A non-commutative version of Lépingle–Yor martingale inequality

2014 ◽  
Vol 91 ◽  
pp. 52-54 ◽  
Author(s):  
Yanqi Qiu
Keyword(s):  
Martingale Inequality

scholarly journals Exponential Stability of Stochastic Differential Equation with Mixed Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Wenli Zhu ◽  
Jiexiang Huang ◽  
Xinfeng Ruan ◽  
Zhao Zhao
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Stochastic Differential Equation ◽  
Exponential Stability ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Martingale Inequality ◽  
Mixed Delay ◽  
Exponentially Stable ◽  
Theoretical Results

This paper focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Itô formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.

scholarly journals A Class of Stochastic Nonlinear Delay System with Jumps

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Ling Bai ◽  
Kai Zhang ◽  
Wenju Zhao
Keyword(s):  
Sufficient Conditions ◽  
Delay System ◽  
Lévy Noise ◽  
Local Martingale ◽  
Martingale Inequality ◽  
Strong Law ◽  
Delay Differential System ◽  
Delay Differential ◽  
Levy Noise ◽  
Nonlinear Delay

We consider stochastic suppression and stabilization for nonlinear delay differential system. The system is assumed to satisfy local Lipschitz condition and one-side polynomial growth condition. Since the system may explode in a finite time, we stochastically perturb this system by introducing independent Brownian noises and Lévy noise feedbacks. The contributions of this paper are as follows. (a) We show that Brownian noises or Lévy noise may suppress potential explosion of the solution for some appropriate parameters. (b) Using the exponential martingale inequality with jumps, we discuss the fact that the sample Lyapunov exponent is nonpositive. (c) Considering linear Lévy processes, by the strong law of large number for local martingale, sufficient conditions for a.s. exponentially stability are investigated in Theorem 13.

Inheritance

1980 ◽  
Vol 12 (3) ◽  
pp. 574-590
Author(s):  
David Stirzaker
Keyword(s):  
Convex Function ◽  
Linear Approximation ◽  
Wealth Distribution ◽  
Upper Bounds ◽  
Limiting Distribution ◽  
Martingale Inequality ◽  
Distribution Of Wealth ◽  
Random Line

We consider a population of reproducing individuals who inherit, earn, consume, and bequeath wealth. A model is constructed to describe the wealth of an individual selected from the nth generation by following a random line of descent from the initial individual. It is shown that bequests are commonly a convex function of wealth. Considering a linear approximation to the bequest function enables us to obtain estimates of the limiting distribution of wealth as the number of generations increases, when earnings of parent and offspring are independent. More generally when earnings of parent and offspring are not independent we obtain upper bounds for the tail of the wealth distribution using a martingale inequality.

scholarly journals A STOCHASTIC GRONWALL LEMMA

2013 ◽  
Vol 16 (02) ◽  
pp. 1350019 ◽  
Author(s):  
MICHAEL SCHEUTZOW
Keyword(s):  
Simple Proof ◽  
Integral Inequality ◽  
Continuous Process ◽  
Main Tool ◽  
Local Martingale ◽  
Gronwall Lemma ◽  
Martingale Inequality ◽  
Optimal Constant ◽  
Linear Integral ◽  
The Right

We prove a stochastic Gronwall lemma of the following type: if Z is an adapted non-negative continuous process which satisfies a linear integral inequality with an added continuous local martingale M and a process H on the right-hand side, then for any p ∈ (0, 1) the pth moment of the supremum of Z is bounded by a constant κp (which does not depend on M) times the pth moment of the supremum of H. Our main tool is a martingale inequality which is due to D. Burkholder. We provide an alternative simple proof of the martingale inequality which provides an explicit numerical value for the constant cp appearing in the inequality which is at most four times as large as the optimal constant.

scholarly journals Asymptotic Stability of Stochastic Differential Equations Driven by Lévy Noise

2009 ◽  
Vol 46 (4) ◽  
pp. 1116-1129 ◽  
Author(s):  
David Applebaum ◽  
Michailina Siakalli
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lévy Noise ◽  
Stochastic Integrals ◽  
Martingale Inequality ◽  
Moment Estimates ◽  
Exponential Martingale ◽  
Exponentially Stable ◽  
Levy Noise

Using key tools such as Itô's formula for general semimartingales, Kunita's moment estimates for Lévy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the stochastic differential equations (SDEs) driven by Lévy noise are stable in probability, almost surely and moment exponentially stable.

Extinction and persistence of a nonautonomous stochastic food-chain system with impulsive perturbations

2016 ◽  
Vol 09 (05) ◽  
pp. 1650077
Author(s):  
Baodan Tian ◽  
Shouming Zhong ◽  
Zhijun Liu
Keyword(s):  
Food Chain ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Martingale Inequality ◽  
Stochastic Permanence ◽  
Chain System ◽  
Persistence In The Mean ◽  
The Mean ◽  
Impulsive Perturbations ◽  
White Noises

In this paper, a nonautonomous stochastic food-chain system with functional response and impulsive perturbations is studied. By using Itô’s formula, exponential martingale inequality, differential inequality and other mathematical skills, some sufficient conditions for the extinction, nonpersistence in the mean, persistence in the mean, and stochastic permanence of the system are established. Furthermore, some asymptotic properties of the solutions are also investigated. Finally, a series of numerical examples are presented to support the theoretical results, and effects of different intensities of white noises perturbations and impulsive effects are discussed by the simulations.

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