2286 CHAPTER 66. THE INTEGRAL∫ t
0 (Y,dM)H
From the above definition, for each ω off a suitable set of measure zero, from Lemma66.0.13,∫ t∧σ
0(Y,dM) ≡ lim
p→∞
∫ t∧σ
0
(X[0,τ p]Y,dMτ p
)= lim
p→∞
∫ t
0
(X[0,τ p]X[0,σ ]Y,dMτ p
)≡∫ t
0
(X[0,σ ]Y,dM
)Finally, consider the claim about the quadratic variation. Using 66.0.6,[(∫ (·)
0(Y,dM)
)]τ p
(t) =
[(∫ (·)
0(Y,dM)
)τ p](t) =
[∫ (·)
0
(X[0,τ p]Y,dMτ p
)](t)
≤∫ t
0
∥∥∥X[0,τ p]Y∥∥∥2
d [M]τ p ≤∫ t
0∥Y∥2 d [M]
Now letting τ p→ ∞, [(∫ (·)
0(Y,dM)
)](t)≤
∫ t
0∥Y∥2 d [M]
Next is the case in which Y is continuous in t but not necessarily bounded nor assumedto be in any kind of L2 space either.
Definition 66.0.16 Let Y be continuous in t and adapted. Let M be a continuous localmartingale M (0) = 0. Then the definition of a local martingale
∫ t0 (Y,dM) is as follows.
Let τ p be an increasing sequence of stopping times for which [M]τ p ,∥Mτ p∥ ,∥∥∥X[0,τ p]Y
∥∥∥are all bounded by p. Then∫ t
0(Y,dM)≡ lim
p→∞
∫ t
0
(X[0,τ p]Y,dMτ p
)Then it is clear that X[0,τ p]Y ∈ G . Therefore, the above Theorem yields the following
corollary.
Corollary 66.0.17 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 ]Y has the same properties as Y, being the pointwise limit on[0,T ] of a bounded seuqence of elementary functions for each ω . In addition to this, thereis an estimate for the quadratic variation[∫ (·)
0(Y,dM)
](t)≤
∫ t
0∥Y∥2 d [M]