2524 CHAPTER 74. A MORE ATTRACTIVE VERSION

Furthermore,off a set of measure zero, t→ X (t) is continuous as a map into H for a.e. ω .In addition to this,

E (⟨X ,X⟩(t)) =

E(|X0|2

)+E

(∫ t

02⟨Y (s) ,X (s)⟩ds

)+E ([M] (t)) (74.6.38)

The quadratic variation of the stochastic integral satisfies[∫ (·)

0(X ,dM)

](t)≤

∫ t

0∥X∥2

H d [M]

It is more attractive to write |X (t)|2H in place of ⟨X ,X⟩(t). However, I guess this is notstrictly right although the discrepancy is only on a set of measure zero so it seems fairlyharmless to indulge in this sloppiness. However, for t /∈ Nω ,

|X (t)|2H = ∑i(X (t) ,ei)

2

where the orthonormal basis {ei} is in V . Then for s ∈ Nω , you can get the following. Lettn→ s where tn ∈ Nω . Then in the above notation,

∑i⟨X (s) ,ei⟩2 ≤ lim inf

n→∞∑

i(X (tn) ,ei)

2H = lim inf

n→∞|X (tn)|2H ≤C (ω)

It follows that in fact X (s) ∈ H and you can take X (s) = ∑i ⟨X (s) ,ei⟩ei ∈ H because∑i ⟨X (s) ,ei⟩2 < ∞. Hence

|X (s)|2 = ∑i⟨X (s) ,ei⟩2 ≤ lim inf

n→∞|X (tn)|2H

so X has values in H and is lower semicontinuous on [0,T ].