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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.
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[2]
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
:
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- 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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References