Stochastic Gronwall inequality

Stochastic Gronwall inequality is a generalization of Gronwall's inequality and has been used for proving the well-posedness of path-dependent stochastic differential equations with local monotonicity and coercivity assumption with respect to supremum norm.

[1][2] Let

( t ) ,

be a non-negative right-continuous

(

-adapted process.

Assume that

is a deterministic non-decreasing càdlàg function with

be a non-decreasing and càdlàg adapted process starting from

- local martingale with

and càdlàg paths.

Assume that for all

{\displaystyle X(t)\leq \int _{0}^{t}X^{*}(u^{-})\,dA(u)+M(t)+H(t),}

and define

Then the following estimates hold for

:[1][2] It has been proven by Lenglart's inequality.