2426 CHAPTER 71. STOCHASTIC O.D.E. ONE SPACE

solution to the integral equation for that ω . It follows from the uniqueness assumption, thatit is not necessary to take a subsequence. Thus

u(t,ω) = limn→∞

un (t,ω)

For t ∈ (tnj−1, t

nj ], it follows that ω → u(t,ω) is Ftn

jmeasurable. Since this is true for each

n and the filtration is assumed to be a normal filtration, we conclude that ω → u(t,ω) isFt measurable.

Why can’t this be generalized to the situation where no uniqueness is known? We havebeen unable to do this. It appears that the difficulty is related to the need to use theoremsabout measurable selections and these theorems pertain to a single σ algebra. Attempts touse the σ - algebra of progressively measurable sets have not been successful either.

71.2 Including Stochastic IntegralsIt is not surprising that Theorem 71.1.1 is sufficient to allow the inclusion of a stochasticintegral. Thus, with the same descriptions of the symbols used in that theorem, one couldconsider the following integral equation.

u(t,ω)−u0(ω)+∫ t

0N(s,u(s,ω),u(s−h,ω) ,w(s,ω) ,ω)ds

=∫ t

0f(s,ω)ds+

∫ t

0ΦdW

where, as usual Φ ∈ L2([0,T ]×Ω;L2

(Q1/2U,Rd

))where U is a Hilbert space. It could

be Rd of course. To include a stochastic integral, you define a new variable.

û(t) = u(t)−∫ t

0ΦdW

Then in terms of this new variable, the integral equation is

û(t,ω)−u0 (ω)+∫ t

0N(

s, û(s,ω)+∫ s

0ΦdW, û(s−h,ω) +

∫ s−h

0ΦdW,

∫ s

0

(û(r)+

∫ r

0ΦdW

)dr,ω

)ds =

∫ t

0f(s,ω)ds

This is in the situation of Theorem 71.1.1 provided N is progressively measurable withrespect to the normal filtration Ft determined by the Wiener process and there exists anestimate of the sort in this theorem and for a given ω there is at most one solution t →û(t,ω) to the above integral equation.

Theorem 71.2.1 Suppose N(t,u,v,w,ω) ∈ Rd for u,v,w ∈ Rd , t ∈ [0,T ] and

(t,u,v,w,ω)→ N(t,u,v,w,ω)