144 8 Arbitrage Theory in Continuous Time: an Overview
Meσ={Q|Sis aQsigma-martingale},is dense inMeswith respect to the norm ofL^1 (Ω,F∞,P).
Let us illustrate this fact for the easy Example 8.3.3 above: fixμ∈R,
σ^2 >0,ε>0 and a probability measureQ 1 ∼P. We want to find a measure
Q∼P,‖Q 1 −Q‖<εsuch thatEQ[X] = 0. Suppose w.l.g. thatEQ 1 [X]<0.
(IfEQ 1 [X] = 0 there is nothing to prove; ifEQ 1 [X]>0 it suffices to reverse
the inequalities in (8.8).)
Define, forξ∈R,α>1, 0<β<1, the measureQ(ξ, α, β)by
dQ(ξ, α, β)
dQ 1=α (^1) {X>ξ}+β (^1) {X≤ξ}. (8.8)
AsXis unbounded under the measureQ 1 , one easily verifies that one may
find (ξ, α, β) such thatQ:=Q(ξ, α, β) is a probability measure and such that
‖Q−Q 1 ‖<εas well asEQ[X] = 0 (compare Lemma 14.3.4).
Summing up, we have shown the validity of Proposition 8.3.4 in the very
special case of Example 8.3.3. This strategy of proof also applies to the proof
of Proposition 8.3.4 in full generality, modulo some delicate technicalities as
we now shall try to explain.
To sketch the idea of the proof of Proposition 8.3.4 suppose thatS =
(St)t≥ 0 is a semi-martingale satisfying(NFLVR),sothatMesis non-empty.
FixQ 1 ∈Mes.
The problem is thatQ 1 may fail to be a sigma-martingale measure; this
is due to the fact that the semi-martingaleS may have “big jumps”. So
let us deal with the jumps in a systematic way. We know from the general
theory [D 72, DM 80] that the jumps of a cadlag processScan be exhausted
by countably many stopping times; in addition, these stopping times can be
classified into the predictable ones and the totally inaccessible ones (Definition
7.2.2 and 7.2.3). More precisely, for a given cadlag processSwe may find
sequences (Tnp)∞n=1and (Tni)∞n=1such thatTnp(resp.Tni) are predictable (resp.
totally inaccessible) stopping times and such that
{(ω, t)∈Ω×R+|∆St(ω)=0}⊆
⋃∞
n=1[[Tnp]]∪⋃∞
n=1[[Tni]]. (8.9)In addition we may assume that the sets ([[Tnp]] )∞n=1and(
[[Tni]])∞
n=1are
mutually disjoint.
To sketch the idea of the proof of Proposition 8.3.4 we start by considering
the case where there is only one predictable jump in (8.9), i.e.
{(ω, t)∈Ω×R+|∆St(ω)=0}⊆[[Tp]], (8.10)for some predictable stopping timeTp. This is the case, e.g., in Example
8.3.3, whereTp≡1. In order to show Proposition 8.3.4 we proceed similarly