{"title":"The Time-Dependent Symbol of a Non-homogeneous Itô Process and Corresponding Maximal Inequalities","authors":"","doi":"10.1007/s10959-023-01308-y","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>The probabilistic symbol is defined as the right-hand side derivative at time zero of the characteristic functions corresponding to the one-dimensional marginals of a time-homogeneous stochastic process. As described in various contributions to this topic, the symbol contains crucial information concerning the process. When leaving time-homogeneity behind, a modification of the symbol by inserting a time component is needed. In the present article, we show the existence of such a time-dependent symbol for non-homogeneous Itô processes. Moreover, for this class of processes, we derive maximal inequalities which we apply to generalize the Blumenthal–Getoor indices to the non-homogeneous case. These are utilized to derive several properties regarding the paths of the process, including the asymptotic behavior of the sample paths, the existence of exponential moments and the finiteness of <em>p</em>-variationa. In contrast to many situations where non-homogeneous Markov processes are involved, the space-time process <em>cannot</em> be utilized when considering maximal inequalities.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10959-023-01308-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The probabilistic symbol is defined as the right-hand side derivative at time zero of the characteristic functions corresponding to the one-dimensional marginals of a time-homogeneous stochastic process. As described in various contributions to this topic, the symbol contains crucial information concerning the process. When leaving time-homogeneity behind, a modification of the symbol by inserting a time component is needed. In the present article, we show the existence of such a time-dependent symbol for non-homogeneous Itô processes. Moreover, for this class of processes, we derive maximal inequalities which we apply to generalize the Blumenthal–Getoor indices to the non-homogeneous case. These are utilized to derive several properties regarding the paths of the process, including the asymptotic behavior of the sample paths, the existence of exponential moments and the finiteness of p-variationa. In contrast to many situations where non-homogeneous Markov processes are involved, the space-time process cannot be utilized when considering maximal inequalities.
摘要 概率符号被定义为与时间均质随机过程的一维边际相对应的特征函数的零时右导数。正如本课题的多篇论文所述，概率符号包含有关过程的重要信息。如果不考虑时间均匀性，就需要通过插入时间分量来修改符号。在本文中，我们证明了对于非均质伊托过程，存在这样一种随时间变化的符号。此外，对于这一类过程，我们还推导出了最大不等式，并将其应用于将布卢门塔尔-盖托指数推广到非均质情况。我们利用这些不等式推导出有关过程路径的若干属性，包括样本路径的渐近行为、指数矩的存在性和 p 变量的有限性a。与许多涉及非均相马尔可夫过程的情况不同，在考虑最大不等式时，不能利用时空过程。
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