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 
 

Stochastic differential equations
Articles containing proofs
Probabilistic inequalities