878 CHAPTER 32. QUADRATIC VARIATION

Theorem 32.5.4 Let {M (t)} be a continuous H valued local martingale, M (0) =0, where H is a separable Hilbert space and t ∈ [0,T ] . Then if F is a function of the sortdescribed in the good lambda inequality, that is,

F (0) = 0, F continuous, F increasing,

F (αx)≤ cα F (x) ,

there are constants, C and c independent of such local 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: Let {τn} be an increasing localizing sequence for M such that Mτn is uniformlybounded. Such a localizing sequence exists from Proposition 32.3.3. Then from Theorem32.5.3 there exist constants c,C independent of τn such that

c∫

Ω

F([Mτn ] (T )1/2

)dP ≤

∫Ω

F((Mτn)∗

)dP

≤ C∫

Ω

F([Mτn ] (T )1/2

)dP

By Corollary 32.4.3, this implies

c∫

Ω

F((

[M]τn)(T )1/2

)dP ≤

∫Ω

F((Mτn)∗

)dP

≤ C∫

Ω

F((

[M]τn)(T )1/2

)dP

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 [46].

Corollary 32.5.5 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

])