The Mathematics of Arbitrage

294 14 The FTAP for Unbounded Stochastic Processes

14.4 The GeneralRd-valued Case

In this sectionS =(St)t∈R+ denotes a general Rd-valued cadlag semi-
martingale based on (Ω,F,(Ft)t∈R+,P) where we assume that the filtration
(Ft)t∈R+satisfies the usual conditions of completeness and right continuity.
Similarly as in Chap. 9 we define anS-integrableRd-valued predictable
processH=(Ht)t∈R+to be anadmissible integrandif the stochastic process

(H·S)t=

∫t

0

(Hu,dSu),t∈R+

is (almost surely) uniformly bounded from below.
It is important to note that, similarly as in Proposition 14.3.7 above, it
may happen that the cone of admissible integrands is rather small and possibly
even reduced to zero: consider, for example, the case whenSis a compound
Poisson process with (two-sided) unbounded jumps, i.e.,St=

∑Nt
i=1Xi,where
(Nt)t∈R+is a Poisson process and (Xi)∞i=1an i.i.d. sequence of real random
variables such that‖Xi+‖∞=‖Xi−‖∞=∞. Clearly, a predictable processH,
such thatH·Sremains uniformly bounded from below, must vanish almost
surely.
Continuing with the general setup we denote byK the convex cone in
L^0 (Ω,F,P)givenby

K={f=(H·S)∞|Hadmissible}

where this definition requires in particular that the random variable (H·
S)∞:= limt→∞(H·S)tis (almost surely) well-defined (compare Definition
9.2.7).
Again we denote byCthe convex cone inL∞(Ω,F,P)formedbythe
uniformly bounded random variables dominated by some element ofK, i.e.,

C=(K−L^0 +(Ω,F,P))∩L∞(Ω,F,P)

={f∈L∞(Ω,F,P)| there isg∈K, f≤g}. (14.1)

We say (see Definition 8.1.2) that the semi-martingaleSsatisfies the con-
dition ofno free lunch with vanishing risk (NFLVR)if the closureCofC,
taken with respect to the norm-topology‖.‖∞ ofL∞(Ω,F,P) intersects
L∞+(Ω,F,P) only in 0, i.e.,

Ssatisfies(NFLVR)⇐⇒C∩L∞+={ 0 }.

For the economic interpretation of this concept, which is a very mild
strengthening of the “no-arbitrage” concept, we refer to Chap. 8.
The subsequent crucial Theorem 14.4.1 was proved in (9.4.2) under the
additional assumption thatSis bounded. An inspection of the proof given in
Chap. 9 reveals that — for the validity of the subsequent Theorem 14.4.1 —
the boundedness assumption onSmay be dropped.