The Local Time of the Classical Risk Process

TL;DR

This paper derives an explicit expression for the local time of the classical risk process, linking it to occupational measure density, and analyzes specific time intervals where the process exhibits certain properties.

Contribution

It provides a novel explicit formula for the local time of the risk process and connects it with occupational measure density, including an application to mean time analysis.

Findings

Explicit expression for local time of the risk process

Connection between local time and occupational measure density

Analysis of mean times where the process satisfies certain inequalities

Abstract

In this paper we give an explicit expression for the local time of the classical risk process and associate it with the density of an occupational measure. To do so, we approximate the local time by a suitable sequence of absolutely continuous random fields. Also, as an application, we analyze the mean of the times $s \in [0,T]$ such that $0\leq X_{s} \leq X_{s+\epsilon} $ for some given $\epsilon>0$.