15.4 A Substitute of Compactness for Bounded Subsets ofH^1345above linear system would only admit the solutionαk=0forallk≤d+1.
Because of the definition of the stopping timesTnkwe immediately obtain
thatgk,k≥γnk. For the non-diagonal elements we distinguish the casesl<k
andl>k.Forl<kwe use the fact that onT ̃nl<∞,wehavethatTnl<Tnk.
It follows that|∆(Hnk·M)Tnl|≤ 2 βnk
(
(trace([M, M]Tnl))(^12)
+1
)
and hence|gl,k|≤βnk.Ifl>kthen|∆(Hnk·M)Tnl|≤κnk
(
(trace([M, M]Tnl))(^12)
+1
)
and
hence|gl,k|≤κnk. We now multiply the last column of the matrixgl,kwith
the fraction βn^1
d+1(d+1)^2
and then we multiply the last row by
βnd+1(d+1)^2
γnd+1.
The result is that the diagonal element at place (d+1,d+1) is equal to 1
and that the other elements of the last row and the last column are bounded
in absolute value by(d+1)^12. We continue in the same way by multiplying the
columndby β^1
nd(d+1)^2
and the rowdby βnd(d+1)
2
γnd. The result is that the
element at place (d, d) is 1 and that the other elements on rowdand column
dare bounded by(d+1)^12. We note that the elements at place (d, d+1) and
(d+1,d) are further decreased by this procedure so that the bound (d+1)^12
will remain valid. We continue in this way and we finally obtain a matrix with
1 on the diagonal and with the off-diagonal elements bounded by(d+1)^12 .By
the classical theorem, due to Hadamard [G 66, Satz 1], such a matrix with
dominant diagonal is non-singular. The proof of the claim is now completed
and so are the proofs of the Theorems 15.A, 15.B and 15.C.
15.4.4A proof of M. Yor’s Theorem 15.1.6 for theL^1 -convergent
Case
We now show how the ideas of the proof given in [Y 78a] fit in the general
framework described above. We will use the generalisation of Theorem 15.A to
processes with jumps (see the remarks following the proof of Theorem 15.A).
In the next theorem we suppose thatMis ad-dimensional local martingale.
Theorem 15.4.7.Let(Hn)n≥ 1 be a sequence ofM-integrable predictable sto-
chastic processes such that each(Hn·M)is a uniformly integrable martingale
and such that the sequence of random variables((Hn·M)∞)n≥ 1 converges
to a random variablef 0 ∈L^1 (Ω,F,P)with respect to theL^1 -norm; (or even
only with respect to theσ(L^1 ,L∞)-topology).
Then there is anM-integrable predictable stochastic processH^0 such that
H^0 ·Mis a uniformly integrable martingale and such that(H^0 ·M)∞=f 0.
Proof.Iffnconverges only weakly tof 0 then we take convex combinations in
order to obtain a strongly convergent sequence. We therefore restrict the proof
to the case wherefnconverges inL^1 -norm tof 0. By selecting a subsequence
we may suppose that‖fn‖L 1 ≤1foreachnand that‖fn−f 0 ‖L 1 ≤ 4 −n.Let
Nbe the cadlag martingale defined asNt=E[f 0 |Ft]. From the maximal
inequality forL^1 -martingales it then follows that: