67.3. A SPECIAL CASE 2291

67.3 A Special CaseTo make it simpler, first consider the situation in which Φ=Φ0 where Φ0 is F0 measurableand has finitely many values in L (U1,H), and φ = φ 0 where φ 0 is F0 measurable and asimple function with values in H. Thus

X (t) = X0 +∫ t

0φ 0ds+

∫ t

0Φ0dW

Now let{

tnk

}mnk=0 denote the nth partition of [0,T ] , referred to as Pn such that

limn→∞

(max

{∣∣tnk − tn

k−1∣∣ ,k = 0,1,2, · · · ,mn

})≡ lim

n→∞||Pn||= 0.

The superscript n will be suppressed to save notation. Then

F (T,X (T ))−F (0,X0) =mn−1

∑k=0

(F (tk+1,X (tk+1))−F (tk,X (tk)))

=mn−1

∑k=0

(F (tk+1,X (tk+1))−F (tk,X (tk+1)))

+mn−1

∑k=0

(F (tk,X (tk+1))−F (tk,X (tk)))

This equalsmn−1

∑k=0

Ft (tk,X (tk+1))(tk+1− tk)+o(|tk+1− tk|) (67.3.3)

+mn−1

∑k=0

FX (tk,X (tk))(X (tk+1)−X (tk)) (67.3.4)

+12

mn−1

∑k=0

(FXX (tk,X (tk))(X (tk+1)−X (tk)) ,(X (tk+1)−X (tk)))H (67.3.5)

+mn−1

∑k=0

O(|X (tk+1)−X (tk)|3H

)(67.3.6)

RecallX (t) = X0 +

∫ t

0φ 0ds+

∫ t

0Φ0dW

From the properties of the Wiener process in 67.0.1, the term in 67.3.6 converges to 0as n→ ∞ since these properties of the Wiener process imply X is Holder continuous withexponent 2/5.

Now consider the term of 67.3.5. All terms converge to 0 except

12

mn−1

∑k=0

(FXX (tk,X (tk))

∫ tk+1

tkΦ0dW,

∫ tk+1

tkΦ0dW

)H

(67.3.7)