Parisian types of ruin probabilities for a class of dependent risk-reserve processes

ABSTRACT

For a rather general class of risk-reserve processes, we provide an exact method for calculating different kinds of ruin probabilities, with particular emphasis on variations over Parisian type of ruin. The risk-reserve processes under consideration have, in general, dependent phase-type distributed claim sizes and inter-arrivals times, whereas the movement between claims can either be linear or follow a Brownian motion with linear drift. For such processes, we provide explicit formulae for classical, Parisian and cumulative Parisian types of ruin (for both finite and infinite time horizons) when the clocks are phase-type distributed. An erlangization scheme provides an efficient algorithmic methods for calculating the aforementioned ruin probabilities with deterministic clocks. Special attention is drawn to the construction of specific dependency structures, and we provide a number of numerical examples to study its effect on probabilities.

KEYWORDS:

• Sparre-Andersen
• Brownian motion
• ruin probability
• (cumulative) Parisian ruin
• fluid flow
• order statistics
• dependency
• Baker copula
• phase-type distributions
• erlangization
• Lévy process

JEL CLASSIFICATION:

• G22

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

OP is supported by the CONACYT PhD Scholarship No. 1410763 sponsored by the Mexican Government.