Self-similarity and Lamperti convergence for families of stochastic processes

We define a new type of self-similarity for one-parameter families of stochastic processes, which applies to certain important families of processes that are not self-similar in the conventional sense. This includes Hougaard Lévy processes such as the Poisson processes, Brownian motions with drift and the inverse Gaussian processes, and some new fractional Hougaard motions defined as moving averages of Hougaard Lévy process. Such families have many properties in common with ordinary self-similar processes, including the form of their covariance functions, and the fact that they appear as limits in a Lamperti-type limit theorem for families of stochastic processes.     

Comparing Fleming-Viot and Dawson-Watanabe processes

Fleming-Viot processes and Dawson-Watanabe processes are two classes of “superprocesses” that have received a great deal of attention in recent years. These processes have many properties in common. In this paper, we prove a result that helps to explain why this is so. It allows one to prove certain theorems for one class when they are true for the other. More specifically, we show that product moments of a Fleming-Viot process can be bounded above by the corresponding moments of the Dawson-Watanabe process with the same “underlying particle motion”, and vice versa except for a multiplicative constant. As an application, we establish existence and continuity properties of local time for certain Fleming-Viot processes.     

Bivariate step processes and random evolutions

A random evolution process constructed from regular step processes with a common state space and indexed on an evolution rule space is shown to be a regular step process on the product space. Conversely, it is shown that under mild conditions, any regular step process on a product space is equivalent to a random evolution process. Conditions are given on the cardinality of the spaces and on the parameters of the process that are sufficient for the process to have various recurrence and ergodicity properties. Applications to birth-death processes are given.     

On some functionals of the first passage times in jump models of stochastic volatility

Abstract

We compute some functionals related to the generalized joint Laplace transforms of the first times at which two-dimensional jump processes exit half strips. It is assumed that the state space components are driven by Cox processes with both independent and common (positive) exponential jump components. The method of proof is based on the solutions of the equivalent partial integro-differential boundary-value problems for the associated value functions. The results are illustrated on several two-dimensional jump models of stochastic volatility which are based on non-affine analogs of certain mean-reverting or diverting diffusion processes representing closed-form solutions of the appropriate stochastic differential equations.     

Linear programming in tector criterion markov and semi-Markov decision processes

Optimality problems in infinite horizon, discrete time, vector criterion Markov and semi-Markov decision processes are expressed as standard problems of multiobjective linear programming. Processes with discounting, absorbing processes and completely ergodie processes without discounting are investigated. The common properties and special structure of derived multiobjective linear programming problems are overviewed. Computational simplicities associated with these problems in comparison with general multiobjective linear programming problems are discussed. Methods for solving these problems are overviewed and simple numerical examples are given.     

Supermodular Comparison of Time-to-Ruin Random Vectors

This paper studies time-to-ruin random vectors for multivariate risk processes. Two cases are considered: risk processes with independent increments and risk processes evolving in a common random environment (e.g., because they share the same economic conditions). As expected, increasing the dependence between the risk processes increases the dependence between their respective time-to-ruin random variables. This article is dedicated to the memory of our beloved friend Benjamin Zeev Levikson who passed away on July 16, 2005.     

A critical comparison of the various plausible inter-echelon gaming processes in supply chain models

An increasing number of supply-chain models are related to the following structure: (i) a manufacturer supplies a product to a retailer—who fixes a retail price and then retails the product to the consumers; (ii) the effect of the retail price on sales volume is dictated by a deterministic demand curve known to both parties. Results from these models depend very much on the ‘gaming process’ that is assumed to govern how the manufacturer and the retailer interact with each other. This paper reviews and compares some basic characteristics of seven seemingly plausible gaming processes; including the two most common ones: the manufacturer-Stackelberg and the fixed-markup-retailer processes. Our results show that: (i) each of the seven processes appear to be no less plausible than the other six; (ii) all seven processes possess some implausible characteristics; (iii) the relationships among the processes are confusing and do not appear to be intuitively logical; (iv) the relationships among these processes are further complicated by the way they are affected by the form of the assumed demand curve. Our results show that in supply-chain modelling more attention should be given to: (i) the proper selection of an appropriate gaming process assumption; (ii) how the model's results change under different gaming processes; (iii) the incorporation of information asymmetry that will allow these gaming-process assumptions to become more realistic.     

Economic lot scheduling problem with imperfect production processes and setup times

Deteriorating production processes are common in reality. Although every production process starts in an ‘in-control’ state to produce items of acceptable quality, it may shift to an ‘out-of-control’ state, owing to ageing, at any random time and produce defective items. In the present article, we study the Economic Lot Scheduling Problem (ELSP) with imperfect production processes having significant changeovers between the products. The mathematical models are developed for the ELSP using both the common cycle approach and the time-varying lot sizes approach, taking into account the effects of imperfect quality and process restoration. Numerical examples are cited to illustrate the solution procedures and to compare the performances of the solution methodologies adopted to solve the ELSP.     

Modeling Dependencies in Operational Risk with Hybrid Bayesian Networks

This paper addresses the problem of quantifying and modeling financial institutions’ operational risk in accordance with the Advanced Measurement Approach put forth in the Basel II Accord. We argue that standard approaches focusing on modeling stochastic dependencies are not sufficient to adequately assess operational risk. In addition to stochastic dependencies, causal topological dependencies between the risk classes are typically encountered. These dependencies arise when risk units have common information- and/or work-flows and when failure of upstream processes imply risk for downstream processes. In this paper, we present a modeling strategy that explicitly captures both topological and stochastic dependencies between risk classes. We represent the operational-risk taxonomy in the framework of a hybrid Bayesian network (BN) and provide an intuitively compelling approach for handling causal relationships and external influences. We demonstrate the use of hybrid BNs as a tool for mapping causal dependencies between frequencies and severities of risk events and for modeling common shocks. Monte-Carlo simulations illustrate that the impact of topological dependencies on triggering overall system breakdowns can be substantial.     

Optimal Mean-Variance Investment-Reinsurance Strategy for a Dependent Risk Model with Ornstein-Uhlenbeck Process

In this paper, we investigate the optimal investment-reinsurance strategy for an insurer with two dependent classes of insurance business, where the claim number processes are correlated through a common shock. It is assumed that the insurer can invest her wealth into one risk-free asset and multiple risky assets, and meanwhile, the instantaneous rates of investment return are stochastic and follow mean-reverting processes. Based on the theory of linear-quadratic control, we adopt a backward stochastic differential equation (BSDE) approach to solve the mean-variance optimization problem. Explicit expressions for both the efficient strategy and efficient frontier are derived. Finally, numerical examples are presented to illustrate our results.

     

Application and evaluation of hyper- and hypoelastic-based plasticity models in the FE simulation of metal forming processes

Metal forming processes are usually accompanied by large plastic strains and rotations of the material elements which emphasizes the need for reliable finite strain elastoplasticity models in corresponding FE simulations. In this work, two specific finite strain hyper- and hypoelastic-based plasticity models with combined nonlinear isotropic and kinematic hardening are presented and compared in numerical FE simulations. Although both models led to remarkably different results in a shear-dominated single element deformation test, the structural simulation of a standard deep drawing process delivered nearly congruent results which suggests that both models are equally well-suited for modeling metals in common forming processes. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positively homogeneous functions in atmospheric radiative transfer theory

Positively homogeneous functions are applied to describe absorption and scattering processes within the framework of atmospheric radiative transfer theory. Solid angle integrations are understood as surface integrations over arbitrary solid angle surfaces which are defined beyond the common picture of atmospheric radiative transfer.     

Fixed-width multiple-comparison procedures using common random numbers for steady-state simulations

Suppose that there are k ? 2 different systems (i.e., stochastic processes), where each system has an unknown steady-state mean performance. We consider the problem of running a two-stage simulation using common random numbers to construct fixed-width confidence intervals for two multiple-comparison problems. Under the assumptions that the stochastic processes representing the simulation output of the different systems satisfy a functional central limit theorem and that the asymptotic covariance matrix satisfies a condition known as sphericity, we prove that our confidence intervals are asymptotically valid (as the desired half-width of the confidence intervals tend to zero). We develop both absolute- and relative-width confidence intervals. Empirical results are presented indicating the procedures’ robustness to violations of the sphericity assumption.     

Discrete time modeling of mean-reverting stochastic processes for real option valuation

In this paper the recombining binomial lattice approach for modeling real options and valuing managerial flexibility is generalized to address a common issue in many practical applications, underlying stochastic processes that are mean-reverting. Binomial lattices were first introduced to approximate stochastic processes for valuation of financial options, and they provide a convenient framework for numerical analysis. Unfortunately, the standard approach to constructing binomial lattices can result in invalid probabilities of up and down moves in the lattice when a mean-reverting stochastic process is to be approximated. There have been several alternative methods introduced for modeling mean-reverting processes, including simulation-based approaches and trinomial trees, however they unfortunately complicate the numerical analysis of valuation problems. The approach developed in this paper utilizes a more general binomial approximation methodology from the existing literature to model simple homoskedastic mean-reverting stochastic processes as recombining lattices. This approach is then extended to model dual correlated one-factor mean-reverting processes. These models facilitate the evaluation of options with early-exercise characteristics, as well as multiple concurrent options.     

Bayesian planning of step‐stress accelerated degradation tests under various optimality criteria

Step‐stress accelerated degradation testing (SSADT) has become a common approach to predicting lifetime for highly reliable products that are unlikely to fail in a reasonable time under use conditions or even elevated stress conditions. In literature, the planning of SSADT has been widely investigated for stochastic degradation processes, such as Wiener processes and gamma processes. In this paper, we model the optimal SSADT planning problem from a Bayesian perspective and optimize test plans by determining both stress levels and the allocation of inspections. Large‐sample approximation is used to derive the asymptotic Bayesian utility functions under 3 planning criteria. A revisited LED lamp example is presented to illustrate our method. The comparison with optimal plans from previous studies demonstrates the necessity of considering the stress levels and inspection allocations simultaneously.     

Iterative procedures for solutions of random operator equations in Banach spaces

We construct random iterative processes for weakly contractive and asymptotically nonexpansive random operators and study necessary conditions for the convergence of these processes. It is shown that they converge to the random fixed points of these operators in the setting of Banach spaces. We also proved that an implicit random iterative process converges to the common random fixed point of a finite family of asymptotically quasi-nonexpansive random operators in uniformly convex Banach spaces.     

Information-based multi-assets artificial stock market with heterogeneous agents

In this paper, an artificial stock market characterized by heterogeneous and informed agents is presented. The heterogeneous agents are seen as nodes of sparsely connected graphs. The agents trade risky assets and are characterized by sentiments, amount of cash and stocks owned. Agents share information and sentiments by means of interactions determined by graphs. A central market maker (clearing house mechanism) determines the price processes for each stock at the intersection of the demand and supply curves. In this framework, the statistical properties of the univariate and multivariate process of prices and returns are studied. Importantly, concerning univariate price processes, the proposed model is able to reproduce unit root, volatility cluster and fat tails of returns. The multivariate price process exhibits both static and dynamic stylized facts, in particular the presence of static factors and common trends. Static factors are studied making reference to the cross-correlation between returns of different stocks, whereas the common trends are investigated considering the variance–covariance matrix of prices. The proposed approach allows to endogenously reproduce the multivariate stylized facts.     

On New Multivariate Probability Distributions and Stochastic Processes with System Reliability and Maintenance Applications

Two distinct methods for construction of some interesting new classes of multivariate probability densities are described and applied. As common results of both procedures three n-variate pdf classes are obtained. These classes are considered as generalizations of the class of univariate Weibullian, gamma, and multivariate normal pdfs. An example of an application of the obtained n-variate pdfs to the problem of modeling the reliability of multicomponent systems with stochastically dependent life-times of their components is given. Obtaining sequences over n = 2, 3, ... of consistent n-variate pdfs, that obey a relatively simple common pattern, for each n, allows us to extend some of the constructions from random vectors to discrete time stochastic processes. Application of one, so obtained, class of highly non-Markovian, but still sufficiently simple, stochastic processes for modeling maintenance of systems with repair, is presented. These models allow us to describe and analyze repaired systems with histories of all past repairs.      

The hyper-Dirichlet process and its discrete approximations: The butterfly model

The aim of this paper is the study of some random probability distributions, called hyper-Dirichlet processes. In the simplest situation considered in the paper these distributions charge the product of three sample spaces, with the property that the first and the last component are independent conditional to the middle one. The law of the marginals on the first two and on the last two components are specified to be Dirichlet processes with the same marginal parameter measure on the common second component. The joint law is then obtained as the hyper-Markov combination, introduced in [A.P. Dawid, S.L. Lauritzen, Hyper-Markov laws in the statistical analysis of decomposable graphical models, Ann. Statist. 21 (3) (1993) 1272-1317], of these two Dirichlet processes. The processes constructed in this way in fact are in fact generalizations of the hyper-Dirichlet laws on contingency tables considered in the above paper. Our main result is the convergence to the hyper-Dirichlet process of the sequence of hyper-Dirichlet laws associated to finer and finer “discretizations” of the two parameter measures, which is proved by means of a suitable coupling construction.     

Implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings

In this paper, we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. These implicit iterative algorithms are based on two well-known methods: extragradient and approximate proximal methods. We obtain some weak convergence theorems for these implicit iterative algorithms. Based on these theorems, we also construct some implicit iterative processes for finding a common fixed point of two mappings, such that one of these two mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive.