15.1 Introduction 321These authors have proved a remarkable decomposition theorem which es-
sentially shows the following (see Theorem 15.2.1 below for a more precise
statement): Given a bounded sequence (fn)n≥ 1 inL^1 (Ω,F,P) we may find
a subsequence (fnk)k≥ 1 which may be split into aregular and asingular
part,fnk=fnrk+fnsk, such that (fnrk)k≥ 1 is uniformly integrable and (fnsk)k≥ 1
tends to zero almost surely.
Admitting this result, Theorem 15.1.3 becomes rather obvious: As regards
theregular part(fnrk)k≥ 1 we can apply Theorem 15.1.2 to find convex combi-
nations converging with respect to the norm ofL^1 and therefore in measure.
As regards thesingular part(fnsk)k≥ 1 we do not have any problems as any
sequence of convex combinations will also tend to zero almost surely.
A similar reasoning allows to deduce the vector-valued case (Theorem 15.1.4
above) from the Kadeˇc-Pelczy ́nski decomposition result (see Sect. 15.2 below).
After this general prelude we turn to the central theme of this paper. Let
(Mt)t∈R+be anRd-valued cadlag local martingale w.r. to (Ω,F,(Ft)t∈R+,P)
and (Hn)n≥ 1 a sequence ofM-integrable processes, i.e., predictableRd-valued
stochastic processes such that the integral
(Hn·M)t=∫t0HundMumakes sense for every t∈R+, and suppose that the resulting processes
((Hn·M)t)t∈R+are martingales. The theme of the present paper is:under
what conditions can we pass to a limitH^0 ?More precisely: by passing to
convex combinations of (Hn)n≥ 1 (still denoted byHn) we would like to en-
sure that the sequence of martingalesHn·Mconverges to some martingale
Nwhich is of the formN=H^0 ·M.
Our motivation for this question comes from applications of stochastic cal-
culus to Mathematical Finance where this question turned out to be of crucial
relevance. For example, in chapter 9 as well as in the work of D. Kramkov
([K 96a]) the passage to the limit of a sequence of integrands is the heart of
the matter. We shall come back to the applications of the results obtained in
this paper to Mathematical Finance in Sect. 15.5 below.
Let us review some known results in the context of the above question.
The subsequent Theorem 15.1.5, going back to the foundations of stochas-
tic integration given by Kunita and Watanabe [KW 67], is a straightforward
consequence of the Hilbert space isometry of stochastic integrands and in-
tegrals (see, e.g., [P 90, p. 153] for the real-valued and Jacod [J 79] for the
vector-valued case).
Theorem 15.1.5 (Kunita-Watanabe).LetMbe anRd-valued cadlag local
martingale,(Hn)n≥ 1 be a sequence ofM-integrable predictable stochastic pro-
cesses such that each(Hn·M)is anL^2 -bounded martingale and such that the
sequence of random variables((Hn·M)∞)n≥ 1 converges to a random variable
f 0 ∈L^2 (Ω,F,P)with respect to the norm ofL^2.