62.4 The Burkholder Davis Gundy Inequality
The Burkholder Davis Gundy inequality is an amazing inequality which involves M∗ and
Before presenting this, here is the good lambda inequality, Theorem 10.7.1 on Page
815 listed here for convenience.
Theorem 62.4.1 Let
be a finite measure space and let F be a continuous
increasing function defined on
) such that F
. Suppose also that for all α >
there exists a constant Cα such that for all x ∈
Also suppose f,g are nonnegative measurable functions and there exists β > 1,0 < r ≤ 1,
such that for all λ > 0 and 1 > δ > 0,
and ϕ is increasing. Under these conditions, there exists a
constant C depending only on β,ϕ,r such that
The proof of this important inequality also will depend on the hitting this before that
theorem which is listed next for convenience.
Theorem 62.4.2 Let
be a continuous real valued martingale adapted to the
normal filtration ℱt and let
Then if a < 0 < b the following inequalities hold.
In words, P
is the probability that M
hits b no later than when it hits a.
(Note that if τa
= τb then you would have
Then the Burkholder Davis Gundy inequality is as follows. Generalizations will be
Theorem 62.4.3 Let
be a continuous H valued martingale which is uniformly
, where H is a separable Hilbert space and t ∈
. Then if F is a
function of the sort described in the good lambda inequality above, there are constants, C
and c independent of such martingales M such that
Proof: Using Corollary 62.3.3, let
Thus N is a martingale and N
In fact N
= 0 as long as
t ≤ τ
. As usual
This is because to say τ < ∞ is to say there exists t < T such that
the same as saying M∗ > λ.
Thus the first two sets are equal. If τ
then from the
formula for N
= 0 for all
and so it can’t happen that
Thus the third set is contained in
Let β > 2 and let δ ∈
Consider the following which is set up to use the good lambda inequality.
where 0 < r < 1.It is shown that Sr corresponds to hitting “this before that” and there is
an estimate for this which involves P
which is bounded above by
as discussed above. This will satisfy the hypotheses of the good lambda
Claim: For ω ∈ Sr, N
Proof of claim: For ω ∈ Sr, there exists a t < T such that
using Corollary 62.3.3
which shows that N
2λ2 − δ2λ2
for ω ∈ Sr
. By the intermediate value
theorem, it also hits λ2
. This proves the claim.
for ω ∈ Sr.
Proof of claim: Suppose t is the first time N
Then t > τ
a contradiction since r <
This proves the claim.
Therefore, for all ω ∈ Sr, N
before it reaches
and because of Theorem 61.11.3 this is no larger than
By the good lambda inequality,
which is one half the inequality.
Now consider the other half. This time define the stopping time τ by
Then there exists t < T such that
This time, let
This is still a martingale since by Corollary 62.3.3
t < T
for ω ∈ Sr.
Proof of claim: Fix such a ω ∈ Sr. Let t < T be such that
t > τ
and so for that ω,
By the intermediate value theorem, it hits λ2
This proves the claim.
for ω ∈ Sr.
Proof of claim: By Corollary 62.3.3, if it did at t, then t > τ because N
= 0 for
t ≤ τ,
a contradiction. This proves the claim.
It follows that for each r ∈
By Theorem 61.11.3 this is no larger than
Now by the good lambda inequality, there is a constant k independent of M such
by the assumptions about F. Therefore, combining this result with the first part,
Of course, everything holds for local martingales in place of martingales.
Theorem 62.4.4 Let
be a continuous H valued local martingale, M
where H is a separable Hilbert space and t ∈
. Then if F is a function of the sort
described in the good lambda inequality, that is,
there are constants, C and c independent of such local martingales M such
be an increasing localizing sequence for
such that Mτn
uniformly bounded. Such a localizing sequence exists from Proposition 62.2.2
from Theorem 62.4.3
there exist constants c,C
independent of τn
By Corollary 62.3.3
, this implies
and now note that
increase in n
respectively. Then the result follows from the monotone convergence theorem.
Here is a corollary [?].
Corollary 62.4.5 Let
be a continuous H valued local martingale and let
. Then there is a constant C, independent of ε,δ such that
Proof: Let the stopping time τ be defined by
On the set where
and so P
By Theorem 62.4.4 and Corollary 62.3.3,
The Burkholder Davis Gundy inequality along with the properties of the covariation
implies the following amazing proposition.
Proposition 62.4.6 The space MT2
is a Hilbert space. Here H is a separable
Proof: We already know from Proposition 61.12.2 that this space is a Banach space.
It is only necessary to exhibit an equivalent norm which makes it a Hilbert space.
However, you can let F
in the Burkholder Davis Gundy theorem and obtain for
M ∈ MT2
the two norms