32.5. THE BURKHOLDER DAVIS GUNDY INEQUALITY 875

The proof of the Burkholder Davis Gundy inequality also will depend on the hittingthis before that theorem which is listed next for convenience.

Theorem 32.5.2 Let {M (t)} be a continuous real valued martingale adapted to thenormal filtration Ft and let

M∗ ≡ sup{|M (t)| : t ≥ 0}

and M (0) = 0. Lettingτx ≡ inf{t > 0 : M (t) = x}

Then if a < 0 < b the following inequalities hold.

(b−a)P([τb ≤ τa])≥−aP([M∗ > 0])≥ (b−a)P([τb < τa])

and(b−a)P([τa < τb])≤ bP([M∗ > 0])≤ (b−a)P([τa ≤ τb]) .

In words, P([τb ≤ τa]) is the probability that M (t) hits b no later than when it hits a. (Notethat if τa = ∞ = τb then you would have [τa = τb] .)

Then the Burkholder Davis Gundy inequality is as follows. Generalizations will bepresented later.

Theorem 32.5.3 Let {M (t)} be a continuous H valued martingale which is uni-formly bounded, M (0) = 0, where H is a separable Hilbert space and t ∈ [0,T ] . Then if Fis 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

c∫

Ω

F(([M] (T ))1/2

)dP≤

∫Ω

F (M∗)dP≤C∫

Ω

F(([M] (T ))1/2

)dP

whereM∗ (ω)≡ sup{∥M (t)(ω)∥ : t ∈ [0,T ]} .

Proof: Using Corollary 32.4.3, let

N (t) ≡ ∥M (t)−Mτ (t)∥2− [M−Mτ ] (t)

= ∥M (t)−Mτ (t)∥2− [M] (t)+ [M]τ (t)

whereτ ≡ inf{t ∈ [0,T ] : ∥M (t)∥> λ}

Thus N is a martingale and N (0)= 0. In fact N (t)= 0 as long as t ≤ τ . As usual inf( /0)≡∞.Note

for some t<T,∥M(t)∥>λ

[τ < ∞] = [M∗ > λ ]⊇ [N∗ > 0]

This is because to say τ < ∞ is to say there exists t < T such that ∥M (t)∥> λ which is thesame as saying M∗ > λ . Thus the first two sets are equal. Either τ < ∞ or τ = ∞. If τ = ∞,then from the formula for N (t) above, N (t) = 0 for all t ∈ [0,T ] and so it can’t happen thatN∗ > 0. Thus [τ = ∞]⊆ [N∗ = 0] so [N∗ > 0]⊆ [τ < ∞].