874 CHAPTER 32. QUADRATIC VARIATION

Now from 32.12 and Corollary 32.4.3, [M,N]τ − [Mτ ,N] = 0. Similarly [M,N]τ −[M,Nτ ] = 0. Now consider the next claim that [M−Mτ ] = [M]− [Mτ ]. From the defi-nition, it follows

[M−Mτ ]− ([M]+ [Mτ ]−2 [M,Mτ ])

= ∥M−Mτ∥2−(∥M∥2 +∥Mτ∥2−2(M,Mτ)

)+ local martingale

= local martingale.

By the first part of the corollary which ensures [M,Mτ ] is of bounded variation, the left sideis the difference of two increasing adapted processes and so by Corollary 32.3.5 again, theleft side equals 0. Thus from the above,

[M−Mτ ] = [M]+ [Mτ ]−2 [M,Mτ ] = [M]+ [Mτ ]−2 [Mτ ,Mτ ]

= [M]+ [Mτ ]−2 [Mτ ] = [M]− [Mτ ]≤ [M]

Finally consider the claim that [M,N] is bilinear. From the definition, letting M1,M2,Nbe H valued local martingales,

(aM1 +bM2,N)H = [aM1 +bM2,N]+ local martingalea(M1,N)+b(M2,N)H = a [M1,N]+b [M2,N]+ local martingale

Hence[aM1 +bM2,N]− (a [M1,N]+b [M2,N]) = local martingale.

The left side can be written as the difference of two increasing functions thanks to [M,N]of bounded variation and so by Lemma 32.2.1 it equals 0. [M,N] is obviously symmetricfrom the definition. ■

32.5 The Burkholder Davis Gundy InequalityDefine

M∗ (ω)≡ sup{∥M (t)(ω)∥ : t ∈ [0,T ]} .The Burkholder Davis Gundy inequality is an amazing inequality which involves M∗ and[M] (T ).

Before presenting this, here is the good lambda inequality, Theorem 10.12.1 on Page299 listed here for convenience.

Theorem 32.5.1 Let (Ω,F ,µ) be a finite measure space and let F be a continuousincreasing function defined on [0,∞) such that F (0) = 0. Suppose also that for all α > 1,there exists a constant Cα such that for all x ∈ [0,∞),

F (αx)≤Cα F (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,

µ ([ f > βλ ]∩ [g≤ rδλ ])≤ φ (δ )µ ([ f > λ ]) (32.13)

where limδ→0+ φ (δ ) = 0 and φ is increasing. Under these conditions, there exists a con-stant C depending only on β ,φ ,r such that∫

Ω

F ( f (ω))dµ (ω)≤C∫

Ω

F (g(ω))dµ (ω) .