73.5. A SIMPLIFICATION OF THE FORMULA 2479
73.5 A Simplification Of The FormulaThis lemma also provides a way to simplify one of the formulas derived earlier in the casethat X0 ∈ Lp (Ω,V ) so that X−X0 ∈ Lp ([0,T ]×Ω,V ). Refer to 73.4.11. One term there is
⟨B(X (t1)−X0−M (t1)) ,X (t1)−X0−M (t1)⟩
Also,⟨B(X (t1)−X0−M (t1)) ,X (t1)−X0−M (t1)⟩
≤ 2⟨B(X (t1)−X0) ,X (t1)−X0⟩+2⟨BM (t1) ,M (t1)⟩
It was observed above that 2⟨BM (t1) ,M (t1)⟩ → 0 a.e. and also in L1 (Ω) as k→∞. Applythe above lemma to ⟨B(X (t1)−X0) ,X (t1)−X0⟩ using [0, t1] instead of [0,T ] . The new X0equals 0. Then from the estimate 73.4.8, it follows that
E (⟨B(X (t1)−X0) ,X (t1)−X0⟩)→ 0
as k→ ∞. Taking a subsequence, we could also assume that
⟨B(X (t1)−X0) ,X (t1)−X0⟩ → 0
a.e. ω as k→ ∞. Then, using this subsequence, it would follow from 73.4.11,
⟨BX (tm) ,X (tm)⟩−⟨BX0,X0⟩= e(k)+2∫ tm
0⟨Y (u) ,X r
k (u)⟩du+
+2∫ tm
0
(Z ◦ J−1)∗BX l
k ◦ JdW
+m−1
∑j=0
⟨B(M(t j+1
)−M (t j)
),M(t j+1
)−M (t j)
⟩−
m−1
∑j=1
⟨B(∆X (t j)−∆M (t j)) ,∆X (t j)−∆M (t j)
⟩(73.5.18)
where e(k)→ 0 in L1 (Ω) and a.e. ω and
∆X (t j)≡ X(t j+1
)−X (t j)
∆M (t j) being defined similarly. Note how this eliminated the need to consider the term
⟨B(X (t1)−X0−M (t1)) ,X (t1)−X0−M (t1)⟩
in passing to a limit. This is a very desirable thing to be able to conclude.Can you obtain something similar even in case X0 is not assumed to be in Lp (Ω,V )?
Let Z0k ∈ Lp (Ω,V )∩ L2 (Ω,W ) ,Z0k → X0 in L2 (Ω,W ) . Then from the usual argumentsinvolving the Cauchy Schwarz inequality,
⟨B(X (t1)−X0) ,X (t1)−X0⟩1/2 ≤ ⟨B(X (t1)−Z0k) ,X (t1)−Z0k⟩1/2
+⟨B(Z0k−X0) ,Z0k−X0⟩1/2