Exact joint laws associated with spectrally negative Levy processes and applications to insurance risk theory

TL;DR

This paper derives exact joint laws for spectrally negative Levy processes and applies these results to insurance risk theory, providing explicit formulas for key risk measures and extending classical results like Dickson's formula.

Contribution

It introduces explicit joint law formulas for spectrally negative Levy processes and applies them to derive new expressions for risk measures in insurance theory.

Findings

Explicit joint laws for passage times, overshoots, undershoots, minima, maxima, and durations.

New formulas for generalized expected discounted penalty functions.

Extended Dickson's formula for insurance risk analysis.

Abstract

We consider the spectrally negative Levy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Further, a new expression for the generalized Dickson's formula is provided.