Narrow Results

This paper characterizes the long term social discount rate (SDR) in terms of indices of variation and identifies a class of discount functions that assign weight to the distant future in terms of the asymptotes of their hazard rates. Let A(t) be a discount function supported on [0, +∞[, whose instantaneous discount rate is δ(t) = -(ln A(t))′. In this paper, we study the limiting behavior of δ(t) and a criterion to relate this limit to so-called long and heavy-tailed discount functions. Moreover, we study the discount functions of regular variation and, more specifically, the rapidly and the slowly-varying (singular) discount functions. In this way, the hyperbolic discounting exhibits long (and then heavy) tailedness, and it is intermediate between rapidly and the slowly-varying discount functions, which makes it a suitable function for long term discounting processes.

We study the asymptotic behavior of the difference between the values at risk (VaR) ${\mathrm{\text{VaR}}}_{\alpha }(L)$ and ${\mathrm{\text{VaR}}}_{\alpha }(L+S)$ for heavy-tailed random variables $L$ and $S$ with $\alpha \uparrow 1$ for application in sensitivity analysis of quantitative operational risk management within the framework of the advanced measurement approach of Basel II (and III). Here, $L$ describes the loss amount of the present risk profile and $S$ describes the loss amount caused by an additional loss factor. We obtain different types of results according to the relative magnitudes of the thicknesses of the tails of $L$ and $S$. In particular, if the tail of $S$ is sufficiently thinner than that of $L$, then the difference between prior and posterior risk amounts ${\mathrm{\text{VaR}}}_{\alpha }(L+S)-{\mathrm{\text{VaR}}}_{\alpha }(L)$ is asymptotically equivalent to the expectation (expected loss) of $S$.

This paper establishes a simple asymptotic formula for the finite time ruin probability of a compound renewal risk model with constant interest force. We assume that the claim sizes are Weakly Negatively Dependent (WND) and identically distributed random variables belonging to the class of regularly varying tails. The results obtained have extended and improved some corresponding results of related papers.

In this paper, we study first exit times from a bounded domain of a gradient dynamical system Ẏ_t = -∇U(Y_t) perturbed by a small multiplicative Lévy noise with heavy tails. A special attention is paid to the way the multiplicative noise is introduced. In particular, we determine the asymptotics of the first exit time of solutions of Itô, Stratonovich and Marcus canonical SDEs.

We study the limiting distribution of the sum as t → ∞, N → ∞, where (X_i) are i.i.d. random variables. Attention to such exponential sums has been motivated by various problems in random media theory. Examples include the quenched mean population size of a colony of branching processes with random branching rates and the partition function of Derrida’s Random Energy Model. In this paper, the problem is considered under the assumption that the log-tail distribution function h(x) = − log P{X_i > x} is regularly varying at infinity with index 1 < ϱ < ∞. An appropriate scale for the growth of N relative to t is of the form e^λH₀(t), where the rate function H₀(t) is a certain asymptotic version of the cumulant generating function H(t) = log E[e^tXi] provided by Kasahara’s exponential Tauberian theorem. We have found two critical points, 0< λ₁< λ₂ < ∞, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. Below λ₂, we impose a slightly stronger condition of normalized regular variation of h. The limit laws here appear to be stable, with characteristic exponent α = α(ϱ, λ) ranging from 0 to 2 and with skewness parameter β = 1. A limit theorem for the maximal value of the sample {e^tXi, i = 1, ..., N} is also proved.