Hidden Markov Filter Estimation of the Occurrence Time of an Event in a Financial Market

Abstract

In this paper, we use filtering techniques to estimate the occurrence time of an event in a financial market. The occurrence time is being viewed as a Markov stopping time with respect to the σ-field generated by a hidden Markov process. We also generalize our result to the Nth occurrence time of that event.     

Bayesian Analysis of a Two-State Markov Modulated Poisson Process

Abstract

We postulate observations from a Poisson process whose rate parameter modulates between two values determined by an unobserved Markov chain. The theory switches from continuous to discrete time by considering the intervals between observations as a sequence of dependent random variables. A result from hidden Markov models allows us to sample from the posterior distribution of the model parameters given the observed event times using a Gibbs sampler with only two steps per iteration.     

Dempster–Shafer fusion of evidential pairwise Markov fields

Hidden Markov fields (HMFs) have been successfully used in many areas to take spatial information into account. In such models, the hidden process of interest X is a Markov field, that is to be estimated from an observable process Y. The possibility of such estimation is due to the fact that the conditional distribution of the hidden process with respect to the observed one remains Markovian. The latter property remains valid when the pairwise process $(X,Y)$ is Markov and such models, called pairwise Markov fields (PMFs), have been shown to offer larger modeling capabilities while exhibiting similar processing cost. Further extensions lead to a family of more general models called triplet Markov fields (TMFs) in which the triplet $(U,X,Y)$ is Markov where U is an underlying process that may have different meanings according to the application. A link has also been established between these models and the theory of evidence, opening new possibilities of achieving Dempster–Shafer fusion in Markov fields context. The aim of this paper is to propose a unifying general formalism allowing all conventional modeling and processing possibilities regarding information imprecision, sensor unreliability and data fusion in Markov fields context. The generality of the proposed formalism is shown theoretically through some illustrative examples dealing with image segmentation, and experimentally on hand-drawn and SAR images.     

Exponential bounds for convergence of entropy rate approximations in hidden Markov models satisfying a path-mergeability condition

A hidden Markov model (HMM) is said to have path-mergeable states   if for any two states i,j$i,j$ there exist a word w$w$ and state k$k$ such that it is possible to transition from both i$i$ and j$j$ to k$k$ while emitting w$w$. We show that for a finite HMM with path-mergeable states the block estimates of the entropy rate converge exponentially fast. We also show that the path-mergeability property is asymptotically typical in the space of HMM topologies and easily testable.     

A genetic filtering problem ∗

Members of a population of fixed size N can be in any one of n states. In discrete time the individuals jump from one state to another, independently of each other, and with probabilities described by a homogeneous Markov chain. At each time a sample of size M is withdrawn, (with replacement). Based on these observations, and using the techniques of Hidden Markov Models, recursive estimates for the distribution of the population are obtained     

Filtering of a Multi-Dimension Stochastic Volatility Model

In this article, we study a stochastic volatility model for a class of risky assets. We assume that the volatilities of the assets are driven by a common state of economy, which is unobservable and represented by a hidden Markov chain. Under this hidden Markov model (HMM), we develop recursively computable filtering equations for certain functionals of the chain. Expectation maximization (EM) parameter estimation is then used. Applications to an optimal asset allocation problem with mean-variance utility are given.     

Cross-entropic learning of a machine for the decision in a partially observable universe

In this paper, we are interested in optimal decisions in a partially observable universe. Our approach is to directly approximate an optimal strategic tree depending on the observation. This approximation is made by means of a parameterized probabilistic law. A particular family of Hidden Markov Models (HMM), with input and output, is considered as a model of policy. A method for optimizing the parameters of these HMMs is proposed and applied. This optimization is based on the cross-entropic (CE) principle for rare events simulation developed by Rubinstein.     

On the distribution of the total number of run lengths

In the present paper, we study the distribution of a statistic utilizing the runs length of “reasonably long” series of alike elements (success runs) in a sequence of binary trials. More specifically, we are looking at the sum of exact lengths of subsequences (strings) consisting ofk or more consecutive successes (k is a given positive integer). The investigation of the statistic of interest is accomplished by exploiting an appropriate generalization of the Markov chain embedding technique introduced by Fu and Koutras (1994,J. Amer. Statist. Assoc.,89, 1050–1058) and Koutras and Alexandrou (1995,Ann. Inst. Statist. Math.,47, 743–766). In addition, we explore the conditional distribution of the same statistic, given the number of successes and establish statistical tests for the detection of the null hypothesis of randomness versus the alternative hypothesis of systematic clustering of successes in a sequence of binary outcomes. Research supported by General Secretary of Research and Technology of Greece under grand PENED 2001.     

Existence and uniqueness of solutions to stochastic heat equations with Markovian switching and Feller property

Abstract

In this paper, we focus on two-component Markov processes which consist of continuous dynamics and discrete events. Using the classical fixed point theorem for contractions to investigate the existence and uniqueness of solutions of stochastic heat equations with Markovian switching, then developing the corresponding Feller property of the solution.     

关于DHMM转化为齐次马尔可夫链的一个定理及其证明

本给出一个将DHMM转化为齐次马尔可夫链的定理，该定理提供了利用在理论上比较完善的齐次马尔可夫链来研究DHMM的一个方法．     

Computational Bayesian Analysis of Hidden Markov Models

Abstract

Versions of the Gibbs Sampler are derived for the analysis of data from hidden Markov chains and hidden Markov random fields. The principal new development is to use the pseudolikelihood function associated with the underlying Markov process in place of the likelihood, which is intractable in the case of a Markov random field, in the simulation step for the parameters in the Markov process. Theoretical aspects are discussed and a numerical study is reported.     

Lumpability and Commutativity of Markov Processes

Abstract

We introduce the concepts of lumpability and commutativity of a continuous time discrete state space Markov process, and provide a necessary and sufficient condition for a lumpable Markov process to be commutative. Under suitable conditions we recover some of the basic quantities of the original Markov process from the jump chain of the lumped Markov process.     

Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory

Multi-dimensional asymptotically quasi-Toeplitz Markov chains with discrete and continuous time are introduced. Ergodicity and non-ergodicity conditions are proven. Numerically stable algorithm to calculate the stationary distribution is presented. An application of such chains in retrial queueing models with Batch Markovian Arrival Process is briefly illustrated. AMS Subject Classifications Primary 60K25 · 60K20     

Hamilton-Jacobi-Bellman inequality for the average control of piecewise deterministic Markov processes

ABSTRACT

The main goal of this paper is to study the infinite-horizon long run average continuous-time optimal control problem of piecewise deterministic Markov processes (PDMPs) with the control acting continuously on the jump intensity λ and on the transition measure Q of the process. We provide conditions for the existence of a solution to an integro-differential optimality inequality, the so called Hamilton-Jacobi-Bellman (HJB) equation, and for the existence of a deterministic stationary optimal policy. These results are obtained by using the so-called vanishing discount approach, under some continuity and compactness assumptions on the parameters of the problem, as well as some non-explosive conditions for the process.     

Bayesian Calibration and Uncertainty Analysis for Computationally Expensive Models Using Optimization and Radial Basis Function Approximation

We presenta Bayesian approach to model calibration when evaluation of the model is computationally expensive. Here, calibration is a nonlinear regression problem: given a data vector Y corresponding to the regression model f(β), find plausible values of β. As an intermediate step, Y and f are embedded into a statistical model allowing transformation and dependence. Typically, this problem is solved by sampling from the posterior distribution of β given Y using MCMC. To reduce computational cost, we limit evaluation of f to a small number of points chosen on a high posterior density region found by optimization.Then,we approximate the logarithm of the posterior density using radial basis functions and use the resulting cheap-to-evaluate surface in MCMC.We illustrate our approach on simulated data for a pollutant diffusion problem and study the frequentist coverage properties of credible intervals. Our experiments indicate that our method can produce results similar to those when the true “expensive” posterior density is sampled by MCMC while reducing computational costs by well over an order of magnitude.     

Parametric Estimation of a Boolean Segment Process with Stochastic Restoration Estimation

Abstract

We propose a stochastic restoration estimation (SRE) algorithm to estimate the parameters of the length distribution of a boolean segment process. A boolean segment process is a stochastic process obtained by considering the union of independent random segments attached to random points independently scattered on the plane. Each iteration of the SRE algorithm has two steps: first, censored segments are restored; second, based on these restored data, parameter estimations are updated. With a usually straightforward implementation, this algorithm is particularly interesting when censoring effects are difficult to take into account. We illustrate this method in two situations where the parameter of interest is either the mean of the segment length distribution or the variance of its logarithm. Its application to vine shoot length distribution estimation is presented.     

Generating Uniform Random Vectors in Z p k : The General Case

We consider the rate of convergence of the Markov chain X _n+1=A X _n +B _n (mod p), where A is an integer matrix with nonzero eigenvalues, and {B _n } _n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Q ^k invariant under A. If for all eigenvalues λ _i of A, then n=O((ln p)²) steps are sufficient and n=O(ln p) steps are necessary to have X _n sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λ _i that are roots of positive integer numbers, |λ ₁|=1 and |λ _i |>1 for all , then O(p ²) steps are necessary and sufficient.      

Conditioning in Markov Chain Monte Carlo

Abstract

The so-called “Rao-Blackwellized” estimators proposed by Gelfand and Smith do not always reduce variance in Markov chain Monte Carlo when the dependence in the Markov chain is taken into account. An illustrative example is given, and a theorem characterizing the necessary and sufficient condition for such an estimator to always reduce variance is proved.     

Population Size Dependent Generalized Multitype Branching Processes

Abstract

In this paper, we introduce the population size dependent generalized multitype branching process. This is a Markovian model that allows us to study homogeneous multitype branching processes in a unified way. The basic properties for this model, transitions between its states, as well as the existence of a stationary limiting distribution, are investigated. Finally, we apply the obtained results to a new controlled multitype branching process.