Citation Link: https://doi.org/10.1007/s10959-023-01308-y
The Time-Dependent Symbol of a Non-homogeneous Itô Process and Corresponding Maximal Inequalities
Publication Type
Article
Subjects
Mathematics
Homogeneity (statistics)
Markov process
Homogeneous
Symbol (formal)
Stochastic process
Exponential function
Time reversibility
Probabilistic logic
Discrete mathematics
Combinatorics
Mathematical analysis
Statistics
Markov property
Markov model
Computer science
Date Issued
2023
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.
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