Ruin probability and joint distributions of some actuarial random vectors in the compound pascal model

The compound negative binomial model,introduced in this paper,is a discrete time version.We discuss the Markov properties of the surplus process,and study the ruin probability and the joint distributions of actuarial random vectors in this model.By the strong Markov property and the mass function of a defective renewal sequence,we obtain the explicit expressions of the ruin probability,the finite-horizon ruin probability,the joint distributions of T,U(T-1),|U(T)| and 0≤inn相似文献   

Banach's Fixed-Point Theorem as a base for data-type equations

For a categoryK of data types, solutions of recursive data-type equationsX T(X), whereT is an endofunctor ofK, can be constructed by iteratingT on the unique arrowT1 1. This is well-known forK enriched over complete posets and forT locally continuous (an application of the Kleene Fixed-Point Theorem). We prove this forK enriched over complete metric spaces and forT contracting (an application of the Banach Fixed-Point Theorem). Moreover, we prove that each such recursive equation has a unique solution. Our results generalize the approach of P. America and J. Rutten.Dedicated to Dieter Pumplün on the occasion of his 60th birthday     

Regular centralizers of idempotent transformations

Denote by T(X) the semigroup of full transformations on a set X. For ε∈T(X), the centralizer of ε is a subsemigroup of T(X) defined by C(ε)={α∈T(X):αε=εα}. It is well known that C(id _X )=T(X) is a regular semigroup. By a theorem proved by J.M. Howie in 1966, we know that if X is finite, then the subsemigroup generated by the idempotents of C(id _X ) contains all non-invertible transformations in C(id _X ).     

Asymptotic behaviour of the finite-time ruin probability under subexponential claim sizes

The paper deals with the Sparre Andersen risk model. We study the tail behaviour of the finite-time ruin probability, Ψ(x,t), in the case of subexponential claim sizes as initial risk reserve x tends to infinity. The asymptotic formula holds uniformly for t in a corresponding region and reestablishes a formula of Tang [Tang, Q., 2004a. Asymptotics for the finite time ruin probability in the renewal model with consistent variation. Stochastic Models 20, 281–297] obtained for the class of claim distributions having consistent variation.     

Torsion Zero-Cycles on a Product of Canonical Lifts of Elliptic Curves

Let X be a surface over a p-adic field with good reduction and let Y be its special fiber. We write T(X) and T(Y) for the kernels of the Albanese maps of X and Y, respectively. Then, F(X) = T(X)/T(X)_div is conjectured to be finite, where T(X)_div is the maximal divisible subgroup of T(X). Furthermore, F(X) is conjectured to be isomorphic to T(Y) modulo p-primary torsion. We show that the p-primary torsion subgroup of F(X) can be arbitrary large even though we fix the special fiber Y.     

On Certain Densely Invariant Quantities of Linear Operators

Let L(X,Y) denote the class of linear transformations T:D(T) ? X → Y where X and Y are normed spaces. A quantity f is called densely invariant if for every system L(X, Y) and every operator T ? L(X,Y) we have f(T/E)= f(T) whenever E is a core of T. The density invariance of certain well known quantities is established. In case Y is complete and T is closable, a stronger property is shown to hold for some of these quantitites, namely invariance under restriction to dense subspaces.     

Special uniform approximations of continuous vector-valued functions. Part II: special approximations in CX(T)CY(S)

In this paper, which is a continuation of Timofte (J. Approx. Theory 119 (2002) 291–299, we give special uniform approximations of functions from C_XY(T×S) and C_∞(T×S,XY) by elements of the tensor products C_X(T)C_Y(S), respectively C₀(T,X)C₀(S,Y), for topological spaces T,S and Γ-locally convex spaces X,Y (all four being Hausdorff).     

Asymptotic behaviour of the finite‐time ruin probability in renewal risk models

In this paper we study the tail behaviour of the probability of ruin within finite time t, as initial risk reserve x tends to infinity, for the renewal risk model with strongly subexponential claim sizes. The asymptotic formula holds uniformly for t∈[f(x), ∞), where f(x) is an infinitely increasing function, and substantially extends the result of Tang (Stoch. Models 2004; 20 :281–297) obtained for the class of claim distributions with consistently varying tails. Two examples illustrate the result. Copyright © 2008 John Wiley & Sons, Ltd.     

The asymptotic distributions of sums of record values for distributions with lognormal-type tails

Let X1,X2,... be a sequence of i.i.d. random variables and let X(1),X(2),... be the associatedrecord value sequence. We focus on the asymptotic distributions of sums of records, Tn=∑nk=1X(k), forX1 ∈ LN(γ). In this case, we find that 2 is a strange point for parameter γ. When γ＞ 2, Tn is asymptoticallynormal, while for 2 ＞γ＞ 1, we prove that Tn cannot converge in distribution to any non-degenerate lawthrough common centralizing and normalizing and log Tn is asymptotically normal.     

A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling

Let (X(t)) be a risk process with reserve-dependent premium rate, delayed claims and initial capital u. Consider a class of risk processes {(X ^ε (t)): ε > 0} derived from (X(t)) via scaling in a slow Markov walk sense, and let Ψ_ε(u) be the corresponding ruin probability. In this paper we prove sample path large deviations for (X ε (t)) as ε → 0. As a consequence, we give exact asymptotics for log Ψ_ε(u) and we determine a most likely path leading to ruin. Finally, using importance sampling, we find an asymptotically efficient law for the simulation of Ψ_ε(u). AMS Subject Classifications 60F10, 91B30 This work has been partially supported by Murst Project “Metodi Stocastici in Finanza Matematica”     

Minimum and Surjectivity Moduli Preserving in Banach Space

Let m(T) and q(T) be respectively the minimum and the surjectivity moduli of T∈ℬ(X), where ℬ(X) denotes the algebra of all bounded linear operators on a complex Banach space X. If there exists a semi-invertible but non-invertible operator in ℬ(X) then, given a surjective unital linear map φ: ℬ(X)⟶ℬ(X), we prove that m(T)=m(φ(T)) for all T∈ℬ(X), if and only if, q(T)=q(φ(T)) for all T∈ℬ(X), if and only if, there exists a bijective isometry U∈ℬ(X) such that φ(T)=UTU ⁻¹ for all T∈ℬ(X).     

Semigroups of Transformations Preserving an Equivalence Relation and a Cross-Section

Abstract

For a set X, an equivalence relation ρ on X, and a cross-section R of the partition X/ρ induced by ρ, consider the semigroup T(X, ρ, R) consisting of all mappings a from X to X such that a preserves both ρ (if (x, y)?∈?ρ then (xa, ya)?∈?ρ) and R (if r?∈?R then ra?∈?R). The semigroup T(X, ρ, R) is the centralizer of the idempotent transformation with kernel ρ and image R. We determine the structure of T(X, ρ, R) in terms of Green's relations, describe the regular elements of T(X, ρ, R), and determine the following classes of the semigroups T(X, ρ, R): regular, abundant, inverse, and completely regular.     

Torus quotients of homogeneous spaces — II

We classify the homogeneous spacesX for which there is aT linearised ample line bundleL onX such thatX _T ^ss(L)=X_T ^s(L).     

Microlocal branes are constructible sheaves

Let X be a compact real analytic manifold, and let T* X be its cotangent bundle. In a recent paper with Zaslow (J Am Math Soc 22:233–286, 2009), we showed that the dg category Sh _c (X) of constructible sheaves on X quasi-embeds into the triangulated envelope F(T* X) of the Fukaya category of T* X. We prove here that the quasi-embedding is in fact a quasi-equivalence. When X is a complex manifold, one may interpret this as a topological analogue of the identification of Lagrangian branes in T* X and regular holonomic D_X{{\mathcal D}_X} -modules developed by Kapustin (A-branes and noncommutative geometry, arXiv:hep-th/0502212) and Kapustin and Witten (Commun Number Theory Phys 1(1):1–236, 2007) from a physical perspective. As a concrete application, we show that compact connected exact Lagrangians in T* X (with some modest homological assumptions) are equivalent in the Fukaya category to the zero section. In particular, this determines their (complex) cohomology ring and homology class in T* X, and provides a homological bound on their number of intersection points. An independent characterization of compact branes in T* X has recently been obtained by Fukaya et al. (Invent Math 172(1):1–27, 2008).     

Cochain model for thickenings and its application to rational LS-category

A thickening of a finite CW-complex X is by definition a compact manifold M of the same simple homotopy type as X. We give a model for the cochain complex of the boundary of that manifold, C ^*(δM), as a module over the cochain algebra C ^*(X). We also show how to construct an algebraic model of the rational homotopy type of δC ^*(M) from a model of X. Using this rational model, we prove a new formula for the rational Lusternik–Schnirelmann category of X. Received: 24 September 1999     

Additive Maps Preserving Similarity of Operators on Banach Spaces

Let X be an infinite-dimensional complex Banach space and denote by B（X） the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from B（X） onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional ψ on B（X） such that either Φ（T） = cATA^-1 ＋ ψ（T）I for all T, or Φ（T） = cAT＊A^-1 ＋ ψ（T）I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite-dimensional Hilbert space, the above similarity invariant additive functional ψ is always zero.     

Linear Maps Preserving Operators of Local Spectral Radius Zero

Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let r_T(x) = limsup_{n ? ￥} || Tⁿx|| ^1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r _T (x) = 0 if and only if r_j(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r _T (By) = 0 if and only if r_{j( T)} (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB^-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}.     

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.     

带息双二项风险模型的破产问题

本文研究了带随机利率的双二项风险模型的破产问题,得到了描述破产严重程度的破产前盈余分布,破产持续时间分布的递推公式,有限时间破产概率的递推公式及终极破产概率满足的积分方程.     

The cardinality of FM(1+1+n)

The aim of this note is to develop a counting formula for the modular lattice FM(1+1+n) freely generated by two single elements and an n-element chain. This answers Problem 44 in Birkhoff [1] which asks one to determine FM(1+1+n). The proof of our recursive formula is based on the scaffolding method developed by R. Wille.