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Of course there is no change in anything if M has its values in a Hilbert space W whileY has its values in its dual space. Then one defines

∫ t0 ⟨Y,dM⟩W ′,W by analogy to the above

for Y an elementary function, step function which is adapted.We use the following definition.

Definition 66.0.18 Let τ p be an increasing sequence of stopping times for which Mτ p is aL2 martingale. If M is already an L2 martingale, simply let τ p ≡ ∞. Let G denote thosefunctions Y which are adapted and for which there is a sequence of elementary functions{Y n} satisfying ∥Y n (t)∥W ′M∗ ∈ L2 (Ω) for each t with

limn→∞

E(∫ T

0∥Y −Y n∥2

W ′ d [M]τ p

)= 0

for each τ p.

Then exactly the same arguments given above yield the following simple generaliza-tions.

Definition 66.0.19 Let Y ∈ G . Then∫ t

0⟨Y,dMτ p⟩W ′,W ≡ lim

n→∞

∫ t

0⟨Y n,dMτ p⟩W ′,W in L2 (Ω)

Lemma 66.0.20 The above definition is well defined. Also,∫ t

0 ⟨Y,dMτ p⟩W ′,W is a continu-ous martingale. The inequality

E

(∣∣∣∣∫ t

0⟨Y,dMτ p⟩W ′,W

∣∣∣∣2)≤ E

(∫ t

0∥Y∥2

W ′ d [M]τ p

)

is also valid. For any sequence of elementary functions {Y n} ,∥Y n (t)∥W ′M∗ ∈ L2 (Ω) ,

∥Y n−Y∥L2(Ω;L2([0,T ];W ′,d[Mτ p ]))→ 0

there exists a subsequence, still denoted as {Y n} of elementary functions for which∫ t

0⟨Y n,dMτ p⟩W ′,W

converges uniformly to∫ t

0 ⟨Y,dMτ p⟩W ′,W on [0,T ] for ω off some set of measure zero. Inaddition, the quadratic variation satisfies the following inequality.[∫ (·)

0⟨Y,dMτ p⟩W ′,W

](t)≤

∫ t

0∥Y∥2

W ′ d [M]τ p ≤∫ t

0∥Y∥2

W ′ d [M]

As before, you can consider the case where you only know X[0,τ p]Y ∈ G . This yieldsa local martingale as before.