On random measures, unordered sums and discontinuities of the first kind
| dc.creator | Oertel, Frank | |
| dc.date | 2006-09-14 | |
| dc.date.accessioned | 2026-07-07T07:24:49Z | |
| dc.date.available | 2026-07-07T07:24:49Z | |
| dc.description | By investigating in detail discontinuities of the first kind of real-valued functions and the analysis of unordered sums, where the summands are given by values of a positive real-valued function, we develop a measure-theoretical framework which in particular allows us to describe \textit{rigorously} the representation and meaning of sums of jumps of type $\sum_{0 < s \leq t} Φ\circ | ΔX_s |$, where $X : Ω\times \R_+ \longrightarrow \R$ is a stochastic process with regulated trajectories, $t \in \R_+$ and $Φ: \R_+ \longrightarrow \R_+$ is a strictly increasing function which maps 0 to 0 (cf. Proposition \ref{prop:sum of jumps on R+ with invertible function}). Moreover, our approach enables a natural extension of the jump measure of càdlàg and adapted processes to an integer-valued random measure of optional processes with regulated trajectories which need not necessarily to be right- or left-continuous (cf. Theorem \ref{thm:optional random measures}). In doing so, we provide a detailed and constructive proof of the fact that the set of all discontinuities of the first kind of a given real-valued function on $\R$ is at most countable (cf. Lemma \ref{lemma:right limits and left limits}, Theorem \ref{thm:at most countably many jumps on compact intervals} and Theorem \ref{thm:at most countably many jumps on R+}). By using the powerful analysis of unordered sums, we hope that our contributions fill an existing gap in the literature, since neither a detailed proof of (the frequently used) Theorem \ref{thm:at most countably many jumps on compact intervals} nor a precise definition of sums of jumps seems to be available yet. | |
| dc.identifier | https://arxiv.org/abs/math/0609395 | |
| dc.identifier | http://arxiv.org/abs/math/0609395 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/116513 | |
| dc.subject | Probability | |
| dc.subject | Functional Analysis | |
| dc.subject | General Mathematics | |
| dc.subject | 28A05, 40G99, 60G05, 60G57 | |
| dc.title | On random measures, unordered sums and discontinuities of the first kind | |
| dc.type | text |