15.5 Application 353sigma-martingale measures. We refer to Chap. 14 for more details and for
a discussion of the concept of sigma-martingales. We also remark that ifSis
a semi-martingale and ifφis strictly positive, bounded and predictable, then
the sets of stochastic integrals with respect toSand with respect toφ·Sare
the same. This follows easily from the formulaH·S=Hφ·(φ·S).
Theorem 15.5.1 (Optional Decomposition Theorem).LetS=(St)t∈R+
be an Rd-valued semi-martingale, such that the setMe(S)=∅,andV =
(Vt)t∈R+ a real-valued semi-martingale, V 0 =0such that, for eachQ∈
Me(S), the processVis aQ-local super-martingale.
Then there is anS-integrableRd-valued predictable processH such that
(H·S)−V is increasing.
Remark 15.5.2.The Optional Decomposition Theorem is proved in [EQ 95] in
the setting ofRd-valued continuous processes. The important — and highly
non-trivial — extension to not necessarily continuous processes was achieved
by D. Kramkov in his beautiful paper [K 96a]. His proof relies on some of
the arguments from Chap. 9 and therefore he was forced to make the follow-
ing hypotheses: The processSis assumed to be a locally boundedRd-valued
semi-martingale andVis assumed to be uniformly bounded from below. Later
H. F ̈ollmer and Y.M. Kabanov [FK 98] gave a proof of the Optional Decompo-
sition Theorem based on Lagrange-multiplier techniques which allowed them
to drop the local boundedness assumption onS.F ̈ ollmer and Kramkov [FK 97]
gave another proof of this result.
In the present paper our techniques — combined with the arguments of
D. Kramkov — allow us to abandon the one-sided boundedness assumption
on the processV and to pass to the — not necessarily locally bounded —
setting for the processS.
For the economic interpretation and relevance of the Optional Decompo-
sition Theorem we refer to [EQ 95] and [K 96a].
We start the proof with some simple lemmas. The first one — which we
state without proof — resumes the well-known fact that a local martingale is
locally inH^1.
Lemma 15.5.3.For aP-local super-martingaleV we may find a sequence
(Tj)j≥ 1 of stopping times increasing to infinity andP-integrable functions
(wj)j≥ 1 such that the stopped super-martingalesVTjsatisfy
|VTj|≤wj a.s., forj∈N.The next lemma is due to D. Kramkov ([K 96a, Lemma 5.1]) and similar
to Lemma 15.4.12 above.
Lemma 15.5.4.In the setting of the Optional Decomposition Theorem 15.5.1
there is a semi-martingaleWwithW−Vincreasing, such thatWis aQ-local
super-martingale, for eachQ∈Me(S)and which is maximal in the following