2298 CHAPTER 67. THE EASY ITO FORMULA

This is just the first hitting time of an open set so it is a stopping time. For t ≤ τ, all of theabove quantities must be no larger than C. In particular,

X[0,τ]Φ ∈ L2([0,T ]×Ω;L2

(Q1/2U,H

)).

ThenXτ (t) = X0 +

∫ t

0X[0,τ]φds+

∫ t

0X[0,τ]ΦdW

and so 67.5.13 holds with X → Xτ ,Φ→X[0,τ]Φ and φ →X[0,τ]φ . Now simply let C→∞

and exploit the continuity of X given by the formula 67.5.12 to obtain the validity of 67.5.13without any reference to the stopping time. Of course arbitrary t can replace T. This leadsto the main result.

Theorem 67.6.1 Let Φ be a progressively measurable with values in L2(Q1/2U,H

)which

is stochastically integrable in [0,T ] because

P([∫ T

0||Φ||2

L2(Q1/2U,H) dt < ∞

])= 1

and let φ : [0,T ]×Ω→ H be progressively measurable and Bochner integrable on [0,T ]for a.e. ω, and let X0 be F0 measurable and H valued. Let

X (t)≡ X0 +∫ t

0φ (s)ds+

∫ t

0ΦdW.

Let F : [0,T ]×H×Ω→ R1 be progressively measurable, have continuous partial deriva-tives Ft ,FX ,FXX which are uniformly continuous on bounded subsets of [0,T ]×H indepen-dent of ω ∈ Ω. Also assume FXX is bounded and let FXXX exist and be bounded. Then thefollowing formula holds for a.e. ω.

F (t,X (t)) = F (0,X0)+∫ t

0FX (·,X (·))ΦdW+

∫ t

0Ft (s,X (s))+FX (s,X (s))φ (s)ds+

12

∫ t

0(FXX (s,X (s))Φ,Φ)L2(Q1/2U,H) ds

The dependence of F on ω is suppressed.

That last term is interesting and can be written differently. Let{

g j}

be an orthonormalbasis for Q1/2U. Then this integrand equals

L

∑i=1

(FXX (s,X (s))Φgi,Φgi)H =L

∑i=1

(Φ∗FXX (s,X (s))Φgi,gi)Q1/2H

and we write this astrace(Φ∗ (s)FXX (s,X (s))Φ(s)) .

A simple special case is where Q = I and then Q1/2U =U . Thus it is only required that Φ

have values in L2 (U,H).