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A class of games for finding a leader among a group of candidates is studied in detail. This class covers games based on coin tossing and rock-paper-scissors as special cases and its complexity exhibits similar stochastic behaviors: either of logarithmic mean and bounded variance or of exponential mean and exponential variance. Many applications are also discussed.

In this note we re-examine the analysis of the paper "On the martingale property of stochastic exponentials" by B. Wong and C.C. Heyde, Journal of Applied Probability, 41(3):654-664, 2004. Some counterexamples are presented and alternative formulations are discussed., Comment: To appear in Journal of Applied Probability, 11 pages

In [5] Doeblin considered some classes of finite non-homogeneous Markov chains and gave without proofs several results concerning their asymptotic behaviour. In the present paper we first attempt to make Doeblin's results precise and try to reconstruct his arguments. Subsequently we investigate more general situations, where a state space decomposition is provided by the sets occurring in the representation of the atomic sets of the tail or-field. We show that Doeblin's notion of an associated chain, as well as considerations regarding the tail ar-field structure of the chain, can be used to solve such cases. FINITE MARKOV CHAIN; FINAL CLASS; CYCLICALLY MOVING SUBCLASS; TAIL ar-FIELD; ATOMIC SET; RECURRENCE; WHIRLPOOL

In a recent paper Weiss (1965) has suggested a simple model for a carrier-borne epidemic such as typhoid. He considers a population (of size m) of susceptibles into which a number (k) of carriers is introduced. These carriers exhibit no overt symptoms and are only detectable by the discovery of infected persons. He supposed that after the initial introduction of the carriers, the population remains entirely closed and no new carriers arise. The epidemic then progresses until either all the carriers have been traced and isolated or until the entire population has succumbed to the disease.

We illustrate a technique for computing certain integrals that arise in probability theory by giving a new derivation of a formula of Kingman. This formula contains the joint distribution of the processes F(t) = inf {s: X(t + s) = b} and B(t) = inf{s: X(t - s) = b} where X is a time homogeneous, continuous parameter, Markov process and b is a fixed point in its state space. We then extend this formula to the situation in which b is replaced by a finite set {b 1, …, b n }.

SUMMARYThe purpose of this study was to apply a novel statistical method for variable selection and a model-based approach for filling data gaps in mortality rates associated with foodborne diseases using the WHO Vital Registration mortality dataset. Correlation analysis and elastic net regularization methods were applied to drop redundant variables and to select the most meaningful subset of predictors. Whenever predictor data were missing, multiple imputation was used to fill in plausible values. Cluster analysis was applied to identify similar groups of countries based on the values of the predictors. Finally, a Bayesian hierarchical regression model was fit to the final dataset for predicting mortality rates. From 113 potential predictors, 32 were retained after correlation analysis. Out of these 32 predictors, eight with non-zero coefficients were selected using the elastic net regularization method. Based on the values of these variables, four clusters of countries were identified. The uncertainty of predictions was large for countries within clusters lacking mortality rates, and it was low for a cluster that had mortality rate information. Our results demonstrated that, using Bayesian hierarchical regression models, a data-driven clustering of countries and a meaningful subset of predictors can be used to fill data gaps in foodborne disease mortality.

The Hodrick–Prescott (HP) filter has been a popular method of trend extraction from economic time series. However, it is impractical without modification if some observations are not available. This paper improves the HP filter so that it can be applied in such situations. More precisely, this paper introduces two alternative generalized HP filters that are applicable for this purpose. We provide their properties and a way of specifying those smoothing parameters that are required for their application. In addition, we numerically examine their performance. Finally, based on our analysis, we recommend one of them for applied studies.

Initial insurance losses are often reported with a textual description of the claim. The claims manager must determine the adequate case reserve for each known claim. In this paper, we present a framework for predicting the amount of loss given a textual description of the claim using a large number of words found in the descriptions. Prior work has focused on classifying insurance claims based on keywords selected by a human expert, whereas in this paper the focus is on loss amount prediction with automatic word selection. In order to transform words into numeric vectors, we use word cosine similarities and word embedding matrices. When we consider all unique words found in the training dataset and impose a generalised additive model to the resulting explanatory variables, the resulting design matrix is high dimensional. For this reason, we use a group lasso penalty to reduce the number of coefficients in the model. The scalable, analytical framework proposed provides for a parsimonious and interpretable model. Finally, we discuss the implications of the analysis, including how the framework may be used by an insurance company and how the interpretation of the covariates can lead to significant policy change. The code can be found in the TAGAM R package (github.com/scottmanski/TAGAM).

Time variation is a fundamental problem in statistical and econometric analysis of macroeconomic and financial data. Recently, there has been considerable focus on developing econometric modelling that enables stochastic structural change in model parameters and on model estimation by Bayesian or nonparametric kernel methods. In the context of the estimation of covariance matrices of large dimensional panels, such data requires taking into account time variation, possible dependence and heavy-tailed distributions. In this paper, we introduce a nonparametric version of regularization techniques for sparse large covariance matrices, developed by Bickel and Levina (2008) and others. We focus on the robustness of such a procedure to time variation, dependence and heavy-tailedness of distributions. The paper includes a set of results on Bernstein type inequalities for dependent unbounded variables which are expected to be applicable in econometric analysis beyond estimation of large covariance matrices. We discuss the utility of the robust thresholding method, comparing it with other estimators in simulations and an empirical application on the design of minimum variance portfolios.

This paper develops an asymptotic theory of nonlinear least squares estimation by establishing a new framework that can be easily applied to various nonlinear regression models with heteroscedasticity. As an illustration, we explore an application of the framework to nonlinear regression models with nonstationarity and heteroscedasticity. In addition to these main results, this paper provides a maximum inequality for a class of martingales, which is of interest in its own right.

This paper establishes a new version of integration by parts formula of Markov chains for sensitivity computation, under much lower restrictions than the existing researches. Our approach is more fundamental and applicable without using Girsanov theorem or Malliavin calculus as did by past papers. Numerically, we apply this formula to compute sensitivity regarding the transition rate matrix and compare with a recent research by an IPA (infinitesimal perturbation analysis) method and other approaches.

Providing optimal strategies for maintaining technical systems in good working condition is an important goal in reliability engineering. The main aim of this paper is to propose some optimal maintenance policies for coherent systems based on some partial information about the status of components in the system. For this purpose, in the first part of the paper, we propose two criteria under which we compute the probability of the number of failed components in a coherent system with independent and identically distributed components. The first proposed criterion utilizes partial information about the status of the components with a single inspection of the system, and the second one uses partial information about the status of component failure under double monitoring of the system. In the computation of both criteria, we use the notion of the signature vector associated with the system. Some stochastic comparisons between two coherent systems have been made based on the proposed concepts. Then, by imposing some cost functions, we introduce new approaches to the optimal corrective and preventive maintenance of coherent systems. To illustrate the results, some examples are examined numerically and graphically.

In this paper, to model cascading failures, a new stochastic failure model is proposed. In a system subject to cascading failures, after each failure of the component, the remaining component suffers from increased load or stress. This results in shortened residual lifetimes of the remaining components. In this paper, to model this effect, the concept of the usual stochastic order is employed along with the accelerated life test model, and a new general class of stochastic failure models is generated.

Recently, Cavaliere, Georgiev, and Taylor (2018, Econometric Theory 34, 302–348) (CGT) considered the augmented Dickey–Fuller (ADF) test for a unit-root model with linear noise driven by i.i.d. infinite variance innovations and showed that ordinary least squares (OLS)-based ADF statistics have the same distribution as in Chan and Tran (1989, Econometric Theory 5, 354–362) for i.i.d. infinite variance noise. They also proposed an interesting question to extend their results to the case with infinite variance GARCH innovations as considered in Zhang, Sin, and Ling (2015, Stochastic Processes and their Applications 125, 482–512). This paper addresses this question. In particular, the limit distributions of the ADF for random walk models with short-memory linear noise driven by infinite variance GARCH innovations are studied. We show that when the tail index $\alpha , the limit distributions are completely different from that of CGT and the estimator of the parameters of the lag terms used in the ADF regression is not consistent. This paper provides a broad treatment of unit-root models with linear GARCH noises, which encompasses the commonly entertained unit-root IGARCH model as a special case.

In this paper, we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of Poisson distribution than {those} of normal distribution. We then show the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in \cite{CFS}. Our estimates contain no unspecified constants and are easy to apply. We illustrate the use of the theorems in six applications: Poisson-binomial distribution, matching problem, occupancy problem, birthday problem, random graphs and 2-runs. The paper complements the works of \cite{CC92,BCC95,CFS}., 29 pages and 5 figures

Large samples from a light-tailed distribution often have a well-defined shape. This paper examines the implications of the assumption that there is a limit shape. We show that the limit shape determines the upper quantiles for a large class of random variables. These variables may be described loosely as continuous homogeneous functionals of the underlying random vector. They play an important role in evaluating risk in a multivariate setting. The paper also looks at various coefficients of tail dependence and at the distribution of the scaled sample points for large samples. The paper assumes convergence in probability rather than almost sure convergence. This results in an elegant theory. In particular, there is a simple characterization of domains of attraction.

This paper generalizes the Kunita–Watanabe decomposition of an $L^2$ space. The generalization comes from using nonlinear stochastic integrals where the integrator is a family of continuous martingales bounded in $L^2$ . This result is also the solution of an optimization problem in $L^2$ . First, martingales are assumed to be stochastic integrals. Then, to get the general result, it is shown that the regularity of the family of martingales with respect to its spatial parameter is inherited by the integrands in the integral representation of the martingales. Finally, an example showing how the results of this paper, with the Clark–Ocone formula, can be applied to polynomial functions of Brownian integrals.

This paper considers two variants of M/G/1 queues with impatient customers, which are denoted by M/G/1+Gw and M/G/1+Gs. In the M/G/1+Gw queue customers have deadlines for their waiting times, and they leave the system immediately if their services do not start before the expiration of their deadlines. On the other hand, in the M/G/1+Gs queue customers have deadlines for their sojourn times, where customers in service also immediately leave the system when their deadlines expire. In this paper we derive comparison results for performance measures of these models. In particular, we show that if the service time distribution is new better than used in expectation, then the loss probability in the M/G/1+Gs queue is greater than that in the M/G/1+Gw queue.

This paper studies a periodic-review single-commodity setup-cost inventory model with backorders and holding/backlog costs satisfying quasiconvexity assumptions. We show that the Markov decision process for this inventory model satisfies the assumptions that lead to the validity of optimality equations for discounted and average-cost problems and to the existence of optimal (s,S) policies. In particular, we prove the equicontinuity of the family of discounted value functions and the convergence of optimal discounted lower thresholds to the optimal average-cost lower threshold for some sequence of discount factors converging to 1. If an arbitrary nonnegative amount of inventory can be ordered, we establish stronger convergence properties: (i) the optimal discounted lower thresholds converge to optimal average-cost lower threshold; and (ii) the discounted relative value functions converge to average-cost relative value function. These convergence results previously were known only for subsequences of discount factors even for problems with convex holding/backlog costs. The results of this paper also hold for problems with fixed lead times.

In this paper, we extend the strong laws of large numbers and entropy ergodic theorem for partial sums for tree-indexed nonhomogeneous Markov chains fields to delayed versions of nonhomogeneous Markov chains fields indexed by a homogeneous tree. At first we study a generalized strong limit theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Then we prove the generalized strong laws of large numbers and the generalized asymptotic equipartition property for delayed sums of finite nonhomogeneous Markov chains indexed by a homogeneous tree. As corollaries, we can get the similar results of some current literatures. In this paper, the problem settings may not allow to use Doob's martingale convergence theorem, and we overcome this difficulty by using Borel–Cantelli Lemma so that our proof technique also has some new elements compared with the reference Yang and Ye (2007).

Monotonicity is a key qualitative prediction of a wide array of economic models derived via robust comparative statics. It is therefore important to design eff ective and practical econometric methods for testing this prediction in empirical analysis. This paper develops a general nonparametric framework for testing monotonicity of a regression function. Using this framework, a broad class of new tests is introduced, which gives an empirical researcher a lot of flexibility to incorporate ex ante information she might have. The paper also develops new methods for simulating critical values, which are based on the combination of a bootstrap procedure and new selection algorithms. These methods yield tests that have correct asymptotic size and are asymptotically nonconservative. It is also shown how to obtain an adaptive rate optimal test that has the best attainable rate of uniform consistency against models whose regression function has Lipschitz-continuous fi rst-order derivatives and that automatically adapts to the unknown smoothness of the regression function. Simulations show that the power of the new tests in many cases signi ficantly exceeds that of some prior tests, e.g. that of Ghosal, Sen, and Van der Vaart (2000). An application of the developed procedures to the dataset of Ellison and Ellison (2011) shows that there is some evidence of strategic entry deterrence in pharmaceutical industry where incumbents may use strategic investment to prevent generic entries when their patents expire.

In this paper, we introduce a new class of set-valued risk measures which satisfies cash sub-additivity. Dual representation for them is provided. Moreover, we also investigate dynamic set-valued cash sub-additive risk measures and discuss the corresponding multi-portfolio time consistency. The equivalent characterization of the multi-portfolio time consistency is given. Finally, an example is also given to illustrate the introduction of set-valued cash sub-additive risk measures. The present paper can be considered as a set-valued extension of scalar cash sub-additive risk measures introduced by El Karouii and Ravanelli [8].

The relation between extropy and variational distance is studied in this paper. We determine the distribution which attains the minimum or maximum extropy among these distributions within a given variation distance from any given probability distribution, obtain the tightest upper bound on the difference of extropies of any two probability distributions subject to the variational distance constraint, and establish an analytic formula for the confidence interval of an extropy. Such a study parallels to that of Ho and Yeung [3] concerning entropy. However, the proofs of the main results in this paper are different from those in Ho and Yeung [3]. In fact, our arguments can simplify several proofs in Ho and Yeung [3].

In this paper, we are going to study the strong limit theorem for the relative entropy density rates between two finite asymptotically circular Markov chains. Firstly, we prove some lammas on which the main result based. Then, we establish two strong limit theorem for non-homogeneous Markov chains. Finally, we obtain the main result of this paper. As corollaries, we get the strong limit theorem for the relative entropy density rates between two finite non-homogeneous Markov chains. We also prove that the relative entropy density rates between two finite non-homogeneous Markov chains are uniformly integrable under some conditions.

This paper weakens the size and moment conditions needed for typical block bootstrap methods (i.e., the moving blocks, circular blocks, and stationary bootstraps) to be valid for the sample mean of Near-Epoch-Dependent (NED) functions of mixing processes; they are consistent under the weakest conditions that ensure the original NED process obeys a central limit theorem (CLT), established by De Jong (1997, Econometric Theory 13(3), 353–367). In doing so, this paper extends De Jong’s method of proof, a blocking argument, to hold with random and unequal block lengths. This paper also proves that bootstrapped partial sums satisfy a functional CLT (FCLT) under the same conditions.

In this paper, we discuss the ordering properties of sample ranges arising from multiple-outlier exponential and proportional hazard rate (PHR) models. The purpose of this paper is twofold. First, sufficient conditions on the parameter vectors are provided for the reversed hazard rate order and the usual stochastic order between the sample ranges arising from multiple-outlier exponential models with common sample size. Next, stochastic comparisons are separately carried out for sample ranges arising from multiple-outlier exponential and PHR models with different sample sizes as well as different hazard rates. Some numerical examples are also presented to illustrate the results established here.

In this paper, we mainly study the moderate deviation principle of sample quantiles and order statistics for stationary m-dependent random variables. The results obtained in this paper extend the corresponding ones for an independent and identically distributed sequence to a stationary m-dependent sequence.

In the first part of this paper we consider a general stationary subcritical cluster model in ℝd. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein–Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE solution in the special case of a Poisson-driven random connection model.

We analyse ruin probabilities for an insurance risk process with a more generalised dependence structure compared to the one introduced in Constantinescu et al. (2016). In this paper, we assume that a random threshold window is generated every time after a claim occurs. By comparing the previous inter-claim time with the threshold window, the distributions of the current threshold window and the inter-arrival time are determined. Furthermore, the statuses for the previous and current inter-arrival times give rise to the current claim size distribution as well. Like Constantinescu et al. (2016), we first identify the embedded Markov additive process where all the randomness takes a general form. Inspired by the Erlangisation technique, the key message of this paper is to analyse such risk process using a Markov fluid flow model where the underlying random variables follow phase-type distributions. This would further allow us to approximate the fixed observation windows by Erlang random variables. Then ruin probabilities under the process with Erlang(n) observation windows are proved to be Erlangian approximations for those related to the process with fixed threshold windows at the limit. An exact form of the limit can be obtained whose application will be illustrated further by a numerical example.

This paper introduces the Fourier Space Time-Stepping algorithm to the valuation of variable annuity (VA) contracts embedded with guaranteed minimum withdrawal benefit (GMWB) riders when the underlying fund dynamics evolve under the influence of a regime-switching model. Mortality risk is introduced to the valuation framework by incorporating a two-factor affine stochastic mortality model proposed in Blackburn and Sherris (2013). The paper considers both, static and dynamic policyholder withdrawal behaviour associated with GMWB riders and assesses how model parameters influence the fees levied on providing such guarantees. Our numerical experiments reveal that the GMWB fees are very sensitive to regime-switching parameters; a percentage increase in the force of interest results in significant decrease in guarantee fees. The guarantee fees increase substantially with increasing volatility levels. Numerical experiments also highlight an increasing importance of mortality as maturity of the VA contract increases. Mortality has less impact on shorter maturity contracts regardless of the policyholder's withdrawal behaviour. As much as mortality influences pricing results for long maturities, the associated guarantee fees are decreasing functions of maturities for the VA contracts. Robustness checks of the Fourier Space Time-Stepping algorithm are performed by making numerical comparisons with several existing valuation approaches.

This paper studies the GMM estimation and inference problem that occurs when the Jacobian of the moment conditions is rank deficient of known forms at the true parameter values. Dovonon and Renault (2013) recently raised a local identification issue stemming from this type of degenerate Jacobian. The local identification issue leads to a slow rate of convergence of the GMM estimator and a nonstandard asymptotic distribution of the over-identification test statistics. We show that the known form of rank-deficient Jacobian matrix contains nontrivial information about the economic model. By exploiting such information in estimation, we provide GMM estimator and over-identification tests with standard properties. The main theory developed in this paper is applied to the estimation of and inference about the common conditionally heteroskedastic (CH) features in asset returns. The performances of the newly proposed GMM estimators and over-identification tests are investigated under the similar simulation designs used in Dovonon and Renault (2013).

In this paper, some results on the complete moment convergence for arrays of rowwise negatively associated (NA, for short) random variables are established. The results obtained in this paper correct the corresponding one obtained in Ko [13] and also improve and generalize the corresponding ones of Kuczmaszewska [14] and Ko [13]. As an application of the main results, we present a result on complete consistency for the estimator in a non-parametric regression model based on NA errors. Finally, we provide a numerical simulation to verify the validity of our result.

In this paper, we develop certain properties for discrete Brownian bridges and Ornstein–Uhlenbeck bridges, which we use in the successor papers Part 3B and Part 3C to analyse real economic data series, with a view to constructing stochastic interpolation models for the Wilkie asset model.

An influential paper by Kleibergen (2005,Econometrica73, 1103–1123) introduces Lagrange multiplier (LM) and conditional likelihood ratio-like (CLR) tests for nonlinear moment condition models. These procedures aim to have good size performance even when the parameters are unidentified or poorly identified. However, the asymptotic size and similarity (in a uniform sense) of these procedures have not been determined in the literature. This paper does so.This paper shows that the LM test has correct asymptotic size and is asymptotically similar for a suitably chosen parameter space of null distributions. It shows that the CLR tests also have these properties when the dimensionpof the unknown parameterθequals 1. Whenp≥ 2, however, the asymptotic size properties are found to depend on how the conditioning statistic, upon which the CLR tests depend, is weighted. Two weighting methods have been suggested in the literature. The paper shows that the CLR tests are guaranteed to have correct asymptotic size whenp≥ 2 when the weighting is based on an estimator of the variance of the sample moments, i.e., moment-variance weighting, combined with the Robin and Smith (2000,Econometric Theory16, 151–175) rank statistic. The paper also determines a formula for the asymptotic size of the CLR test when the weighting is based on an estimator of the variance of the sample Jacobian. However, the results of the paper do not guarantee correct asymptotic size whenp≥ 2 with the Jacobian-variance weighting, combined with the Robin and Smith (2000,Econometric Theory16, 151–175) rank statistic, because two key sample quantities are not necessarily asymptotically independent under some identification scenarios.Analogous results for confidence sets are provided. Even for the special case of a linear instrumental variable regression model with two or more right-hand side endogenous variables, the results of the paper are new to the literature.

The calculation of a fair premium is always a challenging topic in the real-world insurance applications. In this paper, a non-linear premium-reserve (P-R) model is presented and the premium is derived by minimising a quadratic performance criterion. The reserve is a stochastic equation, which includes an additive random non-linear function of the state, premium and not necessarily Gaussian noise, which is, however, independently distributed in time, provided only that the mean value and the covariance of the random function is 0 and a quadratic function of the state, premium and other parameters, respectively. In this quadratic representation of the covariance function, new parameters are implemented and enriched further by the previous linear models, such as the income insurance elasticity of demand, the number of insured and the inflation in addition to the company’s reputation. The quadratic utility function concerns the present value of the reserve. Interestingly, for the very first time, the derived optimal premium in a competitive market environment is also dependent on the company’s reserve among the other parameters. Finally, a numerical application illustrates the main findings of the paper.

In this paper, we project future mortality rates for actuarial use with Chinese data using a modified Continuous Mortality Investigation (CMI) Mortality Projections Model. The model adopts a convergence structure from “initial” to “long-term” rates of mortality improvement as the process of projection. The initial rates of mortality improvement are derived using two-dimensional P-spline methodology. Given the short history of Chinese data, the long-term rates of mortality improvement are determined by borrowing information from international experience. K-means clustering with dynamic time warping distance is used to classify populations, which is novel in the actuarial mortality research field. The original CMI approach is deterministic, however, in this paper we make it stochastic using techniques outlined by Koller and described by Browne et al. Comparing our results with a pure extrapolative approach, we find that the CMI Mortality Projections Model is more suitable for long-term projections for China.

The paper “Kernel estimation when density may not exist” (Zinde-Walsh, 2008) considered density as a generalized function given by a functional on a space of smooth functions; this made it possible to establish the limit properties of the kernel estimator without assuming the existence of the density function. This note corrects an error in that paper in the derivation of the variance of the kernel estimator. The corrected result is that in the space of generalized functions the parametric rate of convergence of the kernel density estimator to the limit Gaussian process is achievable.

This thesis consists of five papers, presented in Chapters A-E, on topics in life and disability insurance. It is naturally divided into two parts, where papers A and B discuss disability rates estimation based on historical claims data, and papers C-E discuss claims reserving, risk management and insurer solvency.In Paper A, disability inception and recovery probabilities are modelled in a generalized linear models (GLM) framework. For prediction of future disability rates, it is customary to combine GLMs with time series forecasting techniques into a two-step method involving parameter estimation from historical data and subsequent calibration of a time series model. This approach may in fact lead to both conceptual and numerical problems since any time trend components of the model are incoherently treated as both model parameters and realizations of a stochastic process. In Paper B, we suggest that this general two-step approach can be improved in the following way: First, we assume a stochastic process form for the time trend component. The corresponding transition densities are then incorporated into the likelihood, and the model parameters are estimated using the Expectation-Maximization algorithm.In Papers C and D, we consider a large portfolio of life or disability annuity policies. The policies are assumed to be independent conditional on an external stochastic process representing the economic-demographic environment. Using the Conditional Law of Large Numbers (CLLN), we establish the connection between claims reserving and risk aggregation for large portfolios. Moreover, we show how statistical multi-factor intensity models can be approximated by one-factor models, which allows for computing reserves and capital requirements efficiently. Paper C focuses on claims reserving and ultimate risk, whereas the focus of Paper D is on the one-year risks associated with the Solvency II directive.In Paper E, we consider claims reserving for life insurance policies with reserve-dependent payments driven by multi-state Markov chains. The associated prospective reserve is formulated as a recursive utility function using the framework of backward stochastic differential equations (BSDE). We show that the prospective reserve satisfies a nonlinear Thiele equation for Markovian BSDEs when the driver is a deterministic function of the reserve and the underlying Markov chain. Aggregation of prospective reserves for large and homogeneous insurance portfolios is considered through mean-field approximations. We show that the corresponding prospective reserve satisfies a BSDE of mean-field type and derive the associated nonlinear Thiele equation.

The Ginibre point process (GPP) is one of the main examples of determinantal point processes on the complex plane. It is a recurring distribution of random matrix theory as well as a useful model in applied mathematics. In this paper we briefly overview the usual methods for the simulation of the GPP. Then we introduce a modified version of the GPP which constitutes a determinantal point process more suited for certain applications, and we detail its simulation. This modified GPP has the property of having a fixed number of points and having its support on a compact subset of the plane. See Decreusefond et al. (2013) for an extended version of this paper.

This paper considers Markov decision processes (MDPs) with unbounded rates, as a function of state. We are especially interested in studying structural properties of optimal policies and the value function. A common method to derive such properties is by value iteration applied to the uniformised MDP. However, due to the unboundedness of the rates, uniformisation is not possible, and so value iteration cannot be applied in the way we need. To circumvent this, one can perturb the MDP. Then we need two results for the perturbed sequence of MDPs: 1. there exists a unique solution to the discounted cost optimality equation for each perturbation as well as for the original MDP; 2. if the perturbed sequence of MDPs converges in a suitable manner then the associated optimal policies and the value function should converge as well. We can model both the MDP and perturbed MDPs as a collection of parametrised Markov processes. Then both of the results above are essentially implied by certain continuity properties of the process as a function of the parameter. In this paper we deduce tight verifiable conditions that imply the necessary continuity properties. The most important of these conditions are drift conditions that are strongly related to nonexplosiveness.

De Bruijn's identity shows a link between two fundamental concepts in information theory: entropy and Fisher information. In the literature, De Bruijn's identity has been stated under the assumption of independence between input signal and an additive noise. However, in the real world, the noise could be highly dependent on the main signal. The main aim of this paper is, firstly, to extend De bruijn's identity for signal-dependent noise channels and, secondly, to study how Stein and heat identities are related to De bruijn's identity. Thus, new versions of De Bruijn's identity are introduced in the case when input signal and additive noise are dependent and are jointly distributed according to Archimedean and Gaussian copulas. It is shown that in this generalized model, the derivatives of the differential entropy can be expressed in terms of a function of Fisher information. Our finding enfolds the conventional De Bruijn's identity as some remarks. Then, the equivalence among the new De Bruijn-type identity, Stein's identity and heat equation identity is established. The paper concludes with an application of the developed results in information theory.

In this paper we use tools from differential topology to give a geometric description of cohomology for Hilbert manifolds. Our model is Quillen’s geometric description of cobordism groups for finite-dimensional smooth manifolds [Quillen, ‘Elementary proofs of some results of cobordism theory using steenrod operations’, Adv. Math., 7 (1971)]. Quillen stresses the fact that this construction allows the definition of Gysin maps for ‘oriented’ proper maps. For finite-dimensional manifolds one has a Gysin map in singular cohomology which is based on Poincaré duality, hence it is not clear how to extend it to infinite-dimensional manifolds. But perhaps one can overcome this difficulty by giving a Quillen type description of singular cohomology for Hilbert manifolds. This is what we do in this paper. Besides constructing a general Gysin map, one of our motivations was a geometric construction of equivariant cohomology, which even for a point is the cohomology of the infinite-dimensional space $BG$, which has a Hilbert manifold model. Besides that, we demonstrate the use of such a geometric description of cohomology by several other applications. We give a quick description of characteristic classes of a finite-dimensional vector bundle and apply it to a generalized Steenrod representation problem for Hilbert manifolds and define a notion of a degree of proper oriented Fredholm maps of index $0$.

In a defaultable market, an investor trades having only partial information about the behavior of the market. Taking into account the intraday stock movements, the risky asset prices are modelled by marked point processes. Their dynamics depend on an unobservable process, representing the amount of news reaching the market. This is a marked point process, which may have common jump times with the risky asset price processes. The problem of hedging a defaultable claim is studied. In order to discuss all these topics, in this paper we examine stochastic control problems using backward stochastic differential equations (BSDEs) and filtering techniques. The goal of this paper is to construct a sequence of functions converging to the value function, each of these is the unique solution of a suitable BSDE.

In this paper we investigate the limiting behavior of the failure rate for the convolution of two or more life distributions. In a previous paper on mixtures, Block, Mi and Savits (1993) showed that the failure rate behaves like the limiting behavior of the strongest component. We show a similar result here for convolutions. We also show by example that unlike a mixture population, the ultimate direction of monotonicity does not necessarily follow that of the strongest component.

To construct forecasts for time series exhibiting breaks, the paper examines long autoregressions, where the number of lags is growing with T, and possible breaks are simply ignored. The paper shows that the OLS estimators are still elementwise consistent for the true autoregressive coefficients when neglecting a break in mean, but the sum of the estimators converges to unity. Thanks to this unit-root like behavior of the fitted model, the resulting conditional forecasts are consistent for the true values. As long as the dynamic structure is invariant, the robustness property of the forecasts holds a) under data-dependent lag length selection, b) for a piecewise smoothly varying mean function, and c) under general autoregressive dynamics of possibly infinite order including stationary long memory. Under breaks in the dynamic structure, however, estimators are asymptotically biased, and the forecasts from long autoregressions are biased themselves even in the limit.

In this paper we study the speed of infection spread and the survival of the contact process in the random geometric graph G = G(n, r n , f) of n nodes independently distributed in S = [-½, ½]2 according to a certain density f(·). In the first part of the paper we assume that infection spreads from one node to another at unit rate and that infected nodes stay in the same state forever. We provide an explicit lower bound on the speed of infection spread and prove that infection spreads in G with speed at least D 1 nr n 2. In the second part of the paper we consider the contact process ξ t on G where infection spreads at rate λ > 0 from one node to another and each node independently recovers at unit rate. We prove that, for every λ > 0, with high probability, the contact process on G survives for an exponentially long time; there exist positive constants c 1 and c 2 such that, with probability at least 1 - c 1 / n 4, the contact process starting with all nodes infected survives up to time t n = exp(c 2 n/logn) for all n.

This paper proposes a semiparametric generalized method of moments estimator (GMM) estimator for a partially parametric spatial model with endogenous spatially dependent regressors. The finite-dimensional estimator is shown to be consistent and root-n asymptotically normal under some reasonable conditions. A spatial heteroscedasticity and autocorrelation consistent covariance estimator is constructed for the GMM estimator. The leading application is nonlinear spatial autoregressions, which arise in a wide range of strategic interaction models. To derive the asymptotic properties of the estimator, the paper also establishes a stochastic equicontinuity criterion and functional central limit theorem for near-epoch dependent random fields.

Let 𝓈 be a finite or countable set. Given a matrix F = (F ij ) i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (q ij ) i,j∈𝓈 , i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑ n≥0 Q n ⊗ F *n associated with Q ⊗ F := (q ij F ij ) i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(M n , S n )} n≥0 with discrete recurrent driving chain {M n } n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

This paper considers the problem of coverage of the elements of the identified set in a class of partially identified econometric models with a prespecified probability. In order to conduct inference in partially identified econometric models defined by moment (in)equalities, the literature has proposed three methods: bootstrap, subsampling, and asymptotic approximation. The objective of this paper is to compare these methods in terms of the rate at which they achieve the desired coverage level, i.e., in terms of the rate at which theerror in the coverage probability(ECP) converges to zero.Under certain conditions, we show that the ECP of the bootstrap and the ECP of the asymptotic approximation converge to zero at the same rate, which is a faster rate than that of the ECP of subsampling methods. As a consequence, under these conditions, the bootstrap and the asymptotic approximation produce inference that is more precise than subsampling. A Monte Carlo simulation study confirms that these results are relevant in finite samples.

Suppose that X 1, …, X n are random variables with the same known marginal distribution F but unknown dependence structure. In this paper we study the smallest possible value of P(X 1 + · · · + X n < s) over all possible dependence structures, denoted by m n,F (s). We show that m n,F (ns) → 0 for s no more than the mean of F under weak assumptions. We also derive a limit of m n,F (ns) for any s ∈ R with an error of at most n -1/6 for general continuous distributions. An application of our result to risk management confirms that the worst-case value at risk is asymptotically equivalent to the worst-case expected shortfall for risk aggregation with dependence uncertainty. In the last part of this paper we present a dual presentation of the theory of complete mixability and give dual proofs of theorems in the literature on this concept.