67.5. THE INTEGRABLE CASE 2295

then

F (T,X (T )) = F (0,X0)+∫ T

0Ft (t,X (t))+FX (t,X (t))φ (t)dt

+∫ T

0FX (t,X (t))ΦdW +

12

∫ T

0(FXX (t,X (t))Φ,Φ)L2(Q1/2U,H) dt (67.4.11)

This has proved the following lemma.

Lemma 67.4.1 Let Φ,φ be elementary functions as described and let

X (t) = X0 +∫ t

0φ (s)ds+

∫ t

0ΦdW

Then 67.4.11 holds.

67.5 The Integrable CaseNow let Φ ∈ L2

([0,T ]×Ω;L2

(Q1/2U,H

)),φ ∈ L1 ([0,T ]×Ω;H) and be progressively

measurable. Let φ be as above, and let

X (t) = X0 +∫ t

0φ (t)dt +

∫ t

0ΦdW (67.5.12)

Suppose also the additional condition that for some M,

|X (t,ω)|< M for all (t,ω) ∈ [0,T ]×NC, P(N) = 0.

Does it follow that 67.4.11 holds?There exists a sequence of elementary functions {Φn} converging to Φ◦ J−1 in

L2([0,T ]×Ω;L2

(JQ1/2U,H

))Similarly let {φ n} converge to φ in L1 ([0,T ]×Ω;H) where φ n is also an elementary func-tion, |φ n| ≤ |φ | at the mesh points. You could use that theorem about approximating withleft and right step functions if desired, Lemma 65.3.1. Let

Xn (t) = X0 +∫ t

0φ n (s)ds+

∫ t

0ΦndW.

Also let τn be the stopping times

τn ≡ inf{t > 0 : |Xn (t)|> M} .

Since Xn is continuous, this is a well defined stopping time. Thus

Xτnn (t) = X0 +

∫ t

0X[0,τn]φ n (t)dt +

∫ t

0X[0,τn]ΦndW

and as noted in the discussion of localization for elementary functions, X[0,τn]Φn is anelementary function.