2706 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALS

79.4 Stochastic Inclusions Without Uniqueness ??We will consider strong solutions to the integral equation

u(t)−u0 +∫ t

0zds =

∫ t

0f +

∫ t

0ΦdW

where z(t,ω)∈ A(u(t,ω) , t,ω) in a situation where there is not necessarily uniqueness forthe integral equation for fixed ω . Uniqueness usually comes from monotonicity consid-erations. I am trying to eliminate these and replace with a monotonicity condition whichcomes from including εF for F a duality map.

79.4.1 EstimatesIn what follows [0,T ] will be a finite interval with no restriction on the size of T .

Definition 79.4.1 Recall a filtration is {Ft} , t ∈ [0,T ] where each Ft is a σ algebra of setsin Ω a probability space and these are increasing in t. Then the progressively measurablesets P are S⊆Ω such that

S∩ [0, t]×Ω is B ([0, t])×Ft measurable

You can verify that this is indeed a σ algebra of sets in [0,T ]×Ω. Here B ([0, t]) is the σ

algebra of Borel measurable sets on [0, t] . We could have used B ([0,T ])×Ft instead ofB ([0, t])×Ft in the above because a set is in B ([0, t]) if and only if it is the intersectionof a Borel set of [0,T ] with [0, t]. We will always assume that each Ft contains all the setsof measure zero of FT .

We will assume all Banach spaces are separable in what follows.We will assume U ⊆ V ⊆ H = H ′ ⊆ V ′ ⊆ U ′ where the inclusion map of V into the

Hilbert space H is compact and V is dense in H and U is a Banach space which is denseand compact in V . One can always obtain such a space. In fact one can always have U bea Hilbert space. In practice this is most easily seen from Sobolev embedding theorems butthe existence of this space follows from general abstract considerations.

Then for p > 1, define

V ≡ Lp ([0,T ]×Ω;V ) , U ≡ Lr ([0,T ]×Ω;V ) , r ≥max(2, p) .

It follows that the dual space V ′ can be identified in the usual way as Lp′ ([0,T ]×Ω;V ).Similarly H will be defined as L2 ([0,T ]×Ω;H). In each instance the relevant σ algebrawill be the progressively measurable sets. A set A⊆Ω× [0,T ] is progressively measurableif

A∩ [0, t]×Ω ∈Ft ×B ([0, t])

where B ([0, t]) denotes the Borel sets of [0, t] , equivalently the intersections of a Borel setof [0,T ] with [0, t].

Also define Vω as Lp ([0,T ] ;V ) where the subscript ω indicates that ω is fixed. Let Hω

and Uω be defined similarly. We will assume the following on A : V× [0,T ]×Ω→P (V ′).