2346 CHAPTER 68. A DIFFERENT KIND OF STOCHASTIC INTEGRATION

Thus E(

δ (u)2)=

m

∑j,k=1

E(δ(F̂jh j

)δ(F̂khk

))=

m

∑j,k=1

E(⟨

D(δ(F̂jh j

)),(F̂khk

)⟩)m

∑j,k=1

E(⟨

F̂jh j +W (h j)D(F̂j)−D

⟨DF̂j,h j

⟩,(F̂khk

)⟩)Separating out the first term this is

= E

(m

∑k=1

∥∥F̂k∥∥2

)+∑

k,kE(⟨

W (h j)D(F̂j), F̂khk

⟩)−∑

j,kE(Dk(D jF̂j

)Fk)

= E

(m

∑k=1

∥∥F̂k∥∥2

)+∑

k,kE(⟨

W (h j)D(F̂j), F̂khk

⟩)−∑

j,kE(Dk(D jF̂j

)F̂k)

= E

(m

∑k=1

∥∥F̂k∥∥2

)+∑

k,kE(W (h j)Dk

(F̂j)

F̂k)

−∑j,k

E(Dk(D jF̂j

)F̂k)

(68.4.24)

By equality of mixed partial derivatives, the third term equals

−∑j,k

E(D j(DkF̂j

)F̂k)= ∑

j,kE((

DkF̂j)(

D jF̂k))−∑

j,kE(Dk(F̂j)

F̂kW (h j))

Therefore, 68.4.24 reduces to

E

δ

(m

∑j=1

Fjh j

)2 = E

(m

∑k=1

∥∥F̂k∥∥2

H

)+

m

∑j,k=1

E((

DkF̂j)(

D jF̂k))

= E

(m

∑k=1

∥∥F̂k∥∥2

H

)+

m

∑j,k=1

E(⟨

DF̂j,hk⟩⟨

DF̂k,h j⟩)

= E

(m

∑k=1∥Fk∥2

H

)+

m

∑j,k=1

E(⟨

DFj,hk⟩⟨

DFk,h j⟩)

because the derivative is well defined. All of this assumes the hk form an orthonormal set.Suppose these are just orthogonal but nonzero. Then

E

δ

(m

∑j=1

Fjh j

)2= E

( m

∑j=1

δ (Fjh j)

)2= E

(∑j,k

δ (Fjh j)δ (Fkhk)

)