Additive processes and stochastic integrals
Additive processes and stochastic integrals
复制标题
DOI：
10.1215/ijm/1258059494
复制
发表时间：
2006
影响因子：
0.6
通讯作者：
Ken-iti Sato
中科院分区：
文献类型：
--
作者：
Ken-iti Sato
研究了关于一个加性过程$X$ （$\mathbb{R}^d$）的非随机$(l\times d)$ -矩阵值函数或非随机实值函数的随机积分。这里的加性过程是指具有独立增量的随机过程，随机连续的，从原点开始，并且具有滞后路径。利用$X$分解的Levy—Khintchine三元组给出了矩阵值函数局部可积的充分必要条件。对于实值函数，给出了$\mathbb{R}^d$上所有半稳定Levy过程和一些自相似加性过程的条件的显式表达式。最后给出了反常随机积分$\int_0^{\infty-}f(s)dX_s$的存在条件及其推广；分析$f(s)\asymp s^{\beta} e^{-cs^{\alpha}}$和$f(s)$与$s=\int_{f(s)}^{\infty} u^{-2} e^{-u} du$相似的情况。
Stochastic integrals of nonrandom $(l\times d)$-matrix-valued functions or nonrandom real-valued functions with respect to an additive process $X$ on $\mathbb{R}^d$ are studied. Here an additive process means a stochastic process with independent increments, stochastically continuous, starting at the origin, and having cadlag paths. A necessary and sufficient condition for local integrability of matrix-valued functions is given in terms of the Levy--Khintchine triplets of a factoring of $X$. For real-valued functions explicit expressions of the condition are presented for all semistable Levy processes on $\mathbb{R}^d$ and some selfsimilar additive processes. In the last part of the paper, existence conditions for improper stochastic integrals $\int_0^{\infty-}f(s)dX_s$ and their extensions are given; the cases where $f(s)\asymp s^{\beta} e^{-cs^{\alpha}}$ and where $f(s)$ is such that $s=\int_{f(s)}^{\infty} u^{-2} e^{-u} du$ are analyzed.