Additive processes and stochastic integrals

Abstract

Stochastic integrals of nonrandom (l×d)-matrix-valued functions or nonrandom real-valued functions with respect to an additive process X on ℝd 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 ℝd and some selfsimilar additive processes. In the last part of the paper, existence conditions for improper stochastic integrals ∫0∞- f(s)dXs and their extensions are given; the cases where f(s) sβe -csα and where f(s) is such that s = ∫f(s)∞u-2e-udu are analyzed. © 2006 University of Illinois.