73.4. THE MAIN ESTIMATE 2473

Each term of the above converges to 0 for a.e. ω as k→ ∞ and in L1 (Ω). This followsright away for the second two terms from the Ito isometry and continuity properties of thestochastic integral. Consider the first term. This term is dominated by(∫ t1

0∥Y (u)∥p′ du

)1/p′(∫ T

0∥X r

k (u)∥p du

)1/p

≤ C (ω)

(∫ t1

0∥Y (u)∥p′ du

)1/p′

,

(∫Ω

C (ω)p dP)1/p

< ∞

Hence this converges to 0 for a.e. ω and also converges to 0 in L1 (Ω).At this time, not much is known about the last term in 73.4.10, but it is negative and is

about to be neglected anyway.The term involving the stochastic integral equals

2m−1

∑j=1

⟨B∫ t j+1

t j

Z (u)dW,X (t j)

⟩By Theorem 73.3.2 this equals

2∫ tm

t1

(Z ◦ J−1)∗BX l

k ◦ JdW

Also note that since ⟨BM (t1) ,M (t1)⟩ converges to 0 in L1 (Ω) and for a.e. ω, the suminvolving ⟨

B(M(t j+1

)−M (t j)

),M(t j+1

)−M (t j)

⟩can be started at 0 rather than 1 at the expense of adding in a term which converges to 0a.e. and in L1 (Ω). Thus 73.4.10 is of the form

⟨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+1

)−X (t j)−

(M(t j+1

)−M (t j)

)),

X(t j+1

)−X (t j)−

(M(t j+1

)−M (t j)

)⟩−⟨B(X (t1)−X0−M (t1)) ,X (t1)−X0−M (t1)⟩ (73.4.11)

where e(k)→ 0 for a.e. ω and also in L1 (Ω).