CHAPTER III - THE INTEGRAL: ESTIMATES AND EXISTENCE

Summary

McSHANE'S INTEGRAL

We shall generally be concerned with the following objects: A subset T ⊂ R with an interval [a, b] ⊂ T;

A probabili.ty space (Ω, F, μ) with a family, or filtration, σ of a-algebras {F_τ:τ ∈ T} such that F_σ ⊂ F_τ ⊂ F whenever σ < τ;

Banach spaces G₁, …, G_q and a Hilbert space H, all separable;

Maps z_ρ:[a, b] → £^o(Ω, F;G_p) ρ = 1, …, q

and a map B:T x Ω → ╙ (G_l, …, G_q;H).

To save space, and brackets, we will sometimes use the notation for z^ρ (t) and B_τ, or B(τ), for B(τ, -), with the corresponding convention for other processes.

We shall usually require our processes to be adapted to the filtration: for z^ρ and similar processes this means that each z^ρ(t) is F_t-measurable, and for B it means that B(τ) is F_τ-measurable for each τ in T. The σ-algebra Fτ can be thought of as consisting of those events which are known about at time τ, or the ‘past’ at time τ. The term non-anticipating is sometimes used instead of ‘adapted’, but it also has a more technical definition; a process adapted to {F_τ:τ ∈ T} is also called an F_*-process.

From time to time our maps will be subjected to some of the following conditions (for positive integers rand p):

Condition A(r)

Each z^ρ is adapted and there exist constants K > 0, δ > 0 such that if a < s < t < b and t-s < δ then, almost everywhere