Chapter 67

The Easy Ito FormulaFirst recall 64.5.26 where it is shown that for every α

E(|W (t)−W (s)|α

)≤Cα |t− s|α/2 ,

and so by Kolmogorov Čentsov continuity theorem

|W (t)−W (s)| ≤Cγ |t− s|γ (67.0.1)

for every γ < 1/2.

67.1 The SituationThe idea is as follows. You have a sufficiently smooth function F : [0,T ]×H → R whereH is a separable Hilbert space. You also have the random variable

X (t) = X0 +∫ t

0φ (s)ds+

∫ t

0ΦdW

where Φ is progressively measurable and in L2([0,T ]×Ω;L2

(Q1/2U,H

))where Q : U→

U is a positive self adjoint operator. Also assume X0 is F0 measurable with values in H.Recall the descriptive diagram.

U↓ Q1/2

U1 ⊇ JQ1/2U J←1−1

Q1/2U

Φn ↘ ↓ Φ

H

Here the Wiener process is in U1 and the filtration with respect to which Φ is progressivelymeasurable is the usual filtration determined by this Wiener process. Then the Ito formulais about writing the random variable F (t,X (t)) in terms of various integrals and derivativesof F .

67.2 Assumptions And A LemmaAssume F : [0,T ]×H×Ω→ R1 has continuous partial derivatives Ft ,FX , and FXX whichare uniformly continuous and bounded on bounded subsets of [0,T ]×H independent ofω ∈Ω. Also assume FXX is uniformly bounded and that FXXX exists. Let φ : [0,T ]×Ω→Hbe progressively measurable and Bochner integrable for each ω . Assume Φ is progressivelymeasurable, and is in L2

([0,T ]×Ω;L2

(Q1/2U,H

)).

Now here is the important lemma which makes the Ito formula possible.

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