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.

Statement

Let be a non-negative right-continuous -adapted process. Assume that is a deterministic non-decreasing càdlàg function with and let be a non-decreasing and càdlàg adapted process starting from . Further, let be an - local martingale with and càdlàg paths.

Assume that for all ,

where .

and define . Then the following estimates hold for and :

• If and is predictable, then ;
• If and has no negative jumps, then ;
• If then ;

Proof

It has been proven by Lenglart's inequality.

References