Parisian ruin of self-similar Gaussian risk processes

In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times

Parisian Ruin for Insurer and Reinsurer under Quata-Share Treaty

In this contribution we study asymptotics of the simultaneous Parisian ruin probability of a two-dimensional fractional Brownian motion risk process. This risk process models the surplus processes of an insurance and a reinsurance companies, where the net loss is distributed between them in given proportions. We also propose an approach for simulation of Pickands and Piterbarg constants appearing in the asymptotics of the ruin probability

Risk Management of Life Insurance Contracts with Interest Rate and Return Guarantees and an Analysis of Chapter 11 Bankruptcy Procedure

Equity-linked life insurance contracts are an example of theinterplay between insurance and finance. By considering some specific equity-linked life insurance contracts, this thesis mainly studies risk management methods, i.e., the insurance company hedges its exposure to risk by using certain conventional hedging criteria for an incomplete market, like risk-minimizing, quantile and efficient hedging. In addition to the untradable insurance risk, different sources of incompleteness are analyzed, such as the incompleteness from trading restrictions or from model misspecification. Furthermore, this thesis provides an insight to the net loss of the insurer, given that the insurer trades in the financial market according to risk-minimizing hedging criterion. However, under no circumstances, the untradable insurance risk can be hedged completely, i.e., there always exists a positive probability that the considered insurance company defaults. In this context, the chapter before last is designed to consider the insurance company as an aggregate and to analyze the market value of this company if default risk and different bankruptcy procedures are taken into consideration. In this analysis, the mortality risk is neglected and no specific contracts are studied

Parisian ruin over a finite-time horizon

For a risk process $R_u(t)=u+ct-X(t), t\ge 0$, where $u\ge 0$ is the initial capital, $c>0$ is the premium rate and $X(t),t\ge 0$ is an aggregate claim process, we investigate the probability of the Parisian ruin $$\mathcal{P}_S(u,T_u)=\mathbb{P}\{\inf_{t\in[0,S]} \sup_{s\in[t,t+T_u]} R_u(s)<0\},$$ with a given positive constant $S$ and a positive measurable function $T_u$. We derive asymptotic expansion of $\mathcal{P}_S(u,T_u)$, as $u\to\infty$, for the aggregate claim process $X$ modeled by Gaussian processes. As a by-product, we derive the exact tail asymptotics of the infimum of a standard Brownian motion with drift over a finite-time interval.Comment: 2

The time of ultimate recovery in Gaussian risk model

We analyze the distance $\mathcal{R}_T(u)$ between the first and the last passage time of $\{X(t)-ct:t\in [0,T]\}$ at level $u$ in time horizon $T\in(0,\infty]$, where $X$ is a centered Gaussian process with stationary increments and $c\in\mathbb{R}$, given that the first passage time occurred before $T$. Under some tractable assumptions on $X$, we find $\Delta(u)$ and $G(x)$ such that $$\lim_{u\to\infty}\mathbb{P}\left(\mathcal{R}_T(u)>\Delta(u)x\right)=G(x),$$ for $x\geq 0$. We distinguish two scenarios: $T<\infty$ and $T=\infty$, that lead to qualitatively different asymptotics. The obtained results provide exact asymptotics of the ultimate recovery time after the ruin in Gaussian risk model.Comment: 21 page