1.3 Stochastic Processes
Throughout this section, (0,F,P) is a complete probability space,
(F ) a filtration satisfying axioms (F1)-(F4) (in particular, one
z 2
zeR
constructed by the method of Meyer), and E will denote a Banach space
equipped with its Borel o-field, denoted B(E). The definitions in
this section are taken from [12], [7].
Definition 1.3.1. A stochastic process Is a measurable (i.e., a
(B(R )xF, B(E))-measurable) function X: R2xn E. We usually denote
X(z,w) by Xz(w), and the mappings w X (w) by Xz
Remark. It will sometimes be convenient to extend the index set to
all of R2 by defining X 0 for z outside the first quadrant, and
F the degenerate o-field for those z. When we wish to consider a
process in this light, we shall say so explicitly.
A brut or raw process is a process X such that X is F-measurable
for all z E R X is called adapted if X is F -measurable for all
z E R2.
2
A process X is called progressive If, for every z E R+, the map
(z,w) Xz(w) from [0,z]xQ E is (B([0,z])xFz, B(E))-measurable.
(Note: This definition comes from Fouque [9]; in Meyer [12]
progressivity is defined using half-open rectangles [0,z). The
definition we give here is the one in common use today.)
1.4 Stopping Points and Lines
1.4.1 Stopping Points
The notion of stopping plays a fundamental role in the theory of
one-parameter processes. The obvious extension of the definition of