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 letbe 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 ;