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 {Fτ:τ ∈ T} such that Fσ ⊂ Fτ ⊂ F whenever σ < τ;

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

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

and a map B:T x Ω → ╙ (Gl, …, Gq;H).

To save space, and brackets, we will sometimes use the notation for zρ (t) and Bτ, 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 zρ and similar processes this means that each zρ(t) is Ft-measurable, and for B it means that B(τ) is Fτ-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 {Fτ: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 zρ is adapted and there exist constants K > 0, δ > 0 such that if a < s < t < b and t-s < δ then, almost everywhere