4.1 Definitions and Preliminaries
Throughout this chapter, (Q,F,P) will denote a complete
probability space, (F ) a filtration of sub-o-fields of F
z R2
satisfying the usual conditions. We also assume (F ) satisfies the
z
axiom (F4) of Cairoli and Walsh [2] (see section 1.2). Throughout
this chapter we shall denote by M the product o-field B(R2)xF. We now
state some definitions we will use in this chapter. (Some are
restatements from Chapter I, but we will give them again here for
completeness.)
Definition 4.1.1.
a) A (two-parameter) stochastic function is a function (not
necessarily M-measurable) X defined on R2x. Here, X will have values
in a Banach space, usually either in a B-space E, or in the space
L(E,F) of continuous linear maps from E into another Banach space F.
We will consider X extended by zero outside the first quadrant, as we
did for functions defined on R2
+
b) A (two-parameter) stochastic process is a function
X: R xn E, measurable with respect to M = B(R2 )xF. A process X is
2
called adapted if X : Q I E is F -measurable for each z c R2 (see
z z
Millet and Sucheston [13] and Chevalier [3] for related notions). We
generally use the term raw or brut to refer to a process that is not
necessarily adapted, i.e., such that X is F-measurable for each
22
z E +.
For fixed w e Q, the map X (w): R2 + E is called a path of the
process. Each path is a function defined on the first quadrant,
process. Each path is a function defined on the first quadrant, so we