2284 CHAPTER 66. THE INTEGRAL∫ t
0 (Y,dM)H
Definition 66.0.14 Let τ p be an increasing sequence of stopping times for which
limp→∞
τ p = ∞
and such that Mτ p is a L2 martingale and X[0,τ p]Y ∈ G . Then the definition of∫ t
0 (Y,dM)
is as follows. For each ω,∫ t
0(Y,dM)≡ lim
p→∞
∫ t
0
(X[0,τ p]Y,dMτ p
)In fact, this is well defined.
Theorem 66.0.15 The above definition is well defined. Also this makes∫ t
0 (Y,dM) a localmartingale. In particular,∫ t∧τ p
0(Y,dM) =
∫ t
0
(X[0,τ p]Y,dMτ p
)In addition to this, if σ is any stopping time,∫ t∧σ
0(Y,dM) =
∫ t
0
(X[0,σ ]Y,dM
)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)
](t)≤
∫ t
0∥Y∥2 d [M]
Proof: Suppose for some ω, t < τ p < τq. Let ω be such that both τ p,τq are larger thant. Then for all ω, and τ a stopping time,∫ t∧τ
0
(X[0,τq]Y,dMτq
)=∫ t
0
(X[0,τq]Y,d
((Mτq)τ
))In particular, for the given ω,∫ t
0
(X[0,τq]Y,d (M
τq)τq)=∫ t
0
(X[0,τq]Y,dMτq
)=∫ t∧τq
0
(X[0,τq]Y,dMτq
)For the particular ω, this equals ∫ t∧τ p
0
(X[0,τq]Y,dMτq
)Now for all ω including the particular one, this equals∫ t
0
(X[0,τq]Y,d
((Mτq)τ p
))=∫ t
0
(X[0,τq]Y,dMτ p
)For the ω of interest, this is ∫ t∧τ p
0
(X[0,τq]Y,dMτ p
)