2288 CHAPTER 66. THE INTEGRAL∫ t
0 (Y,dM)H
Definition 66.0.21 Let τ p be an increasing sequence of stopping times for which
limp→∞
τ p = ∞
and such that Mτ p is a martingale and X[0,τ p]Y ∈ G . Then the definition of∫ t
0 ⟨Y,dM⟩W ′,Wis as follows. For each ω off a set of measure zero,∫ t
0⟨Y,dM⟩W ′,W ≡ lim
p→∞
∫ t
0
⟨X[0,τ p]Y,dMτ p
⟩W ′,W
where∫ t
0
⟨X[0,τ p]Y,dMτ p
⟩W ′,W
is a martingale.
In fact, this is well defined.
Theorem 66.0.22 The above definition is well defined. Also this makes∫ t
0 ⟨Y,dM⟩W ′,W alocal martingale. In particular,∫ t∧τ p
0⟨Y,dM⟩W ′,W =
∫ t
0
⟨X[0,τ p]Y,dMτ p
⟩W ′,W
In addition to this, if σ is any stopping time,∫ t∧σ
0⟨Y,dM⟩W ′,W =
∫ t
0
⟨X[0,σ ]Y,dM
⟩W ′,W
In this last formula, X[0,σ ]X[0,τ p]Y ∈ G . In addition, the following estimate holds for thequadratic variation. [∫ (·)
0⟨Y,dM⟩W ′,W
](t)≤
∫ t
0∥Y∥2
W ′ d [M]
Note that from Definition 66.0.21 it is also true that∫ t
0⟨Y,dM⟩W ′,W ≡ lim
p→∞
∫ t
0
⟨X[0,τ p]Y,dMτ p
⟩W ′,W
in probability. In addition, since τ p → ∞, it follows that for each ω, eventually τ p > T .
Therefore, t →∫ t
0 ⟨Y,dM⟩W ′,W is continuous, being equal to∫ t
0
⟨X[0,τ p]Y,dMτ p
⟩W ′,W
for
that ω .