arXiv math/06072882006-07-12math.PR

随机积分算子域的单调性与非单调性

Monotonicity and non-monotonicity of domains of stochastic integral operators

Ken-iti Sato

备注：: 17 pages

摘要：

在时间1处具有分布$μ$的$R^d$上的Lévy过程用$X^{(μ)}=\{X_t^{(μ)}\}$表示。如果$f$关于$X^{(μ)}$的反常随机积分$\int_0^{\infty-} f(s)dX_s^{(μ)}$是可定义的，则其分布用$Φ_f(μ)$表示。所有在$R^d$上的无限可分分布$μ$使得$Φ_f(μ)$是可定义的类记为$D(Φ_f)$。研究了类$D(Φ_f)$，它的两个扩展$D_c(Φ_f)$和$D_e(Φ_f)$（补偿的和本质的），以及它的限制$D^0(Φ_f)$（绝对可定义的）.证明了$D_e(Φ_f)$相对于$f$是单调的，这意味着$|f_2|\leq |f_1|$蕴涵$D_e(Φ_{f_1})\subset D_e(Φ_{f_2})$。此外，$D^0(Φ_f)$相对于$f$是单调的，但$D(Φ_f)$和$D_c(Φ_f)$都不是相对于$f$单调的。此外，存在$μ$、$f_1$和$f_2$，使得$0\leq f_2\leq f_1$、$μ\in D(Φ_{f_1})$和$μ\not\in D(Φ_{f_2})$。一个明确的例子是关于一类鞅Lévy过程的一些性质。

英文摘要：

A Lévy process on $R^d$ with distribution $μ$ at time 1 is denoted by $X^{(μ)}=\{X_t^{(μ)}\}$. If the improper stochastic integral $\int_0^{\infty-} f(s)dX_s^{(μ)}$ of $f$ with respect to $X^{(μ)}$ is definable, its distribution is denoted by $Φ_f(μ)$. The class of all infinitely divisible distributions $μ$ on $R^d$ such that $Φ_f(μ)$ is definable is denoted by $D(Φ_f)$. The class $D(Φ_f)$, its two extensions $D_c(Φ_f)$ and $D_e(Φ_f)$ (compensated and essential), and its restriction $D^0(Φ_f)$ (absolutely definable) are studied. It is shown that $D_e(Φ_f)$ is monotonic with respect to $f$, which means that $|f_2|\leq |f_1|$ implies $D_e(Φ_{f_1})\subset D_e(Φ_{f_2})$. Further, $D^0(Φ_f)$ is monotonic with respect to $f$ but neither $D(Φ_f)$ nor $D_c(Φ_f)$ is monotonic with respect to $f$. Furthermore, there exist $μ$, $f_1$, and $f_2$ such that $0\leq f_2\leq f_1$, $μ\in D(Φ_{f_1})$, and $μ\not\in D(Φ_{f_2})$. An explicit example for this is related to some properties of a class of martingale Lévy processes.

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