72.2. APPROXIMATING WITH STEP FUNCTIONS 2441

72.2 Approximating With Step FunctionsThis 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 72.2.1 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.

Lemma 72.2.2 Let fn→ f in Lp ([0,T ]×Ω,E) . Then there exists a subsequence nk and aset of measure zero N such that if ω /∈ N, then

fnk (·,ω)→ f (·,ω)

in Lp ([0,T ] ,E) and for a.e. t.