A POWERED GRONWALL-TYPE INEQUALITY AND APPLICATIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS

摘要

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. Mean-while, 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.