On random measures, unordered sums and discontinuities of the first kind

Details On random measures, unordered sums and discontinuities of the first kind On random measures, unordered sums and discontinuities of the first kind

2006-09-14

arxiv.org/abs/math/0609395v1

Mathematics

Probability

Scientific paper

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 $\sum_{0 < s \leq t} \Phi \circ | \Delta X_s |$, where $X : \Omega \times \R_+ \longrightarrow \R$ is a stochastic process with regulated trajectories, $t \in \R_+$ and $\Phi : \R_+ \longrightarrow \R_+$ 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 c\`{a}dl\`{a}g 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.

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