3) The "jump at zero," If(0,0) will play a role later, similar
to that of the jump at zero In the theory of one-parameter
processes. When we associate measures with Ifl, we shall need this
term to get some compatibility between these measures and those
associated with f.
4) The function If| is increasing in both senses of Definition
2.1.3. First of all, if z = (s,t), z' = (s',t'), and z < z', then
Ifl(s',t') Ifl(s,t) = 0 if (s',t') lies outside the first quadrant;
fl (s',t') fl (s,t) Ifl(s',t') > 0 if z' Z 0, z outside R2 If
0 S z < z', then
|f|(s',t') IfI(s,t) = If(0,0)I + Var[o, ]f(.,0)
+ Var[o,t']f(O,.) + Var[(o,0),(s',t')]f)
[|f(o ,0)I + Var[os]f(-,O) + Var [ ]f(0,.)
+ Var[(0,0),(s,t)] ]
= Var f(*,0) + Var [t,t f(0,.)
[ss ^ I[ttf 0
+ (Var[( ,),(s'.t')](f) Var[(o,),(s t)] )
0.
As for the other sense, we have A[z,z] fl = 0 if z' lies outside the
first quadrant. We then deal with the case where 0 5 z'.
If z is in the third quadrant, i.e., if s