Additive processes and stochastic integrals

Abstract

Stochastic integrals of nonrandom $(l\times d)$-matrix-valued functions or nonrandom real-valued functions with respect to an additive process $X$ on $\mathbb{R}^d$ are studied. Here an additive process means a stochastic process with independent increments, stochastically continuous, starting at the origin, and having cadlag paths. A necessary and sufficient condition for local integrability of matrix-valued functions is given in terms of the Lévy--Khintchine triplets of a factoring of $X$. For real-valued functions explicit expressions of the condition are presented for all semistable Lévy processes on $\mathbb{R}^d$ and some selfsimilar additive processes. In the last part of the paper, existence conditions for improper stochastic integrals $\int_0^{\infty-}f(s)dX_s$ and their extensions are given; the cases where $f(s)\asymp s^{\beta} e^{-cs^{\alpha}}$ and where $f(s)$ is such that $s=\int_{f(s)}^{\infty} u^{-2} e^{-u} du$ are analyzed.