Statutory financial reporting for variable annuity guaranteed death benefits: Market practice,mathematical modeling and computation

As more regulatory reporting requirements for equity-linked insurance move towards dependence on stochastic approaches, insurance companies are experiencing increasing difficulty with detailed forecasting and more accurate risk assessment based on Monte Carlo simulations. While there is vast literature on pricing and valuations of various equity-linked insurance products, very few have focused on the challenges of financial reporting for regulatory requirement and internal risk management. Most insurers use either simulation-based spreadsheet calculations or employ third-party vendor software packages. We intend to use a basic variable annuity death benefit as a model example to decipher the common mathematical structure of US statutory financial reporting. We shall demonstrate that alternative deterministic algorithms such as partial differential equation (PDE) methods can also be used in financial reporting, and that a fully quantified model allows us to compare alternatives of risk metrics for financial reporting.     

Optimal behavior of weighted Hardy operators on rearrangement-invariant spaces

The behavior of certain weighted Hardy-type operators on rearrangement-invariant function spaces is thoroughly studied. Emphasis is put on the optimality of the obtained results. First, the optimal rearrangement-invariant function spaces guaranteeing the boundedness of the operators from/to a given rearrangement-invariant function space are described. Second, the optimal rearrangement-invariant function norms being sometimes complicated, the question of whether and how they can be simplified to more manageable expressions is addressed. Next, the relation between optimal rearrangement-invariant function spaces and interpolation spaces is investigated. Last, iterated weighted Hardy-type operators are also studied.     

von Neumann代数中的*-偏序

设H是复Hilbert空间,B(H)是H上的有界线性算子全体组成的代数,M?B(H)是von Neumann代数,"≤"表示M中的*-偏序,即A,B∈M,若A~*A=A~*B,AA~*=BA~*,则A≤B.本文研究了von Neumann代数中*-偏序的上确界和下确界,证明了von Neumann代数M的子集关于*-偏序的上、下确界和B(H)中的上、下确界一致.同时,给出了M的*-偏序遗传子空间的表示,证明了弱~*闭子空间A?M,满足A∈M,B∈A,由A≤B可得A∈A,当且仅当存在唯一具有相同中心投影的投影对E,F∈M,使得A=EMF.     

An overview of Brownian and non-Brownian FCLTs for the single-server queue

We review functional central limit theorems (FCLTs) for the queue-content process in a single-server queue with finite waiting room and the first-come first-served service discipline. We emphasize alternatives to the familiar heavy-traffic FCLTs with reflected Brownian motion (RBM) limit process that arise with heavy-tailed probability distributions and strong dependence. Just as for the familiar convergence to RBM, the alternative FCLTs are obtained by applying the continuous mapping theorem with the reflection map to previously established FCLTs for partial sums. We consider a discrete-time model and first assume that the cumulative net-input process has stationary and independent increments, with jumps up allowed to have infinite variance or even infinite mean. For essentially a single model, the queue must be in heavy traffic and the limit is a reflected stable process, whose steady-state distribution can be calculated by numerically inverting its Laplace transform. For a sequence of models, the queue need not be in heavy traffic, and the limit can be a general reflected Lévy process. When the Lévy process representing the net input has no negative jumps, the steady-state distribution of the reflected Lévy process again can be calculated by numerically inverting its Laplace transform. We also establish FCLTs for the queue-content process when the input process is a superposition of many independent component arrival processes, each of which may exhibit complex dependence. Then the limiting input process is a Gaussian process. When the limiting net-input process is also a Gaussian process and there is unlimited waiting room, the steady-state distribution of the limiting reflected Gaussian process can be conveniently approximated. This revised version was published online in June 2006 with corrections to the Cover Date.     

半序空间中增算子的不动点及其应用

证明了半序空间中增算子的最小最大不动点定理,推广和改进了增算子和混合单调算子的某些最近结果,并应用于没有任何连续性紧性凹性凸性假定下非线性奇异常微分方程的边值问题。     

On path-independent integrals in three-dimensional nonlinear fracture dynamics

This paper deals with the path-independent integrals in non-linear three-dimensional fracture dynamics.Both the nonli-near elastic case and the elastic-plastic case are considered,and some path-independent integrals have been worked out.Forexplaining the physical meaning of these integrals,a speci-men with plane notch is considered,and the relation betweenthe integral and dynamical crack extension force is establish-ed.Thus.such integrals may serve as a fracture criterionin nonlinear fracture dynamics.     

Discontinuous differential games and control systems with supremum cost

This paper concerns with the existence of a value for a zero sum two-player differential game with supremum cost of the form C_t0,x0(u,v)=sup_τ[t0,T]h(x(τ;t₀,x_0,u,v)) under Isaacs' condition. We characterize the value function as the unique solution—in a suitable sense—to a PDE, namely the Hamilton–Jacobi–Isaacs equation. As a byproduct, we obtain a PDE characterization of the value function for control system.     

Moments of the maximum of normed partial sums of ρ^--mixing random variables

Let {Xn,n ≥ 1} be a sequence of identically distributed ρ^--mixing random variables and set Sn =∑i^n=1 Xi,n ≥ 1,the suffcient and necessary conditions for the existence of moments of supn≥1 ｜Sn/n^1/r｜^p（0 〈 r 〈 2,p 〉 0） are given,which are the same as that in the independent case.     

带有“上确界”的二阶脉冲微分方程的无穷边值问题

利用上下解方法及单调迭代技术讨论了一类具有上确界的二阶非线性脉冲微分方程无穷边值问题,给出了其最小最大解存在定理.     

同分布 ρ 混合序列的最大值不等式及其应用

设X_n,n≥1是同分布的ρ混合序列, 记S_n=∑ⁿ_{i=1 Xi}. 该文讨论了$\max\limits_{1\leq i\leq n}\frac{|S_i|}{i}$ $(n\geq1)$的分布函数的上界. 作为应用,获得了随机变量$\sup\limits_{n\geq1}\frac{|S_n|}{n}$的1阶矩及$p(>1)$阶矩分别存在有限的充分必要条件,这是一个与独立同分布场合相一致的结果.