2294 CHAPTER 67. THE EASY ITO FORMULA
because
limn→∞
(mn−1
∑k=0
X(tk,tk+1] (t)FX (tk,X (tk))
)Φ0 = FX (t,X (t))Φ0
in L2([0,T ]×Ω;L2
(Q1/2U,H
)). Next consider the first on the right in 67.3.10. It equals
mn−1
∑k=0
(FX (tk,X (tk))φ 0 (tk+1− tk))
and converges to ∫ T
0FX (t,X (t))φ 0dt.
Finally, it is obviously the case that 67.3.3 converges to∫ T
0Ft (t,X (t))dt
This has shown
F (T,X (T )) = F (0,X0)+∫ T
0Ft (t,X (t))+FX (t,X (t))φ 0dt
+∫ T
0FX (t,X (t))Φ0dW +
12
∫ T
0(FXX (t,X (t))Φ0,Φ0)L2(Q1/2U,H) dt
whenX (t) = X0 +
∫ t
0φ 0ds+
∫ t
0Φ0dW,
φ 0,Φ0F0 measurable as described above. This is the first version of the Ito formula.
67.4 The Case Of Elementary FunctionsOf course there was nothing special about the interval [0,T ] . It follows that for [a,b] ⊆[0,T ] , Φa ∈L (U1,U) and Fa measurable, having finitely many values, φ a a simple func-tion which is Fa measurable,
X (t) = X (a)+∫ t
aφ adt +
∫ t
aΦadW
F (b,X (b)) = F (a,X (a))+∫ b
a(Ft (t,X (t))+FX (t,X (t))φ a)dt
+∫ b
aFX (t,X (t))ΦadW +
12
∫ b
a(FXX (t,X (t))Φa,Φa)L2(Q1/2U,H) dt.
Therefore, if Φ is any elementary function, being a sum of functions like ΦaX(a,b], and φ
a similar sort of elementary fuction with
X (t) = X0 +∫ t
0φds+
∫ t
0ΦdW,