The Mathematics of Arbitrage

15.1 Introduction 325

special assumptions, e.g., one-sided or two-sided bounds on the jumps of the
processes (Hn·M), one may deduce certain features of the processZ(e.g.,
Zbeing monotone or vanishing identically). It is precisely this latter conclu-
sion which has applications in Mathematical Finance and allows to give an
alternative proof of Kramkov’soptional decomposition theorem[K 96a] (see
Theorem 15.5.1 below).
To finish the introduction we shall state the main application of Theorem
15.B. Note that the subsequent statement of Theorem 15.D does not use
the concept ofH^1 (P)-martingales (although the proof heavily relies on this
concept) which makes it more applicable in general situations.

Theorem 15.D.LetMbe anRd-valued local martingale andw≥ 1 an inte-
grable function.
Given a sequence(Hn)n≥ 1 ofM-integrableRd-valued predictable processes
such that
(Hn·M)t≥−w, for alln, t ,

then there are convex combinations

Kn∈conv{Hn,Hn+1,...},

and there is a super-martingale(Vt)t∈R+,V 0 ≤ 0 , such that

lims↘t

s∈Q+

lim

n→∞

(Kn·M)s=Vt fort∈R+,a.s.,

and anM-integrable predictable processH^0 such that

((H^0 ·M)t−Vt)t∈R+ is increasing.

In addition,H^0 ·Mis a local martingale and a super-martingale.

Loosely speaking, Theorem 15.D says that for a sequence (Hn·M)n≥ 1 ,obey-
ing the crucial assumption of uniform lower boundedness with respect to an
integrable weight functionw, we may pass — by forming convex combina-
tions — to a limiting super-martingaleV in a pointwise sense and — more
importantly — to a local martingale of the form (H^0 ·M) which dominatesV.
The paper is organised as follows: Sect. 15.2 introduces notation and fixes
general hypotheses. We also give a proof of the Kadeˇc-Pelczy ́nski decom-
position and we recall basic facts about weak compactness inH^1 .Wegive
additional (and probably new) information concerning the convergence of the
maximal function and the convergence of the square function. Sect. 15.3 con-
tains an example. In Sect. 15.4, we give the proofs of Theorems 15.A, 15.B,
15.C and 15.D. We also reprove M. Yor’s Theorem 15.1.6. In Sect. 15.5 we
reprove Kramkov’s Optional Decomposition Theorem 15.5.1.