8.3 Sigma-martingales and the Non-locally Bounded Case 141Before starting to answer this question, we remark that this question is
not only of “academic” interest. It is also important from the point of view of
applications: if one goes beyond the framework of continuous processesS—
and there are good empirical reasons to do so — it is quite natural to allow
for the jumps of the processes to be unbounded. As a concrete example we
mention L ́evy processes or the family of ARCH (Auto-Regressive Conditional
Heteroskedastic) processes and their relatives (GARCH, EGARCH etc.). The
former find increasing applications in financial engineering. The latter are very
popular in the econometric literature: these are processes in discrete time
where the conditional distribution of the jumps is Gaussian. In particular,
these processes are not locally bounded (compare Example 8.3.3 below). There
are many other examples of processes which fail to be locally bounded, used
in the modelling of financial markets.
The answer to the question, whether Theorem 8.2.1 can be extended to this
setting, is as we expect it to be:mutatis mutandisthe fundamental theorem
of asset pricing as well as the related theorems carry over to the case of not
necessarily locally boundedRd-valued semi-martingalesS. Not coming as a
surprise, the techniques of the proofs have to be refined: in particular, we
cannot entirely reduce to the study of the spaceL∞(Ω,F,P), and the weak-
star and norm topology of this space: there is no possibility anymore to reduce
to the case of (one-sided) bounded stochastic integrals and we therefore have
to use larger spaces thanL∞(Ω,F,P). Yet it turns out — and this is slightly
surprising — that the duality betweenL∞(P)andL^1 (P) still remains the
central issue of the proof.
Here is the statement of the extension of the fundamental theorem of asset
pricing as obtained in Chap. 14.
Theorem 8.3.1 (Main Theorem 14.1.1). The following assertions are
equivalent for anRd-valued semi-martingale modelS=(St)t≥ 0 of a finan-
cial market:
(i) (ESMM), i.e., there is a probability measureQequivalent toPsuch that
Sis a sigma-martingale underQ.
(ii)(NFLVR), i.e.,Ssatisfies the condition of no free lunch with vanishing
risk.
There is a slight change in the statement (i) as compared to the state-
ment of Theorem 8.2.1 above: the term “local martingale” in the definition
of(EMM)was replaced by the term “sigma-martingale” thus replacing the
acronym(EMM)by(ESMM). On the other hand, condition (ii) remained
completely unchanged.
The notion of a sigma-martingale is a generalisation of the notion of a
local martingale:
Definition 8.3.2 (Chap. 14).AnRd-valued semi-martingaleS=(St)t≥ 0
is called a sigma-martingaleif there is a predictable processφ=(φt)t≥ 0 ,