136 8 Arbitrage Theory in Continuous Time: an Overview
Using substantially more refined arguments, M. Yor [Y 78a] was able to
extend this result to the case of Cauchy sequences (Hn·S)∞n=0of martingales
bounded inLp(P)orHp(P), for arbitrary 1≤p≤∞, the most delicate and
interesting case beingp= 1 (compare also Chap. 15 below).
After this review of some of the previous literature on the topic of com-
pleteness of the space of stochastic integrals, let us turn back to Lemma 8.2.3.
Unfortunately the theorems of Kunita-Watanabe and Yor do not apply to
its proof, as we don’t assume thatSis a local martingale. It is precisely the
point, that we finally want toprovethatSis a local martingale with respect
to some measureQequivalent toP.
But in our attempt to build up some motivation for the proof of Lemma
8.2.3, let us cheat for a moment and suppose that we know already thatSis
a local martingale under some equivalent measureQand let (Hn)∞n=0be a
sequence ofS-integrable predictable processes satisfying (8.4). Using again the
theorem of Ansel-Stricker (Theorem 7.3.7 above) we conclude that (Hn·S)∞n=0
is a sequence of local martingales; inequality (8.4) quickly implies that this
sequence is bounded inL^1 (Q)-norm:
‖Hn·S‖L (^1) (Q):= sup{EQ[|(Hn·S)τ|]|τstopping time}≤2, forn≥ 0.
Let us cheat once more and assume that eachHn·Sis in fact a uniformly
integrableQ-martingale (instead of only being a localQ-martingale) and
that ((Hn·S)∞)∞n=0is Cauchy with respect to theL^1 (Q)-norm defined above
(instead of only being bounded with respect to this norm).
Admitting the above “cheating steps” we are in a position to apply Yor’s
theorem to find a limiting processHto the sequence (Hn)∞n=0for which (8.5)
holds true, where we even may replace the inequality by an equality. But, of
course, this is only motivation, why Lemma 8.2.3 should hold true, and we
now have to find a mathematical proof, preferably without cheating.
We have taken some time for the above heuristic considerations to develop
an intuition for the statement of Lemma 8.2.3 and to motivate the general
philosophy underlying its proof:we want to prove results which are — at least
more or less — known for (local) martingalesS, but replacing the martingale
assumption onSby the assumption thatSsatisfies (NFLVR).
As a starter we sketch the proof of a result which shows that, under the
assumption of(NFLVR), the technical condition imposed on the admissible
integrandHin (8.1) is, in fact, automatically satisfied. The lemma is taken
from Theorem 9.3.3 where it is stated for locally bounded semi-martingales,
but it remains valid for general semi-martingalesS.
Lemma 8.2.4 (9.3.3).LetSbe a semi-martingale satisfying (NFLVR) and
letHbe an admissible integrand.
Then
(H·S)∞:= lim
t→∞
(H·S)t
exists and is finite, almost surely.