Multivariate stochastic integrals with respect to independently scattered random measures on $\delta$-rings

Abstract

In this paper we construct general vector-valued infinitely-divisible independently scattered random measures with values in R m and their corresponding stochastic integrals. Moreover, given such a random measure, the class of all integrable matrix-valued deterministic functions is characterized in terms of certain characteristics of the random measure. In addition a general construction principle is presented.

Multi operator-stable random measures and fields

“…This shows that sup s∈S c(s) < ∞. Overall Theorem 3.1 in [10] implies the existence of some suitable probability space (Ω, A, P) with M as asserted.…”

“…Using the general theory of ISRMs developed in [10], we now construct a large class of multivariate independently scattered random measures which are first of all infinitely-divisible. At the same time we refer the reader to [8], [11] and [14] concerning more details about operator-stable distributions which will play a crucial role throughout the paper.…”

“…Turning over to independently scattered random measures recall the notation and results of section 3 in [10]. Consider the set S := {A ∈ Σ : ν(A) < ∞} and observe that this is a so called δ-ring on S whose generated σ-algebra equals Σ (see [10] [14] this means that the characteristic function of M(A) is given by…”

“…For a comprehensive introduction see [14]. Based on the recently developed theory of multivariate independently scattered random measures (short: ISRMs) and their integrals (see [10]), moving-average and harmonizable representations of (E, D)-operator-self-similar random fields with operator-stable marginals are presented in [11]. Our main feature of SαS ISRMs or more generally of operator-stable ISRMs with exponent B (a m × m-matrix) is that they are homogeneous in the time variable t ∈ R d , that is α or B are constant and do not depend on t. Motivated by various applications, in [6] scalarvalued so called multi-stable random measures were constructed.…”

“…An important feature of multi-stable random measures and multi-stable processes is that they behave locally like α(t)-stable random measures and α(t)-stable processes close to time t. That means that the local scaling limits are α(t)-stable, but where the stability index varies with t. See Theorem 2.7 and Theorem 3.2 in [6]. The purpose of this paper is to generalize some of the results in [6] to the operator-stable setting, by allowing the operator-stable exponent B(t) to depend on the time variable t. This paper is organized as follows: In Section 2, by applying the general theory of multivariate ISRMs in [10], multi operator-stable random measures and integrals are constructed. The set of integrable functions is characterized in terms of a quasi-norm.…”