Optimal selection of a portfolio of options under Value-at-Risk constraints: a scenario approach

This paper introduces a multiperiod model for the optimal selection of a financial portfolio of options linked to a single index. The objective of the model is to maximize the expected return of the portfolio under constraints limiting its Value-at-Risk. We rely on scenarios to represent future security prices. The model contains several interesting features, like the consideration of transaction costs, bid-ask spreads, arbitrage-free option pricing, and the possibility to rebalance the portfolio with options introduced at the start of each period. The resulting mixed integer programming model is applied to realistic test instances involving options on the S&P500 index. In spite of the large size and of the numerical difficulty of this model, near-optimal solutions can be computed by a standard branch-and-cut solver or by a specialized heuristic. The structure and the financial features of the selected portfolios are also investigated.     

Generating uniform random vectors over a simplex with implications to the volume of a certain polytope and to multivariate extremes

A uniform random vector over a simplex is generated. An explicit expression for the first moment of its largest spacing is derived. The result is used in a proposed diagnostic tool which examines the validity of random number generators. It is then shown that the first moment of the largest uniform spacing is related to the dependence measure of random vectors following any extreme value distribution. The main result is proved by a geometric proof as well as by a probabilistic one.     

The effective application of a new approach to the generalized orienteering problem

The Orienteering Problem (OP) is an important problem in network optimization in which each city in a network is assigned a score and a maximum-score path from a designated start city to a designated end city is sought that is shorter than a pre-specified length limit. The Generalized Orienteering Problem (GOP) is a generalized version of the OP in which each city is assigned a number of scores for different attributes and the overall function to optimize is a function of these attribute scores. In this paper, the function used was a non-linear combination of attribute scores, making the problem difficult to solve. The GOP has a number of applications, largely in the field of routing. We designed a two-parameter iterative algorithm for the GOP, and computational experiments suggest that this algorithm performs as well as or better than other heuristics for the GOP in terms of solution quality while running faster. Further computational experiments suggest that our algorithm also outperforms the leading algorithm for solving the OP in terms of solution quality while maintaining a comparable solution speed.     

Triebel-Lizorkin Spaces of Para-Accretive Type and a Tb Theorem

In this article, we use a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type $\dot{F}^{\alpha,q}_{b,p}In this article, we use a discrete Calderón-type reproducing formula and Plancherel-P?lya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type , which reduces to the classical Triebel-Lizorkin spaces when the para-accretive function is constant. Moreover, we give a necessary and sufficient condition for the boundedness of paraproduct operators. From this, we show that a generalized singular integral operator T with M _b TM _b ∈WBP is bounded from to if and only if and T ^* b=0 for , where ε is the regularity exponent of the kernel of T. Chin-Cheng Lin supported by National Science Council, Republic of China under Grant #NSC 97-2115-M-008-021-MY3. Kunchuan Wang supported by National Science Council, Republic of China under Grant #NSC 97-2115-M-259-009 and NCU Center for Mathematics and Theoretic Physics.     

Scheduling in a multi-processor environment with deteriorating job processing times and decreasing values: the case of forest fires

In forest fire fighting, time and effort required to control a fire increase if fire containment effort is delayed. The problem of scheduling multiple resources employed as parallel identical or non-identical processors in order to contain N≥2 fires may be tackled using the concept of deteriorating jobs. In this paper, the above problem is stated and a model is formulated, the criterion being to maximize the total remaining value of the burnt areas and a real-time synchronous heuristic algorithm is proposed. The heuristic algorithm’s efficiency was compared to real data provided by Greek Fire Corps from a severe wildfire near Athens in June-July, 2007.     

Simultaneous siting and sizing of distribution centers on a plane

The benefits of simultaneous consideration of siting and sizing of distribution centers have been well acknowledged in supply chain design. Most formulations assume that the potential DC sites are known and the decision on location is to select sites from the finite potential DC sites. However, the quality of this discrete version problem depends on the selection of potential DC sites. In this paper we present a planar version of the problem, which assumes that there is no a priori knowledge of DC sites and DCs can be located anywhere in the plane. The goal of the problem is to simultaneously find locations and sizing of DC sites. The solution of the planar problem provides a lower bound for the discrete problem. The objective of the problem is to minimize the total of inbound and outbound transportation costs and distribution center construction costs—which include its fixed charge cost and concave sizing cost. The problem is initially formulated as a nonlinear programming model. We then reformulate it as a set covering problem after establishing certain key properties. A greedy drop heuristic and a column generation heuristic are developed to solve the problem. Computational experiments are provided.     

The critical groups of a family of graphs and elliptic curves over finite fields

Let q be a power of a prime, and E be an elliptic curve defined over  . Such curves have a classical group structure, and one can form an infinite tower of groups by considering E over field extensions for all k≥1. The critical group of a graph may be defined as the cokernel of L(G), the Laplacian matrix of G. In this paper, we compare elliptic curve groups with the critical groups of a certain family of graphs. This collection of critical groups also decomposes into towers of subgroups, and we highlight additional comparisons by using the Frobenius map of E over  . This work was partially supported by the NSF, grant DMS-0500557 during the author’s graduate school at the University of California, San Diego, and partially supported by an NSF Postdoctoral Fellowship.     

Oscillation and global asymptotic stability of a discrete haematopoiesis model

In this paper, we study a discrete haematopoiesis model $$\Delta y_{n}=-\alpha y_{n}+\frac{\beta y_{n-k}^{m}}{1+y_{n-k}^{p}}\quad\mbox{for}\ n=0,1,\ldots,$$ where α∈(0,1), β∈(α,∞), m∈(1,∞), p∈[m,∞), k∈?, and establish some results about oscillation and global asymptotic stability of the positive solution of this model.     

Interaction of phase-transformations and plasticity – a multi-phase micro-sphere approach

We present an efficient model for the simulation of solid to solid phase-transformations in polycrystalline materials. As a basis, we implement a scalar-valued Gibbs-energy-barrier-based phase-transformation model making use of statistical physics. In this work, we particularly adopt the model for the simulation of phase-transformations between an austenitic parent phase and a martensitic tension and compression phase. The incorporation of plasticity phenomena is established by enhancing the Helmholtz free energy functions of the material phases considered, where the plastic driving forces acting in each phase are derived from the overall free energy potential. The coupled model is embedded into a micro-sphere formulation in order to simulate three-dimensional boundary value problems—a technique well-established in the context of computational inelasticity at small strains. It is shown that the model is capable of reflecting experimentally observed behaviour. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Control of systems with flexible multi-server pools: a shadow routing approach

A general model with multiple input flows (classes) and several flexible multi-server pools is considered. We propose a robust, generic scheme for routing new arrivals, which optimally balances server pools’ loads, without the knowledge of the flow input rates and without solving any optimization problem. The scheme is based on Shadow routing in a virtual queueing system. We study the behavior of our scheme in the Halfin–Whitt (or, QED) asymptotic regime, when server pool sizes and the input rates are scaled up simultaneously by a factor r growing to infinity, while keeping the system load within \(O(\sqrt{r}\,)\) of its capacity.The main results are as follows. (i) We show that, in general, a system in a stationary regime has at least \(O(\sqrt{r}\,)\) average queue lengths, even if the so called null-controllability (Atar et al., Ann. Appl. Probab. 16, 1764–1804, 2006) on a finite time interval is possible; strategies achieving this \(O(\sqrt{r}\,)\) growth rate we call order-optimal. (ii) We show that some natural algorithms, such as MaxWeight, that guarantee stability, are not order-optimal. (iii) Under the complete resource pooling condition, we prove the diffusion limit of the arrival processes into server pools, under the Shadow routing. (We conjecture that result (iii) leads to order-optimality of the Shadow routing algorithm; a formal proof of this fact is an important subject of future work.) Simulation results demonstrate good performance and robustness of our scheme.     

Vector-valued stochastic delay equations—a semigroup approach

Let E be a type 2 umd Banach space, H a Hilbert space and let p∈[1,∞). Consider the following stochastic delay equation in E:

Hypoellipticity and Ergodicity of the Wonham Filter as a Diffusion Process

The ergodicity problem of the Wonham filter as a diffusion process is discussed. Since the Wonham equation is degenerate, we apply an ergodic theorem of degenerate Markov diffusions to the problem. Under a certain condition, the Wonham equation satisfies Hörmander’s condition, and the Wonham filter has a continuous transition density. From these results, we obtain that the Wonham filter is uniformly ergodic as a diffusion process.     

Finite volume element approximation and analysis for a kind of semiconductor device simulation

In this paper, we present a finite volume element scheme for a kind of two dimensional semiconductor device simulation. A general framework is developed for finite volume element approximation of the semiconductor problems. We construct a fully discrete finite volume element scheme based on triangulations with a piecewise linear finite element space and a general type of control volume. Optimal-order convergence in H ¹-norm is derived.     

On the order of a non-abelian representation group of a slim dense near hexagon

In this paper we study the possible orders of a non-abelian representation group of a slim dense near hexagon. We prove that if the representation group R of a slim dense near hexagon S is non-abelian, then R is a 2-group of exponent 4 and |R|=2 ^β , 1+NPdim(S)≤β≤1+dimV(S), where NPdim(S) is the near polygon embedding dimension of S and dimV(S) is the dimension of the universal representation module V(S) of S. Further, if β=1+NPdim(S), then R is necessarily an extraspecial 2-group. In that case, we determine the type of the extraspecial 2-group in each case. We also deduce that the universal representation group of S is a central product of an extraspecial 2-group and an abelian 2-group of exponent at most 4. This work was partially done when B.K. Sahoo was a Research Fellow at the Indian Statistical Institute, Bangalore Center with NBHM fellowship, DAE Grant 39/3/2000-R&D-II, Govt. of India.     

Existence of a periodic solution for continuous and discrete periodic second-order equations with variable potentials

Green’s functions for new second-order periodic differential and difference equations with variable potentials are found, then used as kernels in integral operators to guarantee the existence of a positive periodic solution to continuous and discrete second-order periodic boundary value problems with periodic coefficient functions. A new version of the Leggett-Williams fixed point theorem is employed.     

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

We develop a viscosity solution theory for a system of nonlinear degenerate parabolic integro-partial differential equations (IPDEs) related to stochastic optimal switching and control problems or stochastic games. In the case of stochastic optimal switching and control, we prove via dynamic programming methods that the value function is a viscosity solution of the IPDEs. In our setting the value functions or the solutions of the IPDEs are not smooth, so classical verification theorems do not apply.     

Convexity,moduli of smoothness and a Jackson-type inequality

For a Banach space B of functions which satisfies for some m>0

$ \max ({\|F+G\|}_B,{\|F-G\|}_B)\geqq ({\|F\|}^s_B+m{\|G\|}^s_B)^{1/s},\quad \forall \,F,G\in B $

(?)

a significant improvement for lower estimates of the moduli of smoothness ω ^r (f,t) _B is achieved. As a result of these estimates, sharp Jackson inequalities which are superior to the classical Jackson type inequality are derived. Our investigation covers Banach spaces of functions on ? ^d or \(\mathbb{T}^{d}\) for which translations are isometries or on S ^d?1 for which rotations are isometries. Results for C ₀ semigroups of contractions are derived. As applications of the technique used in this paper, many new theorems are deduced. An L _p space with 1<p<∞ satisfies (?) where s=max??(p,2), and many Orlicz spaces are shown to satisfy (?) with appropriate s.

     

The Real Powers of the Convolution of a Gamma Distribution and a Bernoulli Distribution

In this paper, we essentially compute the set of x,y>0 such that the mapping \(z\longmapsto(1-r+re^{z})^{x}(\frac{\lambda}{\lambda-z})^{y}\) is a Laplace transform. If X and Y are two independent random variables which have respectively Bernoulli and Gamma distributions, we denote by μ the distribution of X+Y. The above problem is equivalent to finding the set of x>0 such that μ ^*x exists.