Integrability conditions for space–time stochastic integrals: Theory and applications

Abstract

We derive explicit integrability conditions for stochastic integrals taken over time and space driven by a random measure. Our main tool is a canonical decomposition of a random measure which extends the results from the purely temporal case. We show that the characteristics of this decomposition can be chosen as predictable strict random measures, and we compute the characteristics of the stochastic integral process. We apply our conditions to a variety of examples, in particular to ambit processes, which represent a rich model class.

Selfdecomposable Fields

“…Now we prove that ν 0 as in (34) is continuous at the empty set. Let (A 0 n ) n≥1 ⊂ B 0 , then there are u n ∈ U and A n ∈ B(R u n ), such that A 0 n = π −1 u n (A n \ 0 u n ) for any n ∈ N. Without loss of generality, we may assume that u n ⊂ u n+1 , otherwise put u n = n i= u n and use that π −1 u n (A n \ 0 u n ) = π −1 u n (π −1 u n u n (A n \ 0 u n )).…”

“…Finally, L denotes a Lévy basis (i.e., an independently scattered and infinitely divisible random measure). For aspects of the theory and applications of ambit processes and fields, see [8,10,12,14,15,23,34,39,52] and [55].…”

Random field solutions to linear SPDEs driven by symmetric pure jump Lévy space-time white noises

“…In a spatio-temporal framework, we refer the reader to [4] for integrability conditions for nondeterministic integrands. A generalized stochastic process (or generalized random field) U is a linear map from a space of test functions DpR d q into L 0 pΩq (the space of a.s. finite random variables with the metric of convergence in probability).…”

Some Recent Developments in Ambit Stochastics

“…Below we briefly discuss how the metatime change is incorporated in the framework of [25]. Suppose that L = {L(A) | A ∈ B b (R d+1 } is a real-valued, homogeneous Lévy basis with associated infinitely divisible law…”

“…Finally, suppose that σ (x, t) is predictable and that L T has no fixed times of discontinuity (see [25]). By rewriting the stochastic integral in the right-hand side of (3) as…”