2228 CHAPTER 65. STOCHASTIC INTEGRATION

Recall the definition of L2 (U,H) ≡L2 the space of Hilbert Schmidt operators. Ψ ∈L2 (U,H) means Ψ has the property that for some (equivalently all) orthonormal basis ofU {ek} , it follows

∞

∑k=1||Ψ(ek)||2 < ∞

and the inner product for two of these, Ψ,Φ is given by

(Ψ,Φ)L2≡∑

k(Ψ(ek) ,Φ(ek))

Then for such a Hilbert Schmidt operator, the norm in L2 is given by(∞

∑k=1||Ψ(ek)||2

)1/2

≡ ||Ψ||L2.

Note this is the same as (∞

∑k=1

∞

∑j=1

(Ψ(ek) , f j)2

)1/2

(65.1.1)

where{

f j}

is an orthonormal basis for H. This is the analog of the Frobenius norm formatrices obtained as

trace(MM∗)1/2 =

(∑

i(MM∗)ii

)1/2

=

(∑i, j

M2i j

)1/2

Also 65.1.1 shows right away that if Ψ ∈L2 (U,H) , then

||Ψ||2L2(U,H) =∞

∑k=1

∞

∑j=1

(Ψek, f j)2H

=∞

∑k=1

∞

∑j=1

(ek,Ψ∗ f j)

2U = ||Ψ∗||2L2(H,U)

and that Ψ and Ψ∗ are Hilbert Schmidt together.The filtration will continue to be denoted by Ft . It will be defined as the following

normal filtration in which

σ (W (s)−W (r) : 0≤ r < s≤ u)

is the completion of σ (W (s)−W (r) : 0≤ r < s≤ u).

Ft ≡ ∩u>tσ (W (s)−W (r) : 0≤ r < s≤ u). (65.1.2)

and σ (W (s)−W (r) : 0≤ r < s≤ u) denotes the σ algebra of all sets of the form

(W (s)−W (r))−1 (Borel)

where 0≤ r < s≤ u.