MC's for MCMC'ists

We develop a minimum amount of theory of Markov chains at as low a level of abstraction as possible in order to prove two fundamental probability laws for standard Markov chain Monte Carlo algorithms:
1. The law of large numbers explains why the algorithm works: it states that the empirical means calculated from the samples converge towards their "true" expected values, viz. expectations with respect to the invariant distribution of the associated Markov chain (=the target distribution of the simulation).
2. The central limit theorem expresses the deviations of the empirical means from their expected values in terms of asymptotically normally distributed random variables. We also present a formula and an estimator for the associated variance.     

A general framework for approximate sampling with an application to generating points on the boundary of bounded convex regions

We consider the problem of generating a sample of points according to some given probability distribution over some region. We give a general framework for constructing approximate sampling algorithms based on the theory of Markov chains. In particular, we show how it can be proven that a Markov chain has a limiting distribution. We apply these results to prove convergence for a class of so-called Shake-and-Bake algorithms, which can be used to approximate any absolutely continuous distribution over the boundary of a full-dimensional convex body.     

基于马氏链的供应链稳定性评价

从供应链运行过程中存在大量不确定因素,相应其产品的生产过程也表现出一定的波动性这一现象入手,根据生产过程具有显著马尔可夫性,以马尔可夫链理论为基础建立模型,对供应链稳定性进行测量评价。     

融合灰色理论与马尔科夫链的青岛市物流需求量预测

物流需求预测的准确性对物流基础设施投入和物流政策制定是至关重要的。根据物流需求受经济、社会发展和环境政策等不确定因素影响的特点,融合灰色理论与马尔科夫链,用灰色预测揭示系统时序变化的总体趋势,马尔科夫方法预测序列的随机波动的范围,能够优化灰色预测结果,提高预测的精度。以货运量为物流需求度量指标,建立灰色状态马尔科夫组合预测模型,对十二五期间青岛市物流需求量进行预测,预测结果可为青岛市十二五期间物流发展战略制定、物流服务体系构建和物流园区建设等工作提供理论参考和数据支撑。     

Probabilistic forecasting of industrial electricity load with regime switching behavior

This paper suggests a novel inhomogeneous Markov switching approach for the probabilistic forecasting of industrial companies’ electricity loads, for which the load switches at random times between production and standby regimes. The model that we propose describes the transitions between the regimes using a hidden Markov chain with time-varying transition probabilities that depend on calendar variables. We model the demand during the production regime using an autoregressive moving-average (ARMA) process with seasonal patterns, whereas we use a much simpler model for the standby regime in order to reduce the complexity. The maximum likelihood estimation of the parameters is implemented using a differential evolution algorithm. Using the continuous ranked probability score (CRPS) to evaluate the goodness-of-fit of our model for probabilistic forecasting, it is shown that this model often outperforms classical additive time series models, as well as homogeneous Markov switching models. We also propose a simple procedure for classifying load profiles into those with and without regime-switching behaviors.     

Exit problems in regime-switching models

This paper provides a general framework for pricing of perpetual American and real options in regime-switching Lévy models. In each state of the Markov chain, which determines switches from one Lévy process to another, the payoff stream is a monotone function of the Lévy process labeled by the state. This allows for additional switching within each state of the Markov chain (payoffs can be different in different regions of the real line). The pricing procedure is efficient even if the number of states is large provided the transition rates are not very large w.r.t. the riskless rates. The payoffs and riskless rates may depend on a state. Special cases are stochastic volatility models and models with stochastic interest rate; both must be modeled as finite-state Markov chains. As an application, we solve exit problems for a price-taking firm, and study the dependence of the exit threshold on the interest rate uncertainty.     

A MOMENT‐MATCHING METHOD FOR APPROXIMATING VECTOR AUTOREGRESSIVE PROCESSES BY FINITE‐STATE MARKOV CHAINS

This paper proposes a moment‐matching method for approximating vector autoregressions by finite‐state Markov chains. The Markov chain is constructed by targeting the conditional moments of the underlying continuous process. The proposed method is more robust to the number of discrete values and tends to outperform the existing methods for approximating multivariate processes over a wide range of the parameter space, especially for highly persistent vector autoregressions with roots near the unit circle. Copyright © 2013 John Wiley & Sons, Ltd.     

Bayesian analysis of nested logit model by Markov chain Monte Carlo

We develop a Markov chain Monte Carlo algorithm for estimating nested logit models in a Bayesian framework. Appropriate “heating target” and reparameterization techniques are adopted for fast mixing. For illustrative purposes, we have implemented the algorithm on two real-life examples involving 3-level structures. The first example involves social security's disability determination process (Soc. Security Bull. 58 (1995) 3). The second one is taken from Amemiya and Shimono's (Econ. Stud. Q. 40 (1989) 14) model of labor supply behavior of the aged. We applied a combination of various convergence criteria to ensure that the chain has converged to its target distribution.     

Optimal consumption and investment under partial information

We consider a stock market model where prices satisfy a stochastic differential equation with a stochastic drift process. The investor’s objective is to maximize the expected utility of consumption and terminal wealth under partial information; the latter meaning that investment decisions are based on the knowledge of the stock prices only. We derive explicit representations of optimal consumption and trading strategies using Malliavin calculus. The results apply to both classical models for the drift process, a mean reverting Ornstein-Uhlenbeck process and a continuous time Markov chain. The model can be transformed to a complete market model with full information. This allows to use results on optimization under convex constraints which are used in the numerical part for the implementation of more stable strategies. Supported by the Austrian Science Fund FWF, project P17947-N12. We thank two anonymous referees for their comments which led to a considerable improvement of the paper.     

A coupled Markov chain approach to credit risk modeling

We propose a Markov chain model for credit rating changes. We do not use any distributional assumptions on the asset values of the rated companies but directly model the rating transitions process. The parameters of the model are estimated by a maximum likelihood approach using historical rating transitions and heuristic global optimization techniques.We benchmark the model against a GLMM model in the context of bond portfolio risk management. The proposed model yields stronger dependencies and higher risks than the GLMM model. As a result, the risk optimal portfolios are more conservative than the decisions resulting from the benchmark model.     

Optimal investment of variance-swaps in jump-diffusion market with regime-switching

We consider a general jump-diffusion market with regime-switching where the jump risk is modeled as a Markov-modulated Poisson random measure. In this incomplete market, we price the variance-swaps using a combination of the Esscher transform and change of measure on time-inhomogeneous Markov chains. We study the dynamic optimal investment problem of the variance-swaps and characterize the optimal feedback strategy. Moreover, a closed-form solution to the HJB PDE associated with the stochastic control problem is established and the verification theorem is proved. The numerical analysis based on a two-state Markov chain uncovers some robust features of the optimal investment strategy.     

Bayesian analysis of a Tobit quantile regression model

This paper develops a Bayesian framework for Tobit quantile regression. Our approach is organized around a likelihood function that is based on the asymmetric Laplace distribution, a choice that turns out to be natural in this context. We discuss families of prior distributions on the quantile regression vector that lead to proper posterior distributions with finite moments. We show how the posterior distribution can be sampled and summarized by Markov chain Monte Carlo methods. A method for comparing alternative quantile regression models is also developed and illustrated. The techniques are illustrated with both simulated and real data. In particular, in an empirical comparison, our approach out-performed two other common classical estimators.     

US deficit sustainability: a new approach based on multiple endogenous breaks

Recent empirical work has questioned the consistency of US fiscal policy with an intertemporal budget constraint. Empirical results have tended to indicate that the deficit process has undergone at least one structural shift during recent decades, with the deficit becoming either unsustainable or sustainable in only a weak sense in the post‐shift period. In this paper, we re‐examine sustainability using a new approach, based on a cointegration model with multiple endogenous breaks. A Bayesian methodology is applied, incorporating Markov chain Monte Carlo simulators. In contrast to previous analyses, we find evidence of a sustainable deficit process over the 1947–1992 period, despite the occurrence of breaks during the 1970s and 1980s. Copyright © 2000 John Wiley & Sons, Ltd.     

On a bivariate Markov process arising in the theory of single-server retrial queues

We introduce a bivariate Markov process which can be seen as the joint process of the channel state and the number of customers in orbit of a Markovian single-server retrial queue with state dependent intensities. We obtain a necessary and sufficient condition for the process to be regular, and necessary and sufficient conditions for ergodicity and recurrence. A product-form formula for the stationary distribution is obtained. Besides, we study the busy period, the number of served customers and other related quantities. We show that for all the above problems there exist "equivalent" birth-and-death processes. However, a "uniformly equivalent" birth-and-death process does not exist.     

Randomly evolving tastes and delayed commitment

We consider a decision maker with randomly evolving tastes who faces dynamic decision situations that involve intertemporal tradeoffs, such as those in consumption savings problems. We axiomatize a recursive representation of choice that features uncertain consumption utilities, which evolve according to a subjective Markov process. The parameters of the representation, which are the subjective Markov process governing the evolution of utilities, and the discount factor, are uniquely identified from behavior. We relate the correlation of tastes over time and the desire to delay commitment to future consumption.     

Input- and Output-Oriented Technical Efficiency of Ukrainian Collective Farms, 1989–1992: Bayesian Analysis of a Stochastic Production Frontier Model

We propose estimation of a stochastic production frontier model within a Bayesian framework to obtain the posterior distribution of single-input-oriented technical efficiency at the firm level. All computations are carried out using Markov chain Monte Carlo methods. The approach is illustrated by applying it to production data obtained from a survey of Ukrainian collective farms. We show that looking at the changes in single-input-oriented technical efficiency in addition to the changes in output-oriented technical efficiency improves the understanding of the dynamics of technical efficiency over the first years of transition in the former Soviet Union.     

基于GSPN的装备器材供应链建模与分析

研究了广义随机Petri网（GSPN）的建模及分析方法,建立了基于GSPN的装备器材供应链流程模型,并将Petri网模型转化为等价的马尔可夫链,得出了供应链模型的主要性能指标,据此分析制约供应链的瓶颈。     

The position of agriculture in kibbutz economy—an attempt at quantitative projection

The Israeli Kibbutz Movement adopted the ideology of the “Return to Nature” and aspired to especially develop agricultural labour. But in the past decade there has been a process of “internal migration” of workers from agriculture to industrial occupations, which was compared to the “Industrial Revolution”. The ratio of agricultural workdays to other productive branches workdays decline from 62% in 1967 to 46.5% in 1974. Kibbutz' management has many reasons to evaluate its future direction, (investment in land, in education, etc.). We shall try to describe the expected developments, using the Markov chain. We shall also try to analyse the yearly interchanges and transfer from agriculture to industry and vice versa. We shall attempt to show that the process of decreasing agricultural manpower has almost ceased, and it will become stable on a fixed level of about 43%.     

Consistency and asymptotic normality of maximum likelihood estimation for Gaussian Markov processes from discrete observations

In this paper we prove the weak consistency and the asymptotic normality of the maximum likelihood estimation based on discrete observations ofn independent Gaussian Markov processes. The Ornstein Uhlenbeck process is a special Gaussian Markov process. We derive asymptotic simultaneous confidence regions for the parameters of the Ornstein Uhlenbeck process as an application.     

Empirical Bayes analysis of log-linear models for a generalized finite stationary Markov chain

This article presents the empirical Bayes method for estimation of the transition probabilities of a generalized finite stationary Markov chain whose ith state is a multi-way contingency table. We use a log-linear model to describe the relationship between factors in each state. The prior knowledge about the main effects and interactions will be described by a conjugate prior. Following the Bayesian paradigm, the Bayes and empirical Bayes estimators relative to various loss functions are obtained. These procedures are illustrated by a real example. Finally, asymptotic normality of the empirical Bayes estimators are established.