8.3 Sigma-martingales and the Non-locally Bounded Case 145as in this example (supposing for the moment thatSis real-valued and that
FT−pis trivial to simplify even further). If the jump ∆STpis unbounded from
above as well as from below we may proceed just as in (8.8) above to find
a probalility measureQ∈Meswith‖Q−Q 1 ‖ 1 <εsuch thatEQ[∆STp|
FT−p]=EQ[∆STp]=0.HenceQ∈Meσas we now have that the (conditional)
expectation at the jump timeTpassumes the correct value, namely zero.
Now suppose that the jump ∆STpis only one-sided bounded, say bounded
from below, but unbounded from above. This is a slight variation of the situa-
tion of Example 8.3.3. In this case it is important to note that, forQ 1 ∈Mes,
we must have that
EQ 1 [∆STp|FT−p]=EQ 1 [∆STp]≤ 0. (8.11)Indeed, the predictable processH= (^1) [[Tp]] then is admissible and we have
(H·S)∞=∆STpwhich implies (8.11).
In the case when we have strict inequality in (8.11) (otherwise there is
nothing to prove) we can again change the measureQ 1 to a probability mea-
sureQsimilarly as in (8.8) above toincreasethe valueEQ[∆STp] by putting
more mass of the probability on the upper tail of the distribution of ∆STp
(recall that ∆STpis assumed to be unbounded from above). Hence we may
increase this value until we have equality in (8.11).
Summing up, we have managed to pass from a givenQ 1 ∈MestoQ∈Meσ
such that‖Q−Q 1 ‖ 1 <ε; but we have used some very restrictive assumptions.
Now let us get rid of them. The most obvious step is to drop the assumption
thatFT−pis trivial: it suffices to apply the above arguments conditionally on
FT−p. More delicate is the passage fromR-valued processesStoRd-valued
ones. In theR-valued case there are only two possibilities of one-sided bound-
edness of ∆XTp, i.e., either from above or from below. Ford≥2wehave
to consider general cones of directions inRdinto which the jump ∆XTpis
bounded. The corresponding arguments are worked out in Chap. 14 below.
Finally, we can generalise the assumption (8.10) to the case when the jumps
ofSare exhausted not by one stopping timeTpbut by a sequence (Tnp)∞n=1of
predictable stopping times: repeat inductively the above argument and apply
an 2 εn-argument. This program takes care of the predictable stopping times
in (8.9).
We still have to deal with the totally inaccessible stopping times (Tni)∞n=1
in (8.9). They form a different league as in this case we cannot argue con-
ditionally onF(Tni)− as in the predictable case above. Instead we have to
consider the compensators of the jumps ofSat the totally inaccessible stop-
ping timesTni. This requires some machinery from semi-martingale theory as
developed, e.g., in [JS 87]. The technicalities are more complicated than in the
case of predictable stopping times; nevertheless it is possible to proceed in a
similar spirit and to argue inductively on (Tni)∞n=1to make sure that all the
relevant conditional expectations are well-defined and have the desired value,