Asymptotic results for renewal risk models with risky investments

We consider a renewal jump–diffusion process, more specifically a renewal insurance risk model with investments in a stock whose price is modeled by a geometric Brownian motion. Using Laplace transforms and regular variation theory, we introduce a transparent and unifying analytic method for investigating the asymptotic behavior of ruin probabilities and related quantities, in models with light- or heavy-tailed jumps, whenever the distribution of the time between jumps has rational Laplace transform.     

On The Randomized Schmitter Problem

Methodology and Computing in Applied Probability - We revisit the classical Schmitter problem in ruin theory and consider it for randomly chosen initial surplus level U. We show that the...     

An asymptotic expansion for the tail of compound sums of Burr distributed random variables

In this paper we show that it is possible to write the Laplace transform of the Burr distribution as the sum of four series. This representation is then used to provide a complete asymptotic expansion of the tail of the compound sum of Burr distributed random variables. Furthermore it is shown that if the number of summands is fixed, this asymptotic expansion is actually a series expansion if evaluated at sufficiently large arguments.     

Ruin problems under IBNR dynamics

In this paper, we consider a discrete‐time risk process allowing for delay in claim settlement, which introduces a certain type of dependence in the process. From martingale theory, an expression for the ultimate ruin probability is obtained, and Lundberg‐type inequalities are derived. The impact of delay in claim settlement is then investigated. To this end, a convex order comparison of the aggregate claim amounts is performed with the corresponding non‐delayed risk model, and numerical simulations are carried out with Belgian market data. Copyright © 2011 John Wiley & Sons, Ltd.     

The Tax Identity For Markov Additive Risk Processes

Taxed risk processes, i.e. processes which change their drift when reaching new maxima, represent a certain type of generalizations of Lévy and of Markov additive processes (MAP), since the times at which their Markovian mechanism changes are allowed to depend on the current position. In this paper we study generalizations of the tax identity of Albrecher and Hipp (2007) from the classical risk model to more general risk processes driven by spectrally-negative MAPs. We use the Sparre Andersen risk processes with phase-type interarrivals to illustrate the ideas in their simplest form.     

Optimal dividend strategies for a risk process under force of interest

In the classical Cramér–Lundberg model in risk theory the problem of maximizing the expected cumulated discounted dividend payments until ruin is a widely discussed topic. In the most general case within that framework it is proved [Gerber, H.U., 1968. Entscheidungskriterien fuer den zusammengesetzten Poisson-prozess. Schweiz. Aktuarver. Mitt. 1, 185–227; Azcue, P., Muler, N., 2005. Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15 (2) 261–308; Schmidli, H., 2008. Stochastic Control in Insurance. Springer] that the optimal dividend strategy is of band type. In the present paper we discuss this maximization problem in a generalized setting including a constant force of interest in the risk model. The value function is identified in the set of viscosity solutions of the associated Hamilton–Jacobi–Bellman equation and the optimal dividend strategy in this risk model with interest is derived, which in the general case is again of band type and for exponential claim sizes collapses to a barrier strategy. Finally, an example is constructed for Erlang(2)-claim sizes, in which the bands for the optimal strategy are explicitly calculated.     

Tail asymptotics for the sum of two heavy-tailed dependent risks

Let X ₁ , X ₂ denote positive heavy-tailed random variables with continuous marginal distribution functions F ₁ and F ₂, respectively. The asymptotic behavior of the tail of X ₁ +X ₂ is studied in a general copula framework and some bounds and extremal properties are provided. For more specific assumptions on F ₁ , F ₂ and the underlying dependence structure of X ₁ and X ₂, we survey explicit asymptotic results available in the literature and add several new cases.Supported by the Austrian Science Fund Project P-18392.     

Tail asymptotics for dependent subexponential differences

We study the asymptotic behavior of ?(X ? Y > u) as u → ∞, where X is subexponential, Y is positive, and the random variables X and Y may be dependent. We give criteria under which the subtraction of Y does not change the tail behavior of X. It is also studied under which conditions the comonotonic copula represents the worst-case scenario for the asymptotic behavior in the sense of minimizing the tail of X ? Y. Some explicit construction of the worst-case copula is provided in other cases.     

On a Markovian Game Model for Competitive Insurance Pricing

Methodology and Computing in Applied Probability - In this paper, we extend the non-cooperative one-period game of Dutang et al. (Journal of Operational...     

An algebraic operator approach to the analysis of Gerber-Shiu functions

We introduce an algebraic operator framework to study discounted penalty functions in renewal risk models. For inter-arrival and claim size distributions with rational Laplace transform, the usual integral equation is transformed into a boundary value problem, which is solved by symbolic techniques. The factorization of the differential operator can be lifted to the level of boundary value problems, amounting to iteratively solving first-order problems. This leads to an explicit expression for the Gerber-Shiu function in terms of the penalty function.