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.
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