scholarly journals Absolute regularity of semi-contractive GARCH-type processes

2019 ◽  
Vol 56 (01) ◽  
pp. 91-115 ◽  
Author(s):  
Paul Doukhan ◽  
Michael H. Neumann
Keyword(s):  
Decay Rate ◽  
Stationary Distribution ◽  
Existence And Uniqueness ◽  
Contractive Condition ◽  
Absolute Regularity ◽  
Mixing Coefficients ◽  
Drift Condition ◽  
Geometric Decay

AbstractWe prove existence and uniqueness of a stationary distribution and absolute regularity for nonlinear GARCH and INGARCH models of order (p, q). In contrast to previous work we impose, besides a geometric drift condition, only a semi-contractive condition which allows us to include models which would be ruled out by a fully contractive condition. This results in a subgeometric rather than the more usual geometric decay rate of the mixing coefficients. The proofs are heavily based on a coupling of two versions of the processes.

scholarly journals COUNT AND DURATION TIME SERIES WITH EQUAL CONDITIONAL STOCHASTIC AND MEAN ORDERS

2020 ◽  
pp. 1-33
Author(s):  
Abdelhakim Aknouche ◽  
Christian Francq
Keyword(s):  
Time Series ◽  
Conditional Distribution ◽  
Stochastic Orders ◽  
Conditional Mean ◽  
Contraction Condition ◽  
Mixing Coefficients ◽  
The Mean ◽  
Quasi Maximum Likelihood ◽  
Geometric Decay ◽  
Mean Function

We consider a positive-valued time series whose conditional distribution has a time-varying mean, which may depend on exogenous variables. The main applications concern count or duration data. Under a contraction condition on the mean function, it is shown that stationarity and ergodicity hold when the mean and stochastic orders of the conditional distribution are the same. The latter condition holds for the exponential family parametrized by the mean, but also for many other distributions. We also provide conditions for the existence of marginal moments and for the geometric decay of the beta-mixing coefficients. We give conditions for consistency and asymptotic normality of the Exponential Quasi-Maximum Likelihood Estimator of the conditional mean parameters. Simulation experiments and illustrations on series of stock market volumes and of greenhouse gas concentrations show that the multiplicative-error form of usual duration models deserves to be relaxed, as allowed in this paper.

scholarly journals Implicit-Relation-Type Cyclic Contractive Mappings and Applications to Integral Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Hemant Kumar Nashine ◽  
Zoran Kadelburg ◽  
Poom Kumam
Keyword(s):  
Integral Equations ◽  
Existence And Uniqueness ◽  
Contractive Condition ◽  
Uniqueness Of Solutions ◽  
Nonlinear Integral Equations ◽  
Implicit Relation ◽  
Existence And Uniqueness Results ◽  
Contractive Mappings ◽  
Uniqueness Results ◽  
Relation Type

We introduce an implicit-relation-type cyclic contractive condition for a map in a metric space and derive existence and uniqueness results of fixed points for such mappings. Examples are given to support the usability of our results. At the end of the paper, an application to the study of existence and uniqueness of solutions for a class of nonlinear integral equations is presented.

A semilinear heat equation with singular initial data

1998 ◽  
Vol 128 (4) ◽  
pp. 745-758 ◽  
Author(s):  
Minkyu Kwak
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Asymptotic Behaviour ◽  
Decay Rate ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Semilinear Heat Equation ◽  
Singular Initial Data ◽  
Self Similar ◽  
Image Position

We first prove existence and uniqueness of non-negative solutions of the equationin in the range 1 < p < 1 + 2/N, when initial data u(x, 0) = a|x|−2(p−1), x ≠ 0, for a > 0. It is proved that the maximal and minimal solutions are self-similar with the formwhere g = ga satisfiesAfter uniqueness is proved, the asymptotic behaviour of solutions ofis studied. In particular, we show thatThe case for a = 0 is also considered and a sharp decay rate of the above equation is derived. In the final, we reveal existence of solutions of the first and third equations above, which change sign.

scholarly journals On the Power of Simulation and Admissible Functions in Metric Fixed Point Theory

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Areej S. S. Alharbi ◽  
Hamed H. Alsulami ◽  
Erdal Karapinar
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Admissible Function ◽  
Point Theory ◽  
Contractive Condition ◽  
Simulation Function ◽  
The Given ◽  
Admissible Functions

We investigate the existence and uniqueness of certain operators which form a new contractive condition via the combining of the notions of admissible function and simulation function contained in the context of completeb-metric spaces. The given results not only unify but also generalize a number of existing results on the topic in the corresponding literature.

scholarly journals Stationary distribution and extinction of a stochastic SEIQ epidemic model with a general incidence function and temporary immunity

2021 ◽  
Vol 6 (11) ◽  
pp. 12359-12378
Author(s):  
Yuhuai Zhang ◽  
◽  
Xinsheng Ma ◽  
Anwarud Din ◽  
◽  
...  
Keyword(s):  
Stationary Distribution ◽  
Epidemic Model ◽  
Existence And Uniqueness ◽  
Noise Intensity ◽  
Epidemic Spreading ◽  
General Incidence Rate ◽  
Global Positive Solution ◽  
Simulation Results ◽  
Theoretical Results ◽  
Temporary Immunity

<abstract><p>In this paper, we propose a novel stochastic SEIQ model of a disease with the general incidence rate and temporary immunity. We first investigate the existence and uniqueness of a global positive solution for the model by constructing a suitable Lyapunov function. Then, we discuss the extinction of the SEIQ epidemic model. Furthermore, a stationary distribution for the model is obtained and the ergodic holds by using the method of Khasminskii. Finally, the theoretical results are verified by some numerical simulations. The simulation results show that the noise intensity has a strong influence on the epidemic spreading.</p></abstract>

scholarly journals Some Generalised Fixed Point Theorems Applied to Quantum Operations

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 759
Author(s):  
Umar Batsari Yusuf ◽  
Poom Kumam ◽  
Sikarin Yoo-Kong
Keyword(s):  
Fixed Point ◽  
Quantum State ◽  
Metric Space ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Quantum States ◽  
Contractive Condition ◽  
Quantum Operation ◽  
Quantum Operations ◽  
Finite Dimensional

In this paper, we consider an order-preserving mapping T on a complete partial b-metric space satisfying some contractive condition. We were able to show the existence and uniqueness of the fixed point of T. In the application aspect, the fidelity of quantum states was used to establish the existence of a fixed quantum state associated to an order-preserving quantum operation. The method we presented is an alternative in showing the existence of a fixed quantum state associated to quantum operations. Our method does not capitalise on the commutativity of the quantum effects with the fixed quantum state(s) (Luders’s compatibility criteria). The Luders’s compatibility criteria in higher finite dimensional spaces is rather difficult to check for any prospective fixed quantum state. Some part of our results cover the famous contractive fixed point results of Banach, Kannan and Chatterjea.

scholarly journals Stationary Distribution and Extinction of a Stochastic SIQR Model with Saturated Incidence Rate

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Qiuhua Zhang ◽  
Kai Zhou
Keyword(s):  
Lyapunov Function ◽  
Incidence Rate ◽  
Stationary Distribution ◽  
Epidemic Model ◽  
Existence And Uniqueness ◽  
Sufficient Condition ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Dynamical Properties ◽  
Theoretical Results

In this paper, we consider a stochastic SIQR epidemic model with saturated incidence rate. By constructing a proper Lyapunov function, we obtain the existence and uniqueness of positive solution for this SIQR model. Furthermore, we study the dynamical properties of this stochastic SIQR model; that is, (i) we establish the sufficient condition for the existence of ergodic stationary distribution of the model; (ii) we obtain the extinction of the disease under some conditions. At last, numerical simulations are introduced to illustrate our theoretical results.

scholarly journals Fixed-Point Theorems in Complete Gauge Spaces and Applications to Second-Order Nonlinear Initial-Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Meryam Cherichi ◽  
Bessem Samet ◽  
Calogero Vetro
Keyword(s):  
Fixed Point ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Fixed Point Theorems ◽  
Second Order ◽  
Uniqueness Of Solution ◽  
Contractive Condition ◽  
Initial Value ◽  
Gauge Space

We establish fixed-point results for mappings and cyclic mappings satisfying a generalized contractive condition in a complete gauge space. Our theorems generalize and extend some fixed-point results in the literature. We apply our obtained results to the study of existence and uniqueness of solution to a second-order nonlinear initial-value problem.

scholarly journals Stationary distribution of a stochastic SIRD epidemic model of Ebola with double saturated incidence rates and vaccination

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Panpan Wang ◽  
Jianwen Jia
Keyword(s):  
Stochastic Model ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Stationary Distribution ◽  
Epidemic Model ◽  
Existence And Uniqueness ◽  
Lyapunov Functions ◽  
Incidence Rates ◽  
Saturated Incidence ◽  
Global Positive Solution

Abstract In this paper, a stochastic SIRD model of Ebola with double saturated incidence rates and vaccination is considered. Firstly, the existence and uniqueness of a global positive solution are obtained. Secondly, by constructing suitable Lyapunov functions and using Khasminskii’s theory, we show that the stochastic model has a unique stationary distribution. Moreover, the extinction of the disease is also analyzed. Finally, numerical simulations are carried out to portray the analytical results.

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