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Register a new userRuin Probabilities for Risk Process in a Regime Switching Environment
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Authors
Zbigniew Palmowski
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In this paper we give few expressions and asymptotics of ruin probabilities for a Markov modulated risk process for various regimes of a time horizon, initial reserves and a claim size distribution. We also consider f
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This paper develops asymptotics and approximations for ruin probabilities in a multivariate risk setting. We consider a model in which the individual reserve processes are driven by a common Markovian environmental process. We subsequently consider a regime in which the claim arrival intensity and transition rates of the environmental process are jointly sped up, and one in which there is (with overwhelming probability) maximally one transition of the environmental process in the time interval considered. The approximations are extensively tested in a series of numerical experiments.
Our work aims to study the tail behaviour of weighted sums of the form $sum_{i=1}^{infty} X_{i} prod_{j=1}^{i}Y_{j}$, where $(X_{i}, Y_{i})$ are independent and identically distributed, with common joint distribution bivariate Sarmanov. Such quantities naturally arise in financial risk models. Each $X_{i}$ has a regularly varying tail. With sufficient conditions similar to those used by Denisov and Zwart (2007) imposed on these two sequences, and with certain suitably summable bounds similar to those proposed by Hazra and Maulik (2012), we explore the tail distribution of the random variable $sup_{n geq 1}sum_{i=1}^{n} X_i prod_{j=1}^{i}Y_{j}$. The sufficient conditions used will relax the moment conditions on the ${Y_{i}}$ sequence.
The classical Cramer-Lundberg risk process models the ruin probability of an insurance company experiencing an incoming cash flow - the premium income, and an outgoing cash flow - the claims. From a systems viewpoint, the web of insurance agents and risk objects can be represented by a bipartite network. In such a bipartite network setting, it has been shown that joint ruin of a group of agents may be avoided even if individual agents would experience ruin in the classical Cramer-Lundberg model. This paper describes and examines a phase transition phenomenon for these ruin probabilities.
Based on a discrete version of the Pollaczeck-Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber-Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.
The study deals with the ruin problem when an insurance company having two business branches, life insurance and non-life insurance, invests its reserve into a risky asset with the price dynamics given by a geometric Brownian motion. We prove a result on smoothness of the ruin probability as a function of the initial capital and obtain for it an integro-differential equation understood in the classical sense. For the case of exponentially distributed jumps we show that the survival probability is a solution of an ordinary differential equation of the 4th order. Asymptotic analysis of the latter leads to the conclusion that the ruin probability decays to zero in the same way as in the already studied cases of models with one-side jumps.
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