Transformations of infinitely divisible distributions via improper stochastic integrals

Abstract

Let $X^{(\mu)}(ds)$ be an $\mathbb{R}^d$-valued homogeneous independently scattered random measure over $\mathbb{R}$ having $\mu$ as the distribution of $X^{(\mu)}((t,t+1])$. Let $f(s)$ be a nonrandom measurable function on an open interval $(a,b)$ where $-\infty\leqslant a<b\leqslant\infty$. The improper stochastic integral $\int_{a+}^{b-} f(s)X^{(\mu)}(ds)$ is studied. Its distribution $\Phi_f(\mu)$ defines a mapping from $\mu$ to an infinitely divisible distribution on $\mathbb{R}^d$. Three modifications (compensated, essential, and symmetrized) and absolute definability are considered. After their domains are characterized, necessary and sufficient conditions for the domains to be very large (or very small) in various senses are given. The concept of the dual in the class of purely non-Gaussian infinitely divisible distributions on $\mathbb{R}^d$ is introduced and employed in studying some examples. The $\tau$-measure $\tau$ of function $f$ is introduced and whether $\tau$ determines $\Phi_f$ is discussed. Related transformations of L\'evy measures are also studied.