From Wikipedia, the free encyclopedia
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
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
:[1][2]
- If
and
is predictable, then
;
- If
and
has no negative jumps, then
;
- If
then
;
It has been proven by Lenglart's inequality.[1]