Abstract
In this paper, we propose and prove several different forms of backward stochastic Bellman–Bihari’s inequality. Then, as two applications, two different types of the comparison theorems for backward stochastic differential equation with stochastic non-Lipschitz condition are presented.
Highlights
In [9], Gronwall was the first to provide Gronwall’s inequality under the framework of differential form
In order to meet the needs of the development of stochastic differential equations, many scholars tried to generalize Gronwall’s inequality
The behavior of solution of the fractional stochastic differential equation has been investigated. [8] analyzed some new nonlinear Gronwall–Bellman–Bihari type inequalities with singular kernel via k-fractional integral of Riemann–Liouville, which can be used to study some properties of solution for fractional differential equations
Summary
In [9], Gronwall was the first to provide Gronwall’s inequality under the framework of differential form. Bellman [2] put forward the integral form of Gronwall’s inequality as the following proposition. Bihari [3] put forward a useful generalization of Gronwall–Bellman’s inequality, called Bihari’s inequality, which provides explicit bounds on unknown functions This inequality had many applications in the fields of differential equations. Fan [6] used Bihari’s inequality to study existence, uniqueness, and stability of L1 solutions for multidimensional backward stochastic differential equations with generators of one-sided Osgood type. [8] analyzed some new nonlinear Gronwall–Bellman–Bihari type inequalities with singular kernel via k-fractional integral of Riemann–Liouville, which can be used to study some properties of solution for fractional differential equations. It is necessary to point out that the proof method in [3, 14, 15] for Bihari’s inequality is no longer applicable since β(s) in the following theorems in this paper depends on ω, while the β(s) in Proposition 1.3 is independent of ω
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