63.4. THE BURKHOLDER DAVIS GUNDY INEQUALITY 2135

and now note that([M]τn

)(T )1/2 and (Mτn)∗ increase in n to [M] (T )1/2 and M∗ respec-

tively. Then the result follows from the monotone convergence theorem.Here is a corollary [108].

Corollary 63.4.5 Let {M (t)} be a continuous H valued local martingale and let ε,δ ∈(0,∞) . Then there is a constant C, independent of ε,δ such that

P

 M∗(T )︷ ︸︸ ︷

supt∈[0,T ]

||M (t)|| ≥ ε

≤ C

εE([M]1/2 (T )∧δ

)+P

([M]1/2 (T )> δ

)

Proof: Let the stopping time τ be defined by

τ ≡ inf{

t > 0 : [M]1/2 (t)> δ

}Then

P([M∗ ≥ ε]) = P([M∗ ≥ ε]∩ [τ = ∞])+P([M∗ ≥ ε]∩ [τ < ∞])

On the set where [τ = ∞] , Mτ = M and so P([M∗ ≥ ε])≤

≤ 1ε

∫Ω

(Mτ)∗ dP+P([M∗ ≥ ε]∩

[[M]1/2 (T )> δ

])By Theorem 63.4.4 and Corollary 63.3.3,

≤ 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

])≤ C

ε

∫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 63.4.6 The space M2T (H) is a Hilbert space. Here H is a separable Hilbert

space.

Proof: We already know from Proposition 62.12.2 that this space is a Banach space. Itis only necessary to exhibit an equivalent norm which makes it a Hilbert space. However,you can 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