Functional Laws for Trimmed Levy Processes

TL;DR

This paper develops a theoretical framework for analyzing how different trimming methods affect Levy processes, establishing conditions for continuity and applying results to limit theorems and reinsurance ruin problems.

Contribution

It introduces conditions for the continuity of trimming operators on Levy processes and applies the theory to limit theorems and reinsurance risk analysis.

Findings

Continuity of trimming operators depends on no ties among largest jumps.

Limit theorems for trimmed Levy processes are established.

Applications to reinsurance ruin time problems are demonstrated.

Abstract

Two different ways of trimming the sample path of a stochastic process in D[0, 1]: global ("trim as you go") trimming and record time ("lookback") trimming are analysed to find conditions for the corresponding operators to be continuous with respect to the (strong) J1-topology. A key condition is that there should be no ties among the largest ordered jumps of the limit process. As an application of the theory, via the continuous mapping theorem we prove limit theorems for trimmed Levy processes, using the functional convergence of the underlying process to a stable process. The results are applied to a reinsurance ruin time problem.