72.6. THE ITO FORMULA 2453

+2∫ t

0R((

Z (u)◦ J−1)∗X lk (u)

)◦ JdW +

qk−1

∑j=0

∣∣M (t j+1)−M (t j)

∣∣2−

qk−1

∑j=1

∣∣X (t j+1)−X (t j)−

(M(t j+1

)−M (t j)

)∣∣2 + e(k) (72.6.13)

where tqk = t. By Lemma 72.5.1 the second term on the right, the stochastic integral,converges to

2∫ t

0R((

Z (u)◦ J−1)∗ X̄ (u))◦ JdW

in probability. The first term on the right converges to

2∫ t

0⟨Y (u) , X̄ (u)⟩du

in L1 (Ω) because X rk → X in K. Therefore, this also happens in probability. Consider the

next term.

E

(qk−1

∑j=0

∣∣M (t j+1)−M (t j)

∣∣2) .

It is known from the theory of the quadratic variation that this term converges in probabilityto [M] (t) =

∫ t0 ||Z (s)||2 ds. See Theorem 63.6.4 on Page 2149 and the description of the

quadratic variation in Corollary 65.11.1.Thus all the terms in 72.6.13 converge in probability except for the last term which also

must converge in probability because it equals the sum of terms which do. It remains tofind what this last term converges to. Thus

|X (t)|2−|X0|2 = 2∫ t

0⟨Y (u) , X̄ (u)⟩du

+2∫ t

0R((

Z (u)◦ J−1)∗X (u))◦ JdW +

∫ t

0||Z (s)||2

L2(Q1/2U,H) ds−a

where a is the limit in probability of the term

qk−1

∑j=1

∣∣X (t j+1)−X (t j)−

(M(t j+1

)−M (t j)

)∣∣2Let Pn be the projection onto span(e1, · · · ,en) as before where {ek} is an orthonormal basisfor H with each ek ∈V . Then using

X(t j+1

)−X (t j)−

(M(t j+1

)−M (t j)

)=∫ t j+1

t j

Y (s)ds

the troublesome term above is of the form

qk−1

∑j=1

∫ t j+1

t j

⟨Y (s) ,X

(t j+1

)−X (t j)−Pn

(M(t j+1

)−M (t j)

)⟩ds (72.6.14)