74.3. THE MAIN ESTIMATE 2503

Some terms cancel and this yields

=−⟨BX (t)−BX (s) ,X (t)−X (s)⟩+2⟨BX (t) ,M (t)−M (s)⟩

=−⟨BX (t)−BX (s) ,X (t)−X (s)⟩+2⟨B(M (t)−M (s)) ,X (t)⟩

= −⟨B(X (t)−X (s)) ,X (t)−X (s)⟩

+2⟨

BX (t)−BX (s)−∫ t

sY (u)du,X (t)

⟩= −⟨BX (t) ,X (t)⟩−⟨BX (s) ,X (s)⟩

+2⟨BX (t) ,X (s)⟩+2⟨BX (t) ,X (t)⟩

−2⟨BX (s) ,X (t)⟩−2∫ t

s⟨Y (u) ,X (t)⟩du

= ⟨BX (t) ,X (t)⟩−⟨BX (s) ,X (s)⟩−2∫ t

s⟨Y (u) ,X (t)⟩du

Therefore,

⟨BX (t) ,X (t)⟩−⟨BX (s) ,X (s)⟩

= 2∫ t

s⟨Y (u) ,X (t)⟩du+ ⟨B(M (t)−M (s)) ,M (t)−M (s)⟩

−⟨BX (t)−BX (s)− (M (t)−M (s)) ,X (t)−X (s)− (M (t)−M (s))⟩+2⟨BX (s) ,M (t)−M (s)⟩

The following phenomenal estimate holds and it is this estimate which is the main ideain proving the Ito formula. The last assertion about continuity is like the well known resultthat if y ∈ Lp (0,T ;V ) and y′ ∈ Lp′ (0,T ;V ′) , then y is actually continuous a.e. with valuesin H, for V,H,V ′ a Gelfand triple. Later, this continuity result is strengthened further togive strong continuity.

Lemma 74.3.2 In the Situation 74.1.1, the following holds for all t /∈ N̂,

E (⟨BX (t) ,X (t)⟩)

< C(||Y ||K′ , ||X ||K ,E ([M] (T )) ,∥⟨BX0,X0⟩∥L1(Ω)

)< ∞. (74.3.8)

where K,K′ were defined earlier. In fact,

E

(sup

t∈[0,T ]∑

i⟨BX (t) ,ei⟩2

)≤C

(||Y ||K′ , ||X ||K ,E ([M] (T )) ,∥⟨BX0,X0⟩∥L1(Ω)

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

supt /∈NC

ω

⟨BX (t,ω) ,X (t,ω)⟩ ≤C (ω)< ∞.

Also for ω off a set of measure zero described earlier, t→ BX (t)(ω) is weakly continuouswith values in W ′ on [0,T ] . Also t→ ⟨BX (t) ,X (t)⟩ is lower semicontinuous on NC

ω .