+
proof for p 0; if u has finite variation, we write p = p ,
and associate X with p and X with We have, then, for A
bounded, measurable: u(A) = UC(A) u (A) = E( 2A dX ) -
R2 z z
E( 2A dX ) = E( 2A dX f A dX ) = E( 2A d(X4 X)). Setting
z z R2 z z R z z R2 z z z
+ 2 +
X = X X p(A) = E(f 2AzdXz), and |XI = X+ X-| I |X~I + X-I,
so X has integrable variation, and the theorem is proved. I
Remark. In Meyer [12] a version of this theorem (without proof) is
given for P-measures and random measures on R2xH.
4.2 Measures Associated With Vector-Valued
Stochastic Functions
In this section we shall show that, starting with a stochastic
function, we can associate a P-measure, with finite variation if the
function has integrable variation. Our first theorem is for
measurable processes with integrable variation.
Theorem 4.2.1. Let E be a Banach space, X an E-valued, raw, right
continuous process such that X is integrable for every z R ,
z +
with raw integrable variation
l (s,t) = X(0o,)1 + Var[0,s](X(.,0)) + Var t] (Xo.)
+ Var[(,),(s,t)]X).
There is a stochastic measure (P-measure) X: B(R2)xF + E with finite
variation satisfying the following.