The Mathematics of Arbitrage

324 15 A Compactness Principle

For general martingales, not necessarily of the formHn·Mfor a fixed
local martingaleM, we can prove the following theorem:

Theorem 15.C.Let(Mn)n≥ 1 be anH^1 -bounded sequence ofRd-valued mar-
tingales. Then there is a subsequence, for simplicity still denoted by(Mn)n≥ 1
and an increasing sequence of stopping times(Tn)n≥ 1 with the following prop-
erties:

(1)Tnincreases to∞,

(2)the martingalesNn=(Mn)Tn−∆MTn (^1) [[Tn,∞[[+Cnform a relatively
weakly compact sequence inH^1 .HereCndenotes the compensator (dual
predictable projection) of the process∆MTn (^1) [[Tn,∞[[,
(3)there are convex combinations

∑

k≥nα

k

nN

kthat converge to anH (^1) ,mar-
tingaleN^0 in the norm ofH^1 ,
(4)there is a cadl ag optional process of finite variationZsuch that almost
everywhere for eacht∈R:Zt= lims↘t;s∈Qlimn→∞

∑

k≥nα

k

nC

k

s.

If, in addition, the set

{

∆(Mn)−T

∣

∣n∈N;Tstopping time}

resp.
{|∆(Mn)T||n∈N;Tstopping time}

is uniformly integrable, e.g. there is an integrable functionw≥ 0 such that

∆(Mn)≥−w resp. |∆(Mn)|≤w, a.s.

then the process(Zt)t∈R+is increasing (resp. vanishes identically).

Let us comment on these theorems. Theorem 15.A shows that in the con-
tinuous case we may cut off somesmall singular parts in order to obtain
a relatively weakly compact sequence ((Mn)Tn)n≥ 1 inH^1. By taking convex
combinations we then obtain a sequence that converges in the norm ofH^1 .The
singular parts are small enough so that they do not influence the almost sure
passage to the limit. Note that — in general — there is no hope to get rid of the
singular parts. Indeed, a Banach spaceEsuch that for each bounded sequence
(xn)n≥ 1 ∈Ethere is a norm-convergent sequenceyn∈conv{xn,xn+1,...}is
reflexive; and, of course,H^1 is only reflexive if it is finite dimensional.
The general situation of local martingalesM(possibly with jumps) de-
scribed in Theorem 15.B is more awkward. As regards the convex combina-
tions of the form (

∑

k≥nα

k
nH
k 1
[[ 0,Tn]]∩(En)c·M)n≥^1 we have convergence in
H^1 but for thesingular parts (Vn)n≥ 1 we cannot assert that they tend to
zero. Nevertheless there is some control on these processes. We may assert
that the processes (Vn)n≥ 1 tend, in a certain pointwise sense, to a process
(Zt)t∈R+of integrable variation. We shall give an example (Sect. 15.3 below)
which illustrates that in general one cannot do better than that. But under