Suppose we have a first-order ordinary differential equation for some function ,
and assume that we can prove from this equation
that the derivative is bounded as follows:
Where is known.
Then Grönwall’s inequality states that the solution is bounded:
We define as equal to the upper bounds above
on both and itself:
Then the goal is to show the following for all :
For , this is trivial, since by definition.
For , we want to grow at least as fast as
in order to satisfy the inequality.
We thus calculate:
Since as a condition,
the above derivative is always negative.
Grönwall’s inequality can be generalized to non-differentiable functions.
Suppose we know:
Where and are known.
Then the Grönwall-Bellman inequality states that:
We start by defining as follows,
which will act as shorthand:
Its derivative is then straightforwardly calculated to be given by:
The parenthesized expression is bounded from above by ,
thanks to the condition that is assumed to satisfy,
for the Grönwall-Bellman inequality to be true:
Integrating this to find yields the following result:
In the initial definition of ,
we now move the exponential to the other side,
and rewrite it using the above inequality for :
This yields the desired result after inserting it
into the condition under which the Grönwall-Bellman inequality holds.
In the special case where is non-decreasing with ,
the inequality reduces to:
Starting from the “ordinary” Grönwall-Bellman inequality,
the fact that is non-decreasing tells us that
for all , so:
Now, consider the following straightforward identity, involving the exponential:
By inserting this into normal Grönwall-Bellman inequality, we arrive at:
Where we have converted the outer integral from definite to indefinite.
- U.H. Thygesen,
Lecture notes on diffusions and stochastic differential equations,
2021, Polyteknisk Kompendie.