Weak convergence of martingale problem solutions

1976 ◽  
Vol 15 (3) ◽  
pp. 481-483
Author(s):  
R. Morkvenas
Keyword(s):  
Weak Convergence ◽  
Martingale Problem

An invariance principle for solutions to stochastic difference equations

1981 ◽  
Vol 18 (2) ◽  
pp. 548-553
Author(s):  
Harry A. Guess
Keyword(s):  
Differential Equation ◽  
Population Genetics ◽  
Brownian Motion ◽  
Weak Convergence ◽  
Difference Equations ◽  
Continuous Time ◽  
Martingale Problem ◽  
Martingale Differences ◽  
Invariance Principles ◽  
Stochastic Difference Equations

In recent papers, McLeish and others have obtained invariance principles for weak convergence of martingales to Brownian motion. We generalize these results to prove that solutions of discrete-time stochastic difference equations defined in terms of martingale differences converge weakly to continuous-time solutions of Ito stochastic differential equations. Our proof is based on a theorem of Stroock and Varadhan which characterizes the solution of a stochastic differential equation as the unique solution of an associated martingale problem. Applications to mathematical population genetics are discussed.

scholarly journals Intermittent quasistatic dynamical systems: weak convergence of fluctuations

2018 ◽  
Vol 5 (1) ◽  
pp. 8-34 ◽  
Author(s):  
Juho Leppänen
Keyword(s):  
Dynamical Systems ◽  
Weak Convergence ◽  
Time Evolution ◽  
Martingale Problem ◽  
Statistical Properties ◽  
Time Dependent ◽  
Stochastic Diffusion ◽  
Interval Maps ◽  
Well Posed ◽  
Over Time

Abstract This paper is about statistical properties of quasistatic dynamical systems. These are a class of non-stationary systems that model situations where the dynamics change very slowly over time due to external influences. We focus on the case where the time-evolution is described by intermittent interval maps (Pomeau-Manneville maps) with time-dependent parameters. In a suitable range of parameters, we obtain a description of the statistical properties as a stochastic diffusion, by solving a well-posed martingale problem. The results extend those of a related recent study due to Dobbs and Stenlund, which concerned the case of quasistatic (uniformly) expanding systems.

An invariance principle for solutions to stochastic difference equations

1981 ◽  
Vol 18 (02) ◽  
pp. 548-553
Author(s):  
Harry A. Guess
Keyword(s):  
Differential Equation ◽  
Population Genetics ◽  
Brownian Motion ◽  
Weak Convergence ◽  
Difference Equations ◽  
Continuous Time ◽  
Martingale Problem ◽  
Martingale Differences ◽  
Invariance Principles ◽  
Stochastic Difference Equations

In recent papers, McLeish and others have obtained invariance principles for weak convergence of martingales to Brownian motion. We generalize these results to prove that solutions of discrete-time stochastic difference equations defined in terms of martingale differences converge weakly to continuous-time solutions of Ito stochastic differential equations. Our proof is based on a theorem of Stroock and Varadhan which characterizes the solution of a stochastic differential equation as the unique solution of an associated martingale problem. Applications to mathematical population genetics are discussed.

Common Fixed Points in Modular Spaces

2018 ◽  
pp. 502 ◽  
Author(s):  
Salwa Salman Abed ◽  
Karrar Emad Abdul Sada
Keyword(s):  
Fixed Point ◽  
Weak Convergence ◽  
Fixed Point Theorems ◽  
Active Role ◽  
Common Fixed Points ◽  
Weakly Compact ◽  
Expansive Mapping ◽  
Modular Spaces ◽  
Opial’S Property ◽  
Common Fixed Point Theorems

     In this paper,there are   new considerations about the dual of a modular spaces and weak convergence. Two common fixed point theorems for a -non-expansive mapping defined on a star-shaped weakly compact subset are proved,  Here the conditions of affineness, demi-closedness and Opial's property play an active role in the proving our results.  

scholarly journals Weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain

2021 ◽  
Vol 58 (2) ◽  
pp. 372-393
Author(s):  
H. M. Jansen
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Regime Switching ◽  
Sufficient Conditions ◽  
The State ◽  
Stochastic Integrals ◽  
Occupation Measure ◽  
Generator Matrix ◽  
Markov Modulated ◽  
Erlang Loss

AbstractOur aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

scholarly journals A Mean Extragradient Method for Solving Variational Inequalities

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 462
Author(s):  
Apichit Buakird ◽  
Nimit Nimana ◽  
Narin Petrot
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Weak Convergence ◽  
Numerical Experiments ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Mean Value ◽  
Inequality Problem ◽  
The Mean ◽  
Mean Value Method

We propose a modified extragradient method for solving the variational inequality problem in a Hilbert space. The method is a combination of the well-known subgradient extragradient with the Mann’s mean value method in which the updated iterate is picked in the convex hull of all previous iterates. We show weak convergence of the mean value iterate to a solution of the variational inequality problem, provided that a condition on the corresponding averaging matrix is fulfilled. Some numerical experiments are given to show the effectiveness of the obtained theoretical result.

Moderate deviation principle for multivalued stochastic differential equations

2019 ◽  
Vol 20 (03) ◽  
pp. 2050015 ◽  
Author(s):  
Hua Zhang
Keyword(s):  
Brownian Motion ◽  
Weak Convergence ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Moderate Deviation ◽  
Reflected Brownian Motion ◽  
Moderate Deviation Principle ◽  
Deviation Principle ◽  
Weak Convergence Approach

In this paper, we prove a moderate deviation principle for the multivalued stochastic differential equations whose proof are based on recently well-developed weak convergence approach. As an application, we obtain the moderate deviation principle for reflected Brownian motion.

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