Chapter 73

The Hard Ito Formula, Implicit Case73.1 Approximating With Step Functions

This Ito formula seems to be the fundamental idea which allows one to obtain solutions tostochastic partial differential equations using a variational point of view. I am followingthe treatment found in [108]. The following lemma is fundamental to the presentation. Itapproximates a function with a sequence of two step functions X r

k ,Xlk where X r

k has thevalue of X at the right end of each interval and X l

k gives the value X at the left end of theinterval. The lemma is very interesting for its own sake. You can obviously do this sort ofthing for a continuous function but here the function is not continuous and in addition, it isa stochastic process depending on ω also. This lemma was proved earlier, Lemma 65.3.1.

Lemma 73.1.1 Let Φ : [0,T ]×Ω→V, be B ([0,T ])×F measurable and suppose

Φ ∈ K ≡ Lp ([0,T ]×Ω;E) , p≥ 1

Then there exists a sequence of nested partitions, Pk ⊆Pk+1,

Pk ≡{

tk0 , · · · , tk

mk

}such that the step functions given by

Φrk (t) ≡

mk

∑j=1

Φ

(tk

j

)X(tk

j−1,tkj ](t)

Φlk (t) ≡

mk

∑j=1

Φ

(tk

j−1

)X[tk

j−1,tkj )(t)

both converge to Φ in K as k→ ∞ and

limk→∞

max{∣∣∣tk

j − tkj+1

∣∣∣ : j ∈ {0, · · · ,mk}}= 0.

Also, each Φ

(tk

j

),Φ(

tkj−1

)is in Lp (Ω;E). One can also assume that Φ(0) = 0. The mesh

points{

tkj

}mk

j=0can be chosen to miss a given set of measure zero. In addition to this, we

can assume that ∣∣∣tkj − tk

j−1

∣∣∣= 2−nk

except for the case where j = 1 or j = mnk when this might not be so. In the case of the lastsubinterval defined by the partition, we can assume∣∣∣tk

m− tkm−1

∣∣∣= ∣∣∣T − tkm−1

∣∣∣≥ 2−(nk+1)

The following lemma is convenient.

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