Monotonicity and non-monotonicity of domains of stochastic integral operators

Abstract

A Lévy process on $R^d$ with distribution $μ$ at time 1 is denoted by $X^{(μ)}=\{X_t^{(μ)}\}$. If the improper stochastic integral $\int_0^{\infty-} f(s)dX_s^{(μ)}$ of $f$ with respect to $X^{(μ)}$ is definable, its distribution is denoted by $Φ_f(μ)$. The class of all infinitely divisible distributions $μ$ on $R^d$ such that $Φ_f(μ)$ is definable is denoted by $D(Φ_f)$. The class $D(Φ_f)$, its two extensions $D_c(Φ_f)$ and $D_e(Φ_f)$ (compensated and essential), and its restriction $D^0(Φ_f)$ (absolutely definable) are studied. It is shown that $D_e(Φ_f)$ is monotonic with respect to $f$, which means that $|f_2|\leq |f_1|$ implies $D_e(Φ_{f_1})\subset D_e(Φ_{f_2})$. Further, $D^0(Φ_f)$ is monotonic with respect to $f$ but neither $D(Φ_f)$ nor $D_c(Φ_f)$ is monotonic with respect to $f$. Furthermore, there exist $μ$, $f_1$, and $f_2$ such that $0\leq f_2\leq f_1$, $μ\in D(Φ_{f_1})$, and $μ\not\in D(Φ_{f_2})$. An explicit example for this is related to some properties of a class of martingale Lévy processes.