Uniform asymptotics for the tail probability of weighted sums with heavy tails

This paper studies the tail probability of weighted sums of the form $\sum_{i=1}^n c_i X_i$, where random variables $X_i$'s are either independent or pairwise quasi-asymptotical independent with heavy tails. Using $h$-insensitive function, the uniform asymptotic equivalence of the tail probabilities of $\sum_{i=1}^n c_iX_i$, $\max_{1\le k\le n}\sum_{i=1}^k c_iX_i$ and $\sum_{i=1}^n c_iX_i^+$ is established, where $X_i$'s are independent and follow the long-tailed distribution, and $c_i$'s take value in a broad interval. Some further uniform asymptotic results for the weighted sums of $X_i$'s with dominated varying tails are obtained. An application to the ruin probability in a discrete-time insurance risk model is presented

Uniform Asymptotics For the Tail Probability of Weighted Sums With Heavy Tails

This paper studies the tail probability of weighted sums of the form ∑i=1nciXi, where random variables X i\u27s are either independent or pairwise quasi-asymptotically independent with heavy tails. Using the idea of uniform long-tailedness, the uniform asymptotic equivalence of the tail probabilities of ∑i=1nciXi, max1≤k≤n∑i=1kciXi and ∑i=1nciXi+ is established, where X i\u27s are independent and follow the long-tailed distribution, and c i\u27s take value in a broad interval. Some further uniform asymptotic results for the weighted sums of X i\u27s with dominated varying tails are obtained. An application to the ruin probability in a discrete-time insurance risk model is presented. © 2014 Elsevier B.V

Interplay of insurance and financial risks in a discrete-time model with strongly regular variation

Consider an insurance company exposed to a stochastic economic environment that contains two kinds of risk. The first kind is the insurance risk caused by traditional insurance claims, and the second kind is the financial risk resulting from investments. Its wealth process is described in a standard discrete-time model in which, during each period, the insurance risk is quantified as a real-valued random variable $X$ equal to the total amount of claims less premiums, and the financial risk as a positive random variable $Y$ equal to the reciprocal of the stochastic accumulation factor. This risk model builds an efficient platform for investigating the interplay of the two kinds of risk. We focus on the ruin probability and the tail probability of the aggregate risk amount. Assuming that every convex combination of the distributions of $X$ and $Y$ is of strongly regular variation, we derive some precise asymptotic formulas for these probabilities with both finite and infinite time horizons, all in the form of linear combinations of the tail probabilities of $X$ and $Y$. Our treatment is unified in the sense that no dominating relationship between $X$ and $Y$ is required.Comment: Published at http://dx.doi.org/10.3150/14-BEJ625 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

On exceedance times for some processes with dependent increments

Let ${Z_n}_{n\ge 0}$ be a random walk with a negative drift and i.i.d. increments with heavy-tailed distribution and let $M=\sup_{n\ge 0}Z_n$ be its supremum. Asmussen & Kl{\"u}ppelberg (1996) considered the behavior of the random walk given that $M>x$, for $x$ large, and obtained a limit theorem, as $x\to\infty$, for the distribution of the quadruple that includes the time \rtreg=\rtreg(x) to exceed level $x$, position Z_{\rtreg} at this time, position Z_{\rtreg-1} at the prior time, and the trajectory up to it (similar results were obtained for the Cram\'er-Lundberg insurance risk process). We obtain here several extensions of this result to various regenerative-type models and, in particular, to the case of a random walk with dependent increments. Particular attention is given to describing the limiting conditional behavior of $\tau$. The class of models include Markov-modulated models as particular cases. We also study fluid models, the Bj{\"o}rk-Grandell risk process, give examples where the order of $\tau$ is genuinely different from the random walk case, and discuss which growth rates are possible. Our proofs are purely probabilistic and are based on results and ideas from Asmussen, Schmidli & Schmidt (1999), Foss & Zachary (2002), and Foss, Konstantopoulos & Zachary (2007).Comment: 17 page

The Finite-time Ruin Probabilities of a Bidimensional risk model with Constant Interest Force and correlated Brownian Motions

We follow some recent works to study bidimensional perturbed compound Poisson risk models with constant interest force and correlated Brownian Motions. Several asymptotic formulae for three different type of ruin probabilities over a finite-time horizon are established. Our approach appeals directly to very recent developments in the ruin theory in the presence of heavy tails of unidimensional risk models and the dependence theory of stochastic processes and random vectors.Comment: 25page

Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities

This paper is concerned with the compound Poisson risk model and two generalized models with still Poisson claim arrivals. One extension incorporates inhomogeneity in the premium input and in the claim arrival process, while the other takes into account possible dependence between the successive claim amounts. The problem under study for these risk models is the evaluation of the probabilities of (non-)ruin over any horizon of finite length. The main recent methods, exact or approximate, used to compute the ruin probabilities are reviewed and discussed in a unified way. Special attention is then paid to an analysis of the qualitative impact of dependence between claim amounts.compound Poisson model; ruin probability; finite-time horizon; recursive methods; (generalized) Appell polynomials; non-constant premium; non-stationary claim arrivals; interdependent claim amounts; impact of dependence; comonotonic risks; heavy-tailed distributions

Two-dimensional ruin probability for subexponential claim size

We analyse the asymptotics of ruin probabilities of two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions when the initial reserves of both companies tend to infinity and generic claim size is subexponential