A powered Gronwall-type inequality and applications to stochastic differential equations

Jun  Zhou; Jun  Shen; Weinian  Zhang

doi:10.3934/dcds.2016114

Article Contents

2016, Volume 36, Issue 12: 7207-7234. Doi: 10.3934/dcds.2016114

This issue Previous Article On large deviations for amenable group actions Next Article Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion

A powered Gronwall-type inequality and applications to stochastic differential equations

1.
Institute of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan 610066
2.
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064

Received: February 2016

Revised: May 2016

Published: May 2017

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Abstract

In this paper we study a powered integral inequality involving a finite sum, which can be used to solve the inequalities with singular kernels. We present that the solution of the inequality is decided by a finite recursion, whose result is proved to be a continuous, bounded or asymptotic function. Meanwhile, in order to overcome an obstacle from powers of integrals, we modify the method of monotonization into the powered monotonization. Furthermore, relying on the result and our technique of concavification, we discuss a generalized stochastic integral inequality, and give an estimate of the mean square. In the end, as applications, we study uniform boundedness and continuous dependence of solutions for a class of stochastic differential equation in mean square.

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Mathematics Subject Classification: Primary: 26D15; Secondary: 60H10, 45G05.

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