Monotonicity and non-monotonicity of domains of stochastic integral operators

Abstract

A Lévy process on $R^d$ with distribution $\mu$ at time 1 is denoted by $X^{(\mu )}=\{X_t^{(\mu )}\}$. If the improper stochastic integral ${\int }_0^{\infty -}f(s)d{X}_s^{(\mu )}$ of $f$ with respect to $X^{(\mu )}$ is definable, its distribution is denoted by ${\Phi }_f(\mu )$. The class of all infinitely divisible distributions $\mu$ on $R^d$ such that ${\Phi }_f(\mu )$ is definable is denoted by $D({\Phi }_f)$. The class $D({\Phi }_f)$, its two extensions $D_c({\Phi }_f)$ and $D_e({\Phi }_f)$ (compensated and essential), and its restriction $D^0({\Phi }_f)$ (absolutely definable) are studied. It is shown that $D_e({\Phi }_f)$ is monotonic with respect to $f$, which means that $|f_2|\le |f_1|$ implies $D_e({\Phi }_{f_1})\subset D_e({\Phi }_{f_2})$. Further, $D^0({\Phi }_f)$ is monotonic with respect to $f$ but neither $D({\Phi }_f)$ nor $D_c({\Phi }_f)$ is monotonic with respect to $f$. Furthermore, there exist $\mu$, $f_1$, and $f_2$ such that $0\le f_2\le f_1$, $\mu \in D({\Phi }_{f_1})$, and $\mu \notin D({\Phi }_{f_2})$. An explicit example for this is related to some properties of a class of martingale Lévy processes.

17 pages

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• Transformations of infinitely divisible distributions via improper stochastic integrals