1. Introduction
In this paper, 
is assumed to be a filtered probability space where 
is a filtration satisfying 
for all
, the usual condition of right continuity and completeness. The random movement of 
risky assets in the market is modeled via cadlag, nonnegative stochastic processes
, where
. We assume that all wealth processes are discounted by another special asset which is considered a baseline. In the market described above, economic agents can trade in order to reallocate their wealth.
Consider a simple predictable process
.
where
, and for all
, 
is a finite stopping time and 
is 
-measurable.
Each
, 
, is an instance when some give economic agent may trade in the market, then, 
is the number of unit from the ith risky assets that the agent will hold in the trading interval
. This form of trading is called simple, as it comprises of finite number of buy-and-hold strategies, in contrast to continuous trading where one is able to change the position of the assets in a continuous fashion. The last form of trading is only theoretical value, since it cannot be implemented in reality, even if one ignores market frictions.
Starting from initial capital 
and following the strategy described by the simple predictable process
, the agent’s discounted process is given by
.
where
, 
are a.s. finite stoping times with respect to 
and the 
are 
-measurable real random variables. Note that the trader is allowed to trade on an infinite time horizon, because we do not restrict to bounded stoping times for the re-allocation of the capital. Of course trading on a finite time horizon [0, T] is covered by switching to the process
.
Theorem 1.1. [1,2] A real valued, cadlag, adapted process 
the following are equivalent:
1) X is a good integrator.
2) X may be decomposed as
, where 
is a local martingale and 
is an adapted process of finite variation.
Defination 1.1. [1,3] A real valued, cadlag, adapted process 
allows for A Free Lunch With Vanishing Risk for simple integrands if there is a sequence 
of simple integrands such that for
,
.
and
In contrast, X therefore admits No Free Lunch With Vanishing Risk (NFLVR) for simple integrands if for every sequence 
satisfying (VR) we have
(NFL) 
in probability.
A free lunch with vanishing risk (FLVR) for simple integrands indicates that S allows for a sequence of trading schemes
, each 
involving only finitely many rebalancing of the portfolio, such that the losses tend to Zero in the sense that of (VR) ,while the terminal gains (FL) remain substantial as n goes to infinity. It is important to note that the condition (VR) of vanishing risk pertains the maximal losses of the trading strategy 
during the entire interval [0,T]: if the left hand side of (VR) equals 
this implies that, with probability one, the strategy 
never, i.e. for not
, cause an accumulated loss of more than
.
Resently, it has been argued that existence of an Equivalent Martingale Measure(EMM) is not necessary for viability of the market; to see this effect, see [4-6]. In [7], the concept of strictly positive supermartingale deflator which is weaker than the existence of an EMM, that allows for consistent theory to be developed.In this paper, we investigate the relation between the no free lunch with vanishing risk property for simple integands and the semimartingale property.
Theorem 1.2. [1,8] Let 
be a real-valued, cadlag, locally bounded process based on and adepted to a filtered probability space
. If S satisfies the condition of no free lunch with vanishing risk (NFLVR) for simple integrands then S is a semimartingale.
Theorem 1.3. For a locally bounded, adopted, cadlag process X the following are equivalent 1) X satisfies NFLVR + LI(little Investment)
2) X is a classical semimartingale.
Theorem 1.4. For an adapted cadlag process X the following are equivalent.
1) For all sequences 
of simple predictable processesa) 
b) 
together imply 
in probability.
2) X is a classical semimartingale.
Proposition 1.5. Let 
be cadlag and adapted, with X0 and such that 
and X satisfies NFLVR + LI For all 
there is 
and a sequence of stopping times 
such that, for all n 1) 
takes values in
.
2)
.
3) The stopped processes 
and 
satisfyfor all n, 
and
Lemma 1.6. Under the assumptions as in the proposition above with
the sequence 
is bounded in probability.
Proof. For all n, let
a simple predictable process, then 
since 
since X satisfies NFLVR + LI, 
is bounded in
. 
For 
define a sequence of stopping times
.
Given 
there is 
such that
Lemma 1.7. Under the same assumptions as in Proposition 1.5 the stopped martingales 
satisfy
.
Proof. For 
and
, since the 
are predictable and the 
are martingales,
we write 
as a telescoping series and simplifying to get
Lemma 1.8. Let
.
Under the assumption of Proposition 1.5 the sequence 
is bounded in probability.
Proof. Assume for contradiction that 
is not bounded in probability. Then there is 
such that for all k there is 
such that 
For 
define
and
.
Then 
and
and at time t = 1 we have
.
But the second summand is bounded in L2, so we conclude that 
is not bounded in probability.
We defined a sequence of stopping times
.
Because
by Doob’s sub-martingale in-equality,(see [9,10])
is bounded in probability. Therefore there is 
such that
. Note that 
is uniformly bounded below by
. We claim 
is not bounded in probability. Indeed, for any n and any k,
Since
, the probability of the other event is at least
. This gives the desired contradiction because it is now easy to construct a FLVR + LI.
Proof of Proposition 1.5: Defined a sequence of stopping times
.
By Lemma 1.8 there is c2 such that
. Take 
and 
Lemma 1.9. [11]. Let 
be measurable functions, where f is left continuous and takes finitely many values. Say
. Define
where 
is the biggest of the k such that Xk less than or equal to t. Then for all partition
,
Proposition 2.0. Let 
be cadlag and adopted, with 
and such that 
and X satisfies NFLVR + LI. For all 
there is C and a 
valued stopping time 
such that
and sequence 
and 
of continuous time cadlag processes such that for all n1) 
2) 
3) 
is a martingale with 
4) 
Proof. Let 
be given. Let C, 
, 
, and 
be as in proposition 1.5. Extended 
and 
to all 
by defining 
and
. Not that the extended 
is no longer predictable, and currently we only have control of the total variation of 
over
, i.e.
Notice that, for
,
From this and 
it follow that
, so
. How do we fine the limit of the sequence of stopping times
? The trick is to define
, a simple predicator process, and note that stopping at 
is like integrating Rn, i.e. 
and
. We have that
.
Apply Komlos’ Lemma to obtain convex weights 
such that
a.s as 
By the dominated convergence theorem,
. Observe that
Define
. Each 
is left continuous, decreasing process. In particular, 
, so we can divide by this quantity. We claim that 
. In deed, on the event
, 
so
Define new processes
. Then 
and
. Thus we define Mn and An by
The total variation of 
over 
is bounded by 3. By Lemma 1.9,
That 
follows from the fact that
. To finish the proof, we show that there is a subsequence 
such that 
satisfies
. We know 
because
. Since 
a.s there is a subsequence such that
. Finally,
Therefore
, 
and 
have the desired properties.
Proof of the Main Theorems Proof of Theorem 1.3. We may assume the hypothesis of proposition. Let 
and take C, 
, 
, 
as in proposition. Apply komlos lemma to find convex weights 
such that
for all
, where the convergence is a.s. For all n,
so the total variation of A over D is bounded by C. Further, we have
. A is a cadlag on D, so define it on all of [0,1] to make it cadlag. M is 
martingale so it has a cadlag modification. Since 
and 
was arbitrary, and the class of classical semimartingales is local, X must be a classical semimartingale. 
Proof of Theorem 1.4. We no longer assume that X is locally bounded. The trick is to leverage the result for locally bounded processes by subtracting the big jump from X. Assume without loss of generality that 
and defined
. Then X = X − J is an adopted, cadlag locally bounded process. We will show that theorem 1.4 for X implies NFLVR + LI for X, so that we may apply theorem 1.3 to X. Then since J is finite variation, this will then imply X is a classical semimartingale .
Suppose 
are such that 
and
. We need to prove that 
in probability . First we will show that
.
by the assumptions on 
By (1), 
in probability. Since
in probability, we conclude that
in probability. Therefore X satisfies NFLVR + LI. 
NOTES