SO, then the paths f(s,*) have finite variation for all f, and in fact

(same proof)


 Var[ ,t]f(s,.) Var (s ),(s,) (f) + Var 0,t f(s, )0


 Up to now, we have avoided using the phrase "f has finite

variation" because of the weakness of the condition Var (f) < m for R

bounded. We shall reserve this term for functions with the additional

conditions described above. We will see that this is enough to give

the additional properties we need to associate useful measures.
 2
Definition 2.3.1. Let f: R2 + E be right continuous, with
 2
VarR(f) < m for bounded rectangles R C R2. We say that f has finite

variation if the real-valued function


Ifj(s,t) = If(o,0)l+Var[os]f(*,0) arl o,t]f(o -)+ar[( O),(s,t)]() <

 2
for every (s,t) E R We say f has bounded variation if there exists

M>O such that


 If (s,t) < M for all (s,t) E R.


The map If : R2+ R+ is called the variation of f. (Note that we use

the single bars to distinguish it from the norm in E.)

Remarks.

 1) Henceforth, the phrase "f has finite variation" will be
 2
understood to mean that IfI(s,t) < for all (s,t) E R.

 2) We extend Ifl by 0 outside the first quadrant to get a
 function defined on all of R2
function defined on all of R