Monotonicity and non-monotonicity of domains of stochastic integral operators

A L\'evy process on $R^d$ with distribution $\mu$ at time 1 is denoted by $X^{(\mu)}=\{X_t^{(\mu)}\}$. If the improper stochastic integral $\int_0^{\infty-} f(s)dX_s^{(\mu)}$ 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 $|f_2|\leq |f_1|$ 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 $0\leq f_2\leq f_1$, $\mu\in D(\Phi_{f_1})$, and $\mu\not\in D(\Phi_{f_2})$. An explicit example for this is related to some properties of a class of martingale L\'evy processes.