Composition of an efficient portfolio in the Bielecki and Pliska market model

We study the continuous time portfolio optimization model due to Bielecki and Pliska where the mean returns of individual securities or asset categories are explicitly affected by underlying economic factors. We introduce the functional Q _γ featuring the expected earnings yield of portfolio minus a penalty term proportional with a coefficient γ to the variance when we keep the value of the factor levels fixed. The coefficient γ plays the role of a risk-aversion parameter. We find the optimal trading positions that can be obtained as the solution to a maximization problem for Q _γ at any moment of time. The single-factor case is analyzed in more details. We present a simple asset allocation example featuring a Vasicek-type interest rate which affects a stock index and also serves as a second investment opportunity. Then we compare our results with the theory of Bielecki and Pliska where the authors employ the methods of the risk-sensitive control theory thereby using an infinite horizon objective featuring the long run expected growth rate, the asymptotic variance, and a risk-aversion parameter similar to γ.     

Traffic processes in a class of finite Markovian queues

This paper exposes the stochastic structure of traffic processes in a class of finite state queueing systems which are modeled in continuous time as Markov processes. The theory is presented for theM/E _k /φ/L class under a wide range of queue disciplines. Particular traffic processes of interest include the arrival, input, output, departure and overflow processes. Several examples are given which demonstrate that the theory unifies many earlier works, as well as providing some new results. Several extensions to the model are discussed.     

Uniform stability for solutions of a structural acoustics PDE model with no added dissipative feedback

A rate of rational decay is obtained for smooth solutions of a PDE model, which has been used in the literature to describe structural acoustic flows. This structural acoustics model is composed of two distinct PDE systems: (i) a wave equation, to model the interior acoustic flow within the given cavity Ω and (ii) a structurally damped elastic equation, to describe time‐evolving displacements along the flexible portion Γ₀ of the cavity walls. Moreover, the extent of damping in this elastic component is quantified by parameter η∈[0,1]. The coupling between these two distinct dynamics occurs across the boundary interface Γ₀. Our main result is the derivation of uniform decay rates for classical solutions of this particular structural acoustic PDE, decay rates that are obtained without incorporating any additional boundary dissipative feedback mechanisms. In particular, in the case that full Kelvin–Voight damping is present in fourth‐order elastic dynamics, that is, the structural acoustics system as it appears in the literature, solutions that correspond to smooth initial data decay at a rate of . By way of deriving these stability results, necessary a priori inequalities for a certain static structural acoustics PDE model are generated here; these inequalities ultimately allow for an application of a recently derived resolvent criterion for rational decay. Copyright © 2016 John Wiley & Sons, Ltd.     

Heavy and light traffic in fluid models with burst arrivals

We consider the problem of finding a heavy and light traffic limits for the steady-state workload in a fluid model having a continuous burst arrival process. Such a model is useful for describing (among other things) the packetwise transmission of data in telecommunications, where each packet is approximated to be a continuous flow. Whereas in a queueing model, each arrival epoch,t _n , corresponds to a customer with a service timeS _n , the burst model is different: each arrival epoch,t _n , corresponds to a burst of work, that is, a continuous flow of work (fluid, information) to the system at rate 1 during the time interval [t _n ,t _n +S _n ]. In the present paper we show that the burst and queueing models share the same heavy-traffic limit for work, but that their behavior in light traffic is quite different.Research supported by the Japan Society for the Promotion of Science, during the author's fellowship in Tokyo.Research funded by C & C Information Technology Research Laboratories, NEC, and the International Science Foundation.     

On equalities for BLUEs under misspecified Gauss-Markov models

This paper studies relationships between the best linear unbiased estimators （BLUEs） of an estimable parametric functions Kβunder the Gauss-Markov model {y, Xβ, σ^2]E} and its misspecified model {y, X0β,σ^2∑0}. In addition, relationships between BLUEs under a restricted Gauss Markov model and its misspecified model are also investigated.     

Existence and convergence for quasi‐static evolution in brittle fracture

This paper investigates the mathematical well‐posedness of the variational model of quasi‐static growth for a brittle crack proposed by Francfort and Marigo in [15]. The starting point is a time discretized version of that evolution which results in a sequence of minimization problems of Mumford and Shah type functionals. The natural weak setting is that of special functions of bounded variation, and the main difficulty in showing existence of the time‐continuous quasi‐static growth is to pass to the limit as the time‐discretization step tends to 0. This is performed with the help of a jump transfer theorem which permits, under weak convergence assumptions for a sequence {u_n} of SBV‐functions to its BV‐limit u, to transfer the part of the jump set of any test field that lies in the jump set of u onto that of the converging sequence {u_n}. In particular, it is shown that the notion of minimizer of a Mumford and Shah type functional for its own jump set is stable under weak convergence assumptions. Furthermore, our analysis justifies numerical methods used for computing the time‐continuous quasi‐static evolution. © 2003 Wiley Periodicals, Inc.     

A two-fluid storage model with Lévy inputs and alternating outputs

In this paper we consider a storage model with two types of inputs and outputs that are subject to seasonal switching. Inputs are assumed to occur in a fluid fashion whereas outputs occur at a unit rate so long as the corresponding storage is non-empty. The distribution properties of the storage levels {Z ₁(t),Z ₂(t)} are derived at finite time as well as in stationary regime. We first investigate this process embedded at the successive switching points. This process is Markovian with independent components. In continuous time the components {Z ₁(t),Z ₂(t)} are also independent for each finite t, but are dependent in stationary regime.      

Optimal production mix

This paper deals with a production plant in which two different products can be produced. The plant consists of three subsystemsS _i . Before or after a phase of separate processing in subsystemsS ₁ andS ₂, the two products have to be processed in subsystemS ₃. Each of these subsystems has a limited capacity.In the first part, we assume empty stocks at the beginning; at a fixed timeT in the future, certain quantitiesX _i of the two products have to be delivered to the customers. Facing linear holding costs, convex production costs, and stringent capacity constraints, the problem is to decide when to produce which product at what rate.It is shown that the optimal solution consists of up to six different regimes and that the time paths of the production rates need not be monotonic. These results, which can be obtained analytically, are also illustrated in several numerical examples.Finally, the case is considered where the terminal demand at timeT is replaced by a continuous and seasonally fluctuating demand rate. It is demonstrated that the optimal production rates show an interesting and nontrivial behavior. In particular, it may happen that, on intervals where the demand for the one product increases, the optimal production rate decreases. This is also demonstrated by computer plots in some numerical examples.The first author gratefully acknowledges support from the Austrian Science Foundation under Grant S3204 and the second author from Stiftung Volkswagenwerk. An earlier version of this paper was presented at the DGOR-NSOR Joint Conference, Eindhoven, Holland, September 23–25, 1987.     

Inconsistencies in Plane-strain Elasto-plastic Rolling Simulations

Plane strain conditions are frequently used as model assumptions for the approximate calculation of flat rolling processes for sufficiently wide strips. This means that during the entire forming process, the strain rate in strip width direction (z-direction) remains zero. In the case of elasto-plastic material, the transitions from elastic to plastic states and vice versa are of particular interest. If we accept the flow rule of Levy-Mises, the normal stress σ_zz equals the average value of the normal stresses σ_xx and σ_yy inside plastic domains. Using Hooke's law we obtain a different relation for σ_zz in elastic domains. At transition points from elastic to plastic domains and vice versa, above relations for σ_zz have to be fulfilled simultaneously. Generally, this yields a discontinuous behaviour of σ_zz at the transition points and consequently to discontinuities of the stress σ_yy as well as the strains ε_zz and ε_yy, provided a continuous behaviour of σ_xx in rolling direction is postulated. The resulting discontinuity of ε_zz indicates therefore an inconsistency of such a plane strain model. Some key aspects of the underlying theory and consequences of such model assumptions will be pointed out. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong approximations for Markovian service networks

Inspired by service systems such as telephone call centers, we develop limit theorems for a large class of stochastic service network models. They are a special family of nonstationary Markov processes where parameters like arrival and service rates, routing topologies for the network, and the number of servers at a given node are all functions of time as well as the current state of the system. Included in our modeling framework are networks of M _t /M _t /n _t queues with abandonment and retrials. The asymptotic limiting regime that we explore for these networks has a natural interpretation of scaling up the number of servers in response to a similar scaling up of the arrival rate for the customers. The individual service rates, however, are not scaled. We employ the theory of strong approximations to obtain functional strong laws of large numbers and functional central limit theorems for these networks. This gives us a tractable set of network fluid and diffusion approximations. A common theme for service network models with features like many servers, priorities, or abandonment is “non-smooth” state dependence that has not been covered systematically by previous work. We prove our central limit theorems in the presence of this non-smoothness by using a new notion of derivative. This revised version was published online in June 2006 with corrections to the Cover Date.     

Linear sufficiency in a general growth curve model

The notion of linear sufficiency for the whole set of estimable functions in the general Gauss-Markov model is extended to the estimation of any special set of estimable functions in a general growth curve model. Some general results with respect to the concept of linear sufficiency are obtained, from which a necessary and sufficient condition is established for a linear transformation, {F₁,F₂}, of the observation matrix Y to have the property that there exists a linear function of which is the BLUE of the estimable functions .     

Pitfalls of the Fourier Transform Method in Affine Models,and Remedies

We study sources of potentially serious errors of popular numerical realizations of the Fourier method in affine models and explain that, in many cases, a calibration procedure based on such a realization will be able to find a “correct parameter set” only in a rather small region of the parameter space, with a blind spot: an interval of strikes depending on the model and time to maturity, where accurate calculations are extremely time-consuming. We explain how to construct more accurate and faster pricing and calibration procedures. An important ingredient of our method is the study of the analytic continuation of the solution of the associated system of generalized Riccati equations, and contour deformation techniques. As a byproduct, we show that the straightforward application of the Runge–Kutta method may lead to sizable errors, and suggest certain remedies. In the paper, the method is applied to a wide class of stochastic volatility models with stochastic interest rate and interest rate models of A_n(n) class. The methodology of the paper can be applied to other models (e.g., quadratic term structure models, Wishart dynamics, 3/2-model).     

Discrete Abstractions of Continuous Systems – an Input/Output Point of View

This contribution proposes a hierarchy of discrete ions for a given continuous model. It adopts an input/output point of view, and starts from the continuous system behaviour B _c (i.e., 'the set of all pairs of input and output signals which are compatible with the continuous model equations). The first step is to construct a sequence of behaviours B _l , l =0, 1,..., such that B ₀ ? B ₁ ?... ? B _c. In a second step, nondeterministic Moore automata A_l are generated as minimal realizations for the behaviours B _l . Hence, the continuous base system and its discrete abstractions A l form a totally ordered set of models, where ordering is in the sense of set inclusion of model behaviours or, equivalently, in terms of approximation accuracy. Within this set, there exists a uniquely defined “coarsest” (and therefore least complex) model which allows a given set of specifications to be enforced by discrete feedback. The ordering property implies that this discrete feedback also forces the continuous base system to obey the specifications.     

A two-stage stochastic integer programming approach as a mixture of Branch-and-Fix Coordination and Benders Decomposition schemes

We present an algorithmic approach for solving two-stage stochastic mixed 0–1 problems. The first stage constraints of the Deterministic Equivalent Model have 0–1 variables and continuous variables. The approach uses the Twin Node Family (TNF) concept within the so-called Branch-and-Fix Coordination algorithmic framework to satisfy the nonanticipativity constraints, jointly with a Benders Decomposition scheme to solve a given LP model at each TNF integer set. As a pilot case, the structuring of a portfolio of Mortgage-Backed Securities under uncertainty in the interest rate path on a given time horizon is used. Some computational experience is reported.     

Strong convergence rates for the estimation of a covariance operator for associated samples

Let X _n , n ≥ 1, be a strictly stationary associated sequence of random variables, with common continuous distribution function F. Using histogram type estimators we consider the estimation of the two-dimensional distribution function of (X ₁,X _k+1) as well as the estimation of the covariance function of the limit empirical process induced by the sequence X _n , n ≥ 1. Assuming a convenient decrease rate of the covariances Cov(X ₁,X _n+1), n ≥ 1, we derive uniform strong convergence rates for these estimators. The condition on the covariance structure of the variables is satisfied either if Cov(X ₁,X _n+1) decreases polynomially or if it decreases geometrically, but as we could expect, under the latter condition we are able to establish faster convergence rates. For the two-dimensional distribution function the rate of convergence derived under a geometrical decrease of the covariances is close to the optimal rate for independent samples.      

Negative binomial version of the Lee–Carter model for mortality forecasting

Mortality improvements pose a challenge for the planning of public retirement systems as well as for the private life annuities business. For public policy, as well as for the management of financial institutions, it is important to forecast future mortality rates. Standard models for mortality forecasting assume that the force of mortality at age x in calendar year t is of the form exp(α_x + β_xκ_t). The log of the time series of age-specific death rates is thus expressed as the sum of an age-specific component α_x that is independent of time and another component that is the product of a time-varying parameter κ_t reflecting the general level of mortality, and an age-specific component β_x that represents how rapidly or slowly mortality at each age varies when the general level of mortality changes. The parameters are usually estimated via singular value decomposition or via maximum likelihood in a binomial or Poisson regression model. This paper demonstrates that it is possible to take into account the overdispersion present in the mortality data by estimating the parameter in a negative binomial regression model. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimal controllability in viscoelasticity of rate type

Recently, the control engineers interest turns upon material with structural damping, such as hysteretic effects in elastoplastic systems [27] and flexible space structures [30] and in particular with damping due to the viscoelastic nature of the material, in the sense of Kelvin, [7], [12]. The reason for this renewed interest is the possibility of constructing finite rank compensators with small spillover in the observation [1], [5], [11], [25]. However, a control theory for the infinite system does not seem to exist in the literature. It is shown that the trajectories, initially in a certain subspace, can be steered to rest in any time T > 0 using distributed load controls

1 The reesults are partially contained in [19].

. Furthermore, and perhaps more interesting, the L₂ (0, T, H)-norm of the control decreases as T increases. This gives rise to the problem of time-optimal-controllability and the correlation to the problem of minimum norm controllability. It is shown that the time-optimal control is characterized by a weak “bang-bang” principle.     

Are regularly estimable functionals always differentiable?

Summary. According to van der Vaart (1991), regularly estimable functionals k \kappa are necessarily differentiable, if lim_{t? 0}t^-1(k(P_t)-k(P₀)) \lim\limits_{t\to 0}t^{-1}(\kappa(P_t)-\kappa(P_0)) exists for every differentiable path. In the present paper, a comparable result is obtained under slightly weaker conditions. A counterexample shows that these conditions are minimal.     

Finite element methods for nonlinear elliptic systems of second order

This paper deals with the finite element displacement method for approximating isolated solutions of general quasilinear elliptic systems. Under minimal assumptions on the structure of the continuous problems it is shown that the discrete analogues also have locally unique solutions which converge with quasi-optimal rates in L₂ and L∞. The essential tools of the proof are a deformation argument and a technique using weighted L₂-norms.     

A contact process with a semi-infected state on the complete graph

In this paper, we are concerned with a contact process with a semi-infected state on the complete graph C_n with n vertices. Our model is a special case of a general model introduced by Schinazi in 2003. In our model, each vertex is in one of three states, namely, “healthy,” “semi-infected,” or “fully-infected.” Only fully-infected vertices can infect others. A healthy vertex becomes semi-infected when being infected while a semi-infected vertex becomes fully-infected when being further infected. Each (semi- and fully-) infected vertex becomes healthy at constant rate. Our main result shows a phase transition for the waiting time until extinction of the fully-infected vertices. Conditioned on all the vertices are fully-infected when t = 0, we show that fully-infected vertices survive for exp?{O(n)} units of time when the infection rate λ > 4 while they die out in O(log?n) units of time when λ < 4.