72.4. THE MAIN ESTIMATE 2447

The term involving the stochastic integral equals

2m−1

∑j=1

(∫ t j+1

t j

Z (u)dW,X (t j)

)H

By Theorem 65.14.1, this equals

2∫ tm

t1R((

Z (u)◦ J−1)∗X lk (u)

)◦ JdW,

t→∫ t

0 R((

Z (u)◦ J−1)∗X l

k (u))◦ JdW being a local martingale. Therefore, 72.4.7 equals

|X (tm)|2−|X0|2 = 2∫ tm

0⟨Y (u) ,X r

k (u)⟩du+ e(k)

2∫ tm

t1R((

Z (u)◦ J−1)∗X lk (u)

)◦ JdW +

m−1

∑j=1

∣∣M (t j+1)−M (t j)

∣∣2−

m−1

∑j=1

∣∣X (t j+1)−X (t j)−

(M(t j+1

)−M (t j)

)∣∣2−|X (t1)−X0−M (t1)|2

where e(k) converges to 0 in L1 (Ω) and for a.e. ω . Note that X lk (u) = 0 on [0, t1) and so

that stochastic integral equals∫ tm

0R((

Z (u)◦ J−1)∗X lk (u)

)◦ JdW.

Therefore, from the above,

|X (tm)|2−|X0|2 = 2∫ tm

0⟨Y (u) ,X r

k (u)⟩du+ e(k)

2∫ tm

0R((

Z (u)◦ J−1)∗X lk (u)

)◦ JdW +

m−1

∑j=0

∣∣M (t j+1)−M (t j)

∣∣2−|M (t1)|2

−m−1

∑j=1

∣∣X (t j+1)−X (t j)−

(M(t j+1

)−M (t j)

)∣∣2−|X (t1)−X0−M (t1)|2

Then since |M (t1)|2 converges to 0 in L1 (Ω) and for a.e. ω, as discussed above,

|X (tm)|2−|X0|2 = 2∫ tm

0⟨Y (u) ,X r

k (u)⟩du+ e(k)

+2∫ tm

0R((

Z (u)◦ J−1)∗X lk (u)

)◦ JdW +

m−1

∑j=0

∣∣M (t j+1)−M (t j)

∣∣2−|X (t1)−X0−M (t1)|2