76.2. PRELIMINARY RESULTS 2539

Also, C is a continuous function of its arguments and C (0,0,0,0) = 0. Thus for a.e. ω,

supt∈[0,T ]

⟨BX ,X⟩(t)≤C (ω)< ∞.

For a.e. ω, t→ BX (t,ω) is weakly continuous with values in W ′ for t off a set of measurezero. Also t→ ⟨BX (t) ,X (t)⟩ is lower semicontinuous off a set of measure zero.

Then from this fundamental lemma, the following Ito formula is valid. The proof ofthis theorem follows the same methods used for a similar result in [108].

Theorem 76.2.3 Off a set of measure zero, for every t ∈ [0,T ] ,

⟨BX ,X⟩(t) = ⟨BX0,X0⟩+∫ t

0

(2⟨Y (s) ,X (s)⟩+ ⟨BZ,Z⟩L2

)ds

+2∫ t

0

(Z ◦ J−1)∗BX ◦ JdW (76.2.2)

AlsoE (⟨BX ,X⟩(t)) =

E (⟨BX0,X0⟩)+E(∫ t

0

(2⟨Y (s) ,X (s)⟩+ ⟨BZ,Z⟩L2

)ds)

(76.2.3)

The quadratic variation of the stochastic integral is dominated by

C∫ t

0∥Z∥2

L2∥BX∥2

W ′ ds (76.2.4)

for a suitable constant C. Also t→ BX (t) is continuous with values in W ′ for t ∈ NCω .

We will often abuse the notation and write ⟨BX (t) ,X (t)⟩ instead of the more precise⟨BX ,X⟩(t). No harm is done because these two are equal a.e.

In addition to the above, we will use the following basic theorems about nonlinearoperators. This is Proposition 75.1.8 above.

Proposition 76.2.4 Suppose A : V → V ′ is type M, see [91], and suppose L : V → V ′ ismonotone, bounded and linear. Here V is a separable reflexive Banach space. Then L+Ais type M.

As an important example, we give the following definition.

Definition 76.2.5 Let f : [0,T ]×Ω→V

τh f (t,ω)≡{

f (t−h,ω) if t ≥ h0 if t < h