In this paper, we present recursive formulae for the ruin probability at or before a certain claim arrival instant for some particular continuous time risk model. The claim number process underlying this risk model is a renewal process with either Erlang or a mixture of exponentials inter-claim times (ICTs). The claim sizes (CSs) are independent and distributed in Erlang's family, i.e., they can have different parameters, which yields a non-homogeneous risk process. We present the corresponding recursive algorithm used to evaluate the above mentioned ruin probability and we illustrate it on several numerical examples in which we vary the model's parameters to assess the impact of the non-homogeneity on the resulting ruin probability.

We consider a sequence ${\left({\mathbf{Z}}_j\right)}_{j\ge 1}$ of i.i.d. d-dimensional random vectors and for every $n\ge 1$ consider the sample ${\mathcal{S}}_n=\{{\mathbf{Z}}_1,{\mathbf{Z}}_2,\dots ,{\mathbf{Z}}_n\}.$ We say that ${\mathbf{Z}}_j$ is a “leader” in the sample ${\mathcal{S}}_n$ if ${\mathbf{Z}}_j\ge {\mathbf{Z}}_k,$$\forall k\in \{1,2,\dots ,n\}$and ${\mathbf{Z}}_j$ is an “anti-leader” if ${\mathbf{Z}}_j\le {\mathbf{Z}}_k,$$\forall k\in \{1,2,\dots ,n\}$. After all, the leader and the anti-leader are the naive extremes. Let $a_n$ be the probability that ${\mathcal{S}}_n$ has a leader, $b_n$ be the probability that ${\mathcal{S}}_n$ has an anti-leader and $c_n$ be the probability that ${\mathcal{S}}_n$ has both a leader and an anti-leader. One of the aims of the paper is to compute, or, at least to estimate, or if even that is not possible, to estimate the limits of this quantities. Another goal is to find conditions on the distribution F of ${\left({\mathbf{Z}}_j\right)}_{j\ge 1}$ so that the inferior limits of $a_n,b_n,c_n$ are positive. We give examples of distributions for which we can compute these probabilities and also examples when we are not able to do that. Then we establish conditions, unfortunately only sufficient when the limits are positive. Doing that we discovered a lot of open questions and we make two annoying conjectures—annoying because they seemed to be obvious but at a second thought we were not able to prove them. It seems that these problems have never been approached in the literature.

Let Δ_n be the n-dimensional simplex, ξ = (ξ₁, ξ₂,…, ξ_n) be an n-dimensional random vector, and $\mathcal{U}$ be a set of utility functions. A vector x^*$\in$ Δ_n is a $\mathcal{U}$ -absolutely optimal portfolio if $\mathrm{E}\left(u\left({\mathsf{\xi }}^{\mathbf{T}}{x}^{*}\right)\right)\ge \mathrm{E}\left(u\left({\mathsf{\xi }}^{\mathbf{T}}x\right)\right)$ for every x$\in$ Δ_n and u ∈ $\mathcal{U}$. In this paper, we investigate the following problem: For what random vectors, ξ, do $\mathcal{U}$-absolutely optimal portfolios exist? If ${\mathcal{U}}_2$ is the set of concave utility functions, we find necessary and sufficient conditions on the distribution of the random vector, ξ, in order that it admits a ${\mathcal{U}}_2$-absolutely optimal portfolio. The main result is the following: If x⁰ is a portfolio having all its entries positive, then x⁰ is an absolutely optimal portfolio if and only if all the conditional expectations of ${\xi }_i$, given the return of portfolio x⁰, are the same. We prove that if ξ is bounded below then CARA-absolutely optimal portfolios are also ${\mathcal{U}}_2$-absolutely optimal portfolios. The classical case when the random vector ξ is normal is analyzed. We make a complete investigation of the simplest case of a bi-dimensional random vector ξ = (ξ₁, ξ₂). We give a complete characterization and we build two dimensional distributions that are absolutely continuous and admit ${\mathcal{U}}_2$-absolutely optimal portfolios.

A stick of length 1 is broken at random into n smaller sticks. How much inequality does this procedure produce? What happens if, instead of breaking a stick, we break a square? What happens asymptotically? Which is the most egalitarian distribution of the smaller sticks (or rectangles)? Usually, when studying inequality, one uses a Lorenz curve. The more egalitarian a distribution, the closer the Lorenz curve is to the first diagonal of ${\left[0,1\right]}^2$. This is why in the first section we study the space of Lorenz curves. What is the limit of a convergent sequence of Lorenz curves? We try to answer these questions, firstly, in the deterministic case and based on the results obtained there in the stochastic one.