TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts

In this paper, we consider a portfolio of n dependent risks X₁,…,X_n and we study the stochastic behavior of the aggregate claim amount S=X₁+?+X_n. Our objective is to determine the amount of economic capital needed for the whole portfolio and to compute the amount of capital to be allocated to each risk X₁,…,X_n. To do so, we use a top-down approach. For (X₁,…,X_n), we consider risk models based on multivariate compound distributions defined with a multivariate counting distribution. We use the TVaR to evaluate the total capital requirement of the portfolio based on the distribution of S, and we use the TVaR-based capital allocation method to quantify the contribution of each risk. To simplify the presentation, the claim amounts are assumed to be continuously distributed. For multivariate compound distributions with continuous claim amounts, we provide general formulas for the cumulative distribution function of S, for the TVaR of S and the contribution to each risk. We obtain closed-form expressions for those quantities for multivariate compound distributions with gamma and mixed Erlang claim amounts. Finally, we treat in detail the multivariate compound Poisson distribution case. Numerical examples are provided in order to examine the impact of the dependence relation on the TVaR of S, the contribution to each risk of the portfolio, and the benefit of the aggregation of several risks.     

Cube structures and intersection bundles

We define the notion of a hypercube structure on a functor between two commutative Picard categories which generalizes the notion of a cube structure on a G_m-torsor over an abelian scheme. We prove that the determinant functor of a relative scheme X/S of relative dimension n is canonically endowed with a (n+2)-cube structure. We use this result to define the intersection bundle I_X/S(L₁,…,L_n+1) of n+1 line bundles on X/S and to construct an additive structure on the functor I_X/S:PIC(X/S)ⁿ⁺¹→PIC(S). Then, we construct the resultant of n+1 sections of n+1 line bundles on X, and the discriminant of a section of a line bundle on X. Finally we study the relationship between the cube structures on the determinant functor and on the discriminant functor, and we use it to prove a polarization formula for the discriminant functor.     

Joint modelling of the total amount and the number of claims by conditionals

In the risk theory context, let us consider the classical collective model. The aim of this paper is to obtain a flexible bivariate joint distribution for modelling the couple (S,N), where N is a count variable and S=X₁+?+X_N is the total claim amount. A generalization of the classical hierarchical model, where now we assume that the conditional distributions of S|N and N|S belong to some prescribed parametric families, is presented. A basic theorem of compatibility in conditional distributions of the type S given N and N given S is stated. Using a known theorem for exponential families and results from functional equations new models are obtained. We describe in detail the extension of two classical collective models, which now we call Poisson-Gamma and the Poisson-Binomial conditionals models. Other conditionals models are proposed, including the Poisson-Lognormal conditionals distribution, the Geometric-Gamma conditionals model and a model with inverse Gaussian conditionals. Further developments of collective risk modelling are given.     

Non-conflicting ordering cones and vector optimization in inductive limits

Let (E, ξ)= ind (E_n, ξ_n) be an inductive limit of a sequence (E_n, ξ_n)_{n∈ N} of locally convex spaces and let every step (E_n, ξ_n) be endowed with a partial order by a pointed convex (solid) cone S_n. In the framework of inductive limits of partially ordered locally convex spaces, the notions of lastingly efficient points, lastingly weakly efficient points and lastingly globally properly efficient points are introduced. For several ordering cones, the notion of non-conflict is introduced. Under the requirement that the sequence (S_n)_{n∈ N} of ordering cones is non-conflicting, an existence theorem on lastingly weakly efficient points is presented. From this, an existence theorem on lastingly globally properly efficient points is deduced.     

General theorems on rates of convergence in distribution of random variables I. General limit theorems

Let (X_n)_n?$\text{N}$ be a sequence of real, independent, not necessarily identically distributed random variables (r.v.) with distribution functions F_Xn, and S_n = Σ_i=1ⁿX_i. The authors present limit theorems together with convergence rates for the normalized sums ?(n)S_n, where ?: $\text{N}$ → $\text{R}$⁺, ?(n) → 0, n → ∞, towards appropriate limiting r.v. X, the convergence being taken in the weak (star) sense. Thus higher order estimates are given for the expression ∝_$\text{R}$f(x) d[F_?(n)Sn(x) ? F_X(x)] which depend upon the normalizing function ?, decomposability properties of X and smoothness properties of the function f under consideration. The general theorems of this unified approach subsume O- and o-higher order error estimates based upon assumptions on associated moments. These results are also extended to multi-dimensional random vectors.     

Splitting manifolds as M × R where M has a k-fold end structure

Let X be a locally finite simplicial complex of dimension n, n? 5, equipped with a k-fold end structure [4] and consider a piecewise linear (n + 1)-dimensional manifold M that is proper homotopy equivalent to X × R by F:M → X × R, where R is the set of real numbers. The question arises as to whether or not the manifold M can be split, i.e., written as M = N × R where N is a n-manifold and where there is a proper homotopy between F and (p₁ ° F₀) × id:N × R → X × R, preserving the natural (k+1)-fold end structure, where F₀ is F|_N and p₁ is the projection X × R → X. Of particular significance is the fact that X is noncompact. When the construction of such splittings is attempted, algebraic obstructions arise, which vanish if and only if the construction can be completed. This paper develops such an obstruction theory by utilizing methods of L.C. Siebenmann and the k-fold end structures of F. Waldhausen.     

Constructing quasi-cyclic codes from linear algebra theory

Let F _q be a finite field of cardinality q, l and m be positive integers and M _l (F _q ) the F _q -algebra of all l × l matrices over F _q . We investigate the relationship between monic factors of X ^m ? 1 in the polynomial ring M _l (F _q )[X] and quasi-cyclic (QC) codes of length lm and index l over F _q . Then we consider the idea of constructing QC codes from monic factors of X ^m ? 1 in polynomial rings over F _q -subalgebras of M _l (F _q ). This idea includes ideas of constructing QC codes of length lm and index l over F _q from cyclic codes of length m over a finite field F _q l, the finite chain ring F _q  + uF _q  + · · · + u ^{l ? 1} F _q (u ^l  = 0) and other type of finite chain rings.     

On the sectionwise connectedness of a contingent

Let X be a real normed space and let f: ? → X be a continuous mapping. Let T _f (t ₀) be the contingent of the graph G(f) at a point (t ₀, f(t ₀)) and let S ⁺ ? (0,∞) × X be the “right” unit hemisphere centered at (0, 0 _X ). We show that

1. 1.
   If dimX < ∞ and the dilation D(f, t ₀) of f at t ₀ is finite then T _f (t ₀) ∩ S ⁺ is compact and connected. The result holds for \(T_f (t_0 ) \cap \overline {S^ + } \) even with infinite dilation in the case f: [0,∞) → X.
    
2. 2.
   If dimX = ∞, then, given any compact set F ? S ⁺, there exists a Lipschitz mapping f: ? → X such that T _f (t ₀) ∩ S ⁺ = F.
    
3. 3.
   But if a closed set F ? S ⁺ has cardinality greater than that of the continuum then the relation T _f (t ₀) ∩ S ⁺ = F does not hold for any Lipschitz f: ? → X.
    

     

On the rate of convergence in the global central limit theorem for random sums of independent random variables

We present upper bounds of the integral \( {\int}_{-\infty}^{\infty }{\left|x\right|}^l\left|\mathbf{P}\left\{{Z}_N<x\right\}-\varPhi (x)\right|\mathrm{d}x \) for 0 ≤ l ≤ 1 + δ, where 0 < δ ≤ 1, Φ(x) is a standard normal distribution function, and Z _N = \( {S}_N/\sqrt{\mathbf{V}{S}_N} \) is the normalized random sum with variance V S _N > 0 (S _N = X ₁ + · · · + X _N ) of centered independent random variables X ₁ ,X ₂ , . . . . The number of summands N is a nonnegative integer-valued random variable independent of X ₁ ,X ₂ , . . . .     

Fixed points on model solvmanifold pairs

In this paper we define model solvmanifold pairs and their diagonal type selfmaps in the tradition of Heath and Keppelmann. We derive an explicit formula for computing the relative Nielsen number N(F;X,A) on these spaces and selfmaps F:(X,A)→(X,A). We find that model solvmanifold pairs often exhibit interesting Schirmer theory, meaning N(F;X,A)>max{N(F),N(F|_A)}.     

On the normal approximation of a sum of a random number of independent random variables

In this paper, we extend the results obtained in [J. Sunklodas, Some estimates of normal approximation for the distribution of a sum of a random number of independent random variables, Lith. Math. J., 52(3):326?C333, 2012] for a thrice-differentiable function h : ? ?? ? to the case of h ?? BL(?); namely, we estimate the quantity | E h(Z _N )? E h(Y)| where h is a real bounded Lipschitz function, $ {Z_N}={{{\left( {{S_N}-\mathrm{E}{S_N}} \right)}} \left/ {{\sqrt{{\mathrm{D}{S_N}}}}} \right.} $ , S _N = X ₁ + · · · + X _N , X ₁ , X ₂ , . . . are independent, not necessarily identically distributed, real random variables, N is a positive integer-valued r.v. independent of X ₁ , X ₂ , . . . , and Y is a standard normal random variable.     

Maximum likelihood estimation of a survival function under the koziol-green proportional hazards model

Let X and Y be two independent competing lifetimes wih survival functions S_F(t) and S_G(t). In this paper we, study a proportional-hazards competing risks model characterized by Armitage (1959): S_G(t) = S_G(t)]^β for certain positive constant β. The maximum likelihood estimator for S_F is proposed, examined and compared with the product limit estimator of Kaplan and Meier (1958). The MLE is shown to be asymptotically more efficient that the PLE under the specified model.     

Thinning of point processes,revisited

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Characterizing filters by convergence (with respect to filters) in Banach spaces

Let X be a topological space and let F be a filter on N, recall that a sequence (xn)_n∈N in X is said to be F-convergent to the point x∈X, if for each neighborhood U of x, {n∈N:xn∈U}∈F. By using F-convergence in ?₁ and in Banach spaces, we characterize the P-filters, the P-filters⁺, the weak P-filters, the Q-filters, the Q-filters⁺, the weak Q-filters, the selective filters and the selective⁺ filters.     

Some exact distributions of the number of one-sided deviations and the time of the last such deviation in the simple random walk

For every positive integer n, let S_n be the n-th partial sum of a sequence of independent and identically distributed random variables, each assuming the values +1 and −1 with respective probabilities p (0<p<1)) and q (= 1 −p) and having mean μ = p − q. For a fixed positive real number λ, let N⁺[N1] be the total number of values of n for which S_n > (μ + λ)n [S_n⩾(μ + λ)n] and let L⁺[L1] be the supremum of the values of n for which S_n > (μ + λ)n [S_n⩾(μ + λ)n], where sup Oslash; = 0. Explicit expressions for the exact distributions of N⁺, N1, L⁺ and L1 are given when μ + λ = ±k/(k + 2) for any nonnegative integer k.     

Central limit theorems for C(S)-valued random variables

Let C(S) be the space of real-valued continuous functions on a compact metric space S. Let {X_n, n ? 1} be a sequence of independent identically distributed C(S)-valued random variables with mean zero and sup_t?sE[X₁²(t)] = 1. We show that the measures induced by ($\text{X}{}_1\text{+ ··· + X}{}_{\mathrm{n}}\text{) n}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$ converge weakly to a Gaussian measure on C(S) under different conditions on X₁, one of which consolidates and extends results of Strassen and Dudley, Giné, and Dudley. Our method of proof is different from the methods employed by these authors.     

On some chains of fuzzy sets

A partial order relation σ is defined in the set $\text{F}$(X) of the fuzzy sets in X. If this ordering is induced in the subset F(X) of the measurable fuzzy sets in the set X with totally finite positive measure, then fσg implies that the entropy of the fuzyy set f is not less than the entropyof g. By means of this ordering a lattice L on $\text{F}$(X) is defined and a lattice structure is induced in the set of infinite chains in L. Furthermore the set F′(X) of the fuzzy sets of F(X) which assume value in a finite subset of the real interval [0,1] is considered and the following properties are stated: any chain of elements of F′(X) is an infinite sequence of functions convergent in the mean to an integrable function, and the entropy is a valuation of bounded variation on the sublattice of L whose elements are in F′(X). The chains on L can offer a model of a cognitive process in a fuzzy environment when their elements are determined by a sequence of decisions. The limit property traduces the determinism of a such procedure.