880 CHAPTER 32. QUADRATIC VARIATION

The quadratic variation of the Wiener process is just t. This is because if A ∈Fs,s < t,

E(XA

(|W (t)|2− t

))=

E(XA

(|W (t)−W (s)|2 + |W (s)|2 +2(W (s) ,W (t)−W (s))− (t− s+ s)

))Now

E (XA (2(W (s) ,W (t)−W (s)))) = P(A)E (2W (s))E (W (t)−W (s)) = 0

by the independence of the increments. Thus the above reduces to

E(XA

(|W (t)−W (s)|2 + |W (s)|2− (t− s+ s)

))

= E(XA

(|W (t)−W (s)|2− (t− s)

))+E

(XA

(|W (s)|2− s

))= P(A)E

(|W (t)−W (s)|2− (t− s)

)+E

(XA

(|W (s)|2− s

))= E

(XA

(|W (s)|2− s

))and so E

(|W (t)|2− t|Fs

)= |W (s)|2− s showing that t → |W (t)|2− t is a martingale.

Hence, by uniqueness, [W ] (t) = t.

32.6 Approximation With Step FunctionsThere is a really nice result about approximating a function f ∈ Lp ([0,T ] ,E) with stepfunctions. In this we deal with a specific representative of the equivalence class for f ∈Lp ([0,T ] ,E).

Lemma 32.6.1 Let f ∈ L2 ([0,T ] ;E) for E a Banach space. For simplicity let f beBorel measurable. Then there exists a sequence of nested partitions, Pk ⊆Pk+1,

Pk ≡{

tk0 , · · · , tk

mk

}such that the step functions given by

f rk (t) ≡

mk

∑j=1

f(

tkj

)X[tk

j−1,tkj )(t)

f lk (t) ≡

mk

∑j=1

f(

tkj−1

)X[tk

j−1,tkj )(t)

both converge to f in L2 ([0,T ] ;E) as k→ ∞ and

limk→∞

max{∣∣∣tk

j − tkj+1

∣∣∣ : j ∈ {0, · · · ,mk}}= 0.

The mesh points{

tkj

}mk

j=0can be chosen to miss a given set of measure zero N if N does not

contan either 0 or T .