32.5. THE BURKHOLDER DAVIS GUNDY INEQUALITY 879

By Theorem 32.5.4 and Corollary 32.4.3,

≤ Cε

∫Ω

[Mτ ]1/2 (T )dP+P([M∗ ≥ ε]∩

[[M]1/2 (T )> δ

])=

Cε

∫Ω

([M]τ

)1/2(T )dP+P

([M∗ ≥ ε]∩

[[M]1/2 (T )> δ

])≤ C

ε

∫Ω

[M]1/2 (T )∧δdP+P([M∗ ≥ ε]∩

[[M]1/2 (T )> δ

])

≤ Cε

∫Ω

[M]1/2 (T )∧δdP+P([

[M]1/2 (T )> δ

])■

The Burkholder Davis Gundy inequality along with the properties of the covariation impliesthe following amazing proposition.

Proposition 32.5.6 The space M2T (H) is a Hilbert space with respect to an equivalent

norm. Here H is a separable Hilbert space.

Proof: We already know from Proposition 31.7.2 that this space is a Banach space. It isonly necessary to exhibit an equivalent norm which makes it a Hilbert space. However, youcan let F (λ ) = λ

2 in the Burkholder Davis Gundy theorem and obtain for M ∈ M2T (H) ,

the two norms (∫Ω

[M] (T )dP)1/2

=

(∫Ω

[M,M] (T )dP)1/2

and (∫Ω

(M∗)2 dP)1/2

are equivalent. The first comes from an inner product since from Corollary 32.4.3, [·, ·] isbilinear and symmetric and nonnegative. If [M,M] (T ) = [M] (T ) = 0 in L1 (Ω) , then fromthe Burkholder Davis Gundy inequality, M∗ = 0 in L2 (Ω) and so M = 0. Hence∫

Ω

[M,N] (T )dP

is an inner product which yields the equivalent norm. ■Later, the Wiener process will be discussed and the existence of such a process is

proved. For now, the following example shows something about such processes.

Example 32.5.7 An example of a real martingale is the Wiener process W (t). It has theproperty that whenever t1 < t2 < · · · < tn, the increments {W (ti)−W (ti−1)} are indepen-dent and whenever s < t,W (t)−W (s) is normally distributed with mean 0 and variance(t− s). For the Wiener process, we let

Ft ≡ ∩u>tσ (W (s)−W (r) : r < s≤ u)

and it is with respect to this normal filtration that W is a continuous martingale. What isthe quadratic variation of such a process?