2686 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALS

Theorem 79.1.4 Assume the above conditions, 79.1.1 - , 79.1.7 along with the progressivemeasurability condition 79.1.2. Also assume there is at most one solution to 79.1.8 where

q(t, ·)≡∫ t

0ΦndW

Then there exists a P measurable un such that also zn is progressively measurable

Bun (t,ω)−Bu0 (ω)+∫ t

0zn (s,ω)ds =

∫ t

0f (s,ω)ds+B

∫ t

0ΦndW

where for each ω, zn (·,ω) ∈ A(un (·,ω) ,ω). The function Bun (t,ω) = B(un (t,ω)) fora.e. t.

This gives an existence theorem for the inclusion of a stochastic integral. However, it isdesired to get a similar result for Φ rather than Φn. Next is the Ito formula which is useablebecause of the progressive measurability of un,zn. This formula applies to the followingsituation.

Situation 79.1.5 Let X have values in V and satisfy the following

BX (t) = BX0 +∫ t

0Y (s)ds+B

∫ t

0Z (s)dW (s) , (79.1.9)

X0 ∈ L2 (Ω;W ) and is F0 measurable, where Z is L2(Q1/2U,W

)progressively measurable

and∥Z∥L2([0,T ]×Ω,L2(Q1/2U,W)) < ∞.

This is what is needed to define the stochastic integral in the above formula.Assume X ,Y satisfy

BX ,Y ∈ K′ ≡ Lp′ ([0,T ]×Ω;V ′),

the σ algebra of measurable sets defining K′ will be the progressively measurable sets.Here 1/p′+1/p = 1, p > 1.

Also the sense in which the equation holds is as follows. For a.e. ω, the equationholds in V ′ for all t ∈ [0,T ]. Thus we are considering a particular representative X of Kfor which this happens. Also it is only assumed that BX (t) = B(X (t)) for a.e. t. ThusBX is the name of a function having values in V ′ for which BX (t) = B(X (t)) for a.e.t, all t /∈ Nω a set of measure zero. Assume that X is progressively measurable also andX ∈ Lp ([0,T ]×Ω,V ) .

Then in the above situation, we obtain the following integration by parts formula whichis called the Ito formula. This particular version is presented in Theorem 73.7.2 and is ageneralization of work of Krylov. A proof of the case of a Gelfand triple in which B = I isin [108].