A Wavelet Characterization of Continuous-Time Periodically Correlated Processes with Application to Simulation

2016 ◽  
Vol 37 (6) ◽  
pp. 741-762 ◽  
Author(s):  
Mitra Ghanbarzadeh ◽  
Mina Aminghafari
Keyword(s):  
Continuous Time ◽  
Periodically Correlated Processes

scholarly journals Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach

2021 ◽  
Author(s):  
Yves Achdou ◽  
Jiequn Han ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions ◽  
Benjamin Moll
Keyword(s):  
Continuous Time ◽  
Wealth Distribution ◽  
Closed Form Solution ◽  
Form Solution ◽  
Heterogeneous Agent ◽  
Saving Behavior ◽  
Income And Wealth Distribution ◽  
Heterogeneous Agent Models ◽  
Special Case

Abstract We recast the Aiyagari-Bewley-Huggett model of income and wealth distribution in continuous time. This workhorse model – as well as heterogeneous agent models more generally – then boils down to a system of partial differential equations, a fact we take advantage of to make two types of contributions. First, a number of new theoretical results: (i) an analytic characterization of the consumption and saving behavior of the poor, particularly their marginal propensities to consume; (ii) a closed-form solution for the wealth distribution in a special case with two income types; (iii) a proof that there is a unique stationary equilibrium if the intertemporal elasticity of substitution is weakly greater than one. Second, we develop a simple, efficient and portable algorithm for numerically solving for equilibria in a wide class of heterogeneous agent models, including – but not limited to – the Aiyagari-Bewley-Huggett model.

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

2020 ◽  
Vol 52 (4) ◽  
pp. 1249-1283
Author(s):  
Masatoshi Kimura ◽  
Tetsuya Takine
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Continuous Time ◽  
State Transition ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Systems Of Linear Inequalities ◽  
Free Sets ◽  
Practical Implications

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

Strong ergodicity for continuous-time, non-homogeneous Markov chains

1982 ◽  
Vol 19 (3) ◽  
pp. 692-694 ◽  
Author(s):  
Mark Scott ◽  
Barry C. Arnold ◽  
Dean L. Isaacson
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Homogeneous Markov Chain ◽  
Strong Ergodicity

Characterizations of strong ergodicity for Markov chains using mean visit times have been found by several authors (Huang and Isaacson (1977), Isaacson and Arnold (1978)). In this paper a characterization of uniform strong ergodicity for a continuous-time non-homogeneous Markov chain is given. This extends the characterization, using mean visit times, that was given by Isaacson and Arnold.

A Linear Matrix Inequality Solution to the Input Covariance Constraint Control Problem

2013 ◽  
Author(s):  
Andrew White ◽  
Guoming Zhu ◽  
Jongeun Choi
Keyword(s):  
Linear Matrix Inequality ◽  
Control Problem ◽  
Continuous Time ◽  
Covariance Matrices ◽  
Control Synthesis ◽  
Linear Matrix ◽  
Multiple Constraints ◽  
Matrix Inequality ◽  
Output Performance

In this paper, the input covariance constraint (ICC) control problem is solved by a convex optimization with linear matrix inequality (LMI) constraints. The ICC control problem is an optimal control problem that is concerned with finding the best output performance possible subject to multiple constraints on the input covariance matrices. The contribution of this paper is the characterization of the control synthesis LMIs used to solve the ICC control problem. To demonstrate the effectiveness of the proposed approach a numerical example is solved with the control synthesis LMIs. Both discrete and continuous-time problems are considered.

scholarly journals Representation and Characterization of Nonstationary Processes by Dilation Operators and Induced Shape Space Manifolds

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 717 ◽  
Author(s):  
Maël Dugast ◽  
Guillaume Bouleux ◽  
Eric Marcon
Keyword(s):  
Lie Group ◽  
Correlation Coefficients ◽  
Shape Space ◽  
Rotation Matrices ◽  
Dilation Matrix ◽  
Naimark Dilation ◽  
Periodically Correlated Processes ◽  
New Vision

We proposed in this work the introduction of a new vision of stochastic processes through geometry induced by dilation. The dilation matrices of a given process are obtained by a composition of rotation matrices built in with respect to partial correlation coefficients. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, only one dilation matrix is obtained and it corresponds therefore to Naimark dilation. When the process is nonstationary, a set of dilation matrices is obtained. They correspond to Kolmogorov decomposition. In this work, the nonstationary class of periodically correlated processes was of interest. The underlying periodicity of correlation coefficients is then transmitted to the set of dilation matrices. Because this set lives on the Lie group of rotation matrices, we can see them as points of a closed curve on the Lie group. Geometrical aspects can then be investigated through the shape of the obtained curves, and to give a complete insight into the space of curves, a metric and the derived geodesic equations are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices, and because the metric in the space of curve naturally extends to the space of shapes; this enables a comparison between curves’ shapes and allows then the classification of random processes’ measures.

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