ィイD2xィエD2 and BィイD2θィエD2 are operators defined by BィイD2θィエD2φ(x) = φ(x + b(x,θ)).A similar problem for continuous Markov processes was studied in the course of the Malliavin calculus, and the Hormander condition is well-known as a sufficient condition for the hypoellipticity. We carried out the variation for jump type Markov processes by Girzanov transforms of Levy processes, and proved special necessary formulas of integration by parts on the cad-lag space. We also faced the problem to show the exponential decay of the Laplace transform of the distribution of a specific functional associated with the Malliavin covariance. So far, the exponential decay was proved by long complicated arguments. We showed it by a new method where the key lemma is an estimate for general semimartingales. And we proved the smoothness of transition densities of Markov processes with jumps under certain conditions which are weaker than Hormander type conditions in the previous sense. This method also gives a quite simple proof to the Hormander theorem for usual parabolic differential operators.