# Superhedging Supermartingales

C. Bender  Department of Mathematics
Saarland University
Campus E 2 4, 66123 Saarbrücken Germany<EMAIL_ADDRESS>,  S.E.Ferrando
Department of Mathematics
Toronto Metropolitan University
350 Victoria Street<EMAIL_ADDRESS>,  K.Gajewski  Department of
Mathematics
Toronto Metropolitan University
350 Victoria Street<EMAIL_ADDRESS>and  A.L. Gonzalez  Department
of Mathematics
Mar del Plata National University
Mar del Plata, Funes 3350, 7600 Argentina<EMAIL_ADDRESS>
(Date: Received: date / Accepted: date)

###### Abstract.

Supermartingales are here defined on a non-probabilistic setting and can be
interpreted solely in terms of superhedging operations. The classical
expectation operator is replaced by a pair of subadditive operators one of
them providing a class of null sets and the other one acting as an outer
integral. These operators are motivated by a financial theory of no-arbitrage
pricing. Such a setting extends the classical stochastic framework by
replacing the path space of the process by a trajectory set, while also
providing a financial/gambling interpretation based on the notion of
superhedging. The paper proves analogues of the following classical results:
Doob’s supermartingale decomposition and Doob’s pointwise convergence theorem
for nonnegative supermartingales. The approach shows how linearity of the
expectation operator can be circumvented and how integrability properties in
the proposed setting lead to the special case of (hedging) martingales while
no integrability conditions are required for the general supermartingale case.

##  1\. Introduction

The paper introduces a class of non-probabilistic supermartingales in a
setting where a set of price scenarios (also called trajectories) is given
along with the possibility to trade as price trajectories unfold over time.
Trajectories are sequences $S=(S_{i})_{i\geq 0}\in\mathcal{S}$ with a common
origin $S_{0}=s_{0}$, where the set $\mathcal{S}$ substitutes the abstract
sample space $\Omega$ of the probabilistic setting. Following ideas of the
theory of non-lattice integration developed by Leinert [14] and König [12],
one can construct an outer integral operator, denoted by $\overline{\sigma}$,
which corresponds to the superhedging price when trading takes place by means
of some idealized class of linear combinations of buy-and-hold strategies, see
[9]. The idealization facilitates to establish an analogue of Daniell’s
continuity from below condition for this outer superhedging integral operator,
which is a standing assumption for the main results of this paper. Considering
an investor, who enters the market at any later time, we can naturally
introduce a conditional version of this superhedging outer integral operator,
denoted by $\overline{\sigma}_{j}$, which gives rise to the notion of a
superhedging supermartingale via the relation
$\overline{\sigma}_{j}f_{j+1}\leq f_{j}$. More precisely, the latter relation
is only required to hold outside a null set – and it is an important subtlety
of Leinert’s integration theory, that the null sets are determined by a
countably sub-additive operator, $\overline{I}$ say, which is closely related
but, in general, different from the outer integral $\overline{\sigma}$.

Proofs of classical results (i.e. in an stochastic setting) that involve
supermartingales rely, at one point or another, on properties of conditional
expectation operators as well as on some measure theory. In our non-
probabilistic setting, also referred to as trajectorial setting, it turns out
that the space of integrable functions (restricted on which the superhedging
outer integral operator acts linearly) is inconveniently small, see [2] for
details. This fact obstructs the naive strategy of emulating classical proofs
by replacing the expectation operator with the non-classical superhedging
integral, but suggests to work with the superhedging outer integral
instead.This is a sublinear operator with unrestricted domain but its
definition allows to bypass the need for the linearity of the expectation as
well as the non availability of some classical limit theorems.

In this paper, we prove analogues of several classical results (see for
example [15] and [11]) for supermartingales in the trajectorial setting. In
particular, we derive a representation theorem for superhedging
supermartingales. Our representation is of a similar type as the uniform Doob
decomposition in discrete time (Theorem 7.5 in [11]) or the optional
decomposition in continuous time, see [8, 13] in the classical setting or [17,
16] for non-dominated versions. As illustrated by an example, our Doob
decomposition can also be applied to trajectory sets which do not have any
martingale measure and, thus, cannot be recovered by classical robust
supermartingale decompositions.

Combining the supermartingale representation theorem with a convergence result
for martingale transforms in the trajectorial setting (derived in [9]), we
can, moreover, prove an analogue of Doob’s a.e. pointwise convergence theorem
for non-negative supermartingales.

As another application of the supermartingale representation theorem, we
clarify the role of the two superhedging operators $\overline{\sigma}$ and
$\overline{I}$. Theorem 6.1 shows that the superhedging outer integral
$\overline{\sigma}$ indeed provides the ‘correct’ superhedging price in the
sense that for payoffs of finite maturity it coincides with the minimal
superhedging cost within the class of linear combinations of buy-and-hold
strategies up to the null sets induced by the $\overline{I}$ operator.

Our work in the Leinert-König setting provides an independent meaning, purely
financially motivated, to the results listed above. An inspection of our proof
techniques shows also the need to rely on new and independent proof arguments.

###  1.1. Relation to the Literature

The paper could be loosely considered as being part of the literature on
robust financial mathematics that weakens a-priori probabilistic modeling
hypothesis, or dispenses with them altogether. This literature ranges from
discrete-time model-free superhedging dualities (e.g., [3, 5, 4]) to extending
stochastic calculus beyond its original settings (e.g., [23, 18, 1]).

Our setting is, however, more closely related to the game theoretic approach
to probability initiated by Shafer, Vovk and coauthors (see e.g. [19, 20] and
the references therein). On a technical level, a key difference between the
Shafer/Vovk approach and our setting is that their conditional global upper
expectation operator $\mathbb{E}_{s}$ satisfies the axiomatic properties of an
outer expectation in every situation (Proposition 8.3 in [20]), while our
conditional outer integral operator $\overline{\sigma}_{j}$ may assign value
$-\infty$ to any bounded function on a null set, on which the conditional
version of the continuity from below property fails. In view of Theorem 7.1
below such a failure of the conditional continuity from below property can
have two origins: a) The trajectory set may run in a situation (or, node, as
we call it), in which the stock price will move upwards for sure (or will move
downwards for sure) leading to an obvious arbitrage opportunity; b) A sure
arbitrage situation arises at some node by trading up to an unbounded
investment horizon, when the trajectory set turns out to be trajectorially
incomplete (in the sense of Section 7 below). To the best of our knowledge,
the aforementioned types of arbitrage situations are not presently
accommodated into the abstract game-theoretic setting (as presented in Chapter
7 of [20]), but they are detected as null sets by our $\bar{I}$-operator (as
should be), see also Remark 2.17 below. Dealing with these additional null
sets does not only lead to significant technical difficulties, but, in view of
Theorem 6.4, we are required to work with the two different families of
conditional superhedging operators $\overline{\sigma}_{j}$ and
$\overline{I}_{j}$ (as opposed to the single family of global conditional
upper expectations $\mathbb{E}_{s}$ in the game-theoretic approach). In this
sense, our paper builds a bridge between the game-theoretic approach of Shafer
and Vovk and the theory of non-lattice integration developed by Leinert and
König.

###  1.2. Structure of the Paper

The paper is organized as follows; Section 2 provides the definitions of the
basic superhedging operators, the crucial continuity from below property of
the superhedging outer integral (see Definition 2.12) and the notion of
conditionally integrable functions (see Definition 2.14). Section 3 defines
supermartingales and stopping times and provides some examples. Section 4
proves our supermartingale representation theorem. Doob’s pointwise
convergence result for non-negative supermartingales is derived in Section 5.
The relation of the two families of superhedging operators
$\overline{\sigma}_{j}$ and $\overline{I}_{j}$ is discussed in Section 6. In
particular, we show that these operators actually differ, if the conditional
continuity from below property is only asked to hold almost everywhere, see
Theorem 6.4 for the precise statement. Section 7 is dedicated to establish the
validity of the conditional continuity of below property at almost every node
(see Theorem 7.1) – a key hypothesis in several of our results. Section 7 also
provides sufficient condition for a related hypothesis used in Section 5.
Additional examples are also introduced to illustrate our results. Technical
developments and some proofs are collected in Appendices A and B. Appendix C
contains the proof of Theorem 7.1 and provides a more general result, namely
Theorem C.2, establishing continuity from below at a given node.

##  2\. Basic setting and Fundamental operators

###  2.1. Trajectorial Setting

######  Definition 2.1 (Trajectory set).

[9, Definition 1] Given a real number $s_{0}$, a _trajectory set_ , denoted by
$\mathcal{S}=\mathcal{S}(s_{0})$, is a subset of

$\mathcal{S}_{\infty}(s_{0})=\\{S=(S_{i})_{i\geq
0}:S_{i}\in\mathbb{R},~{}S_{0}=s_{0}\\}.$

We make fundamental use of the following _conditional spaces_ ; for
$S\in\mathcal{S}$ and $j\geq 0$ set:

$\mathcal{S}_{(S,j)}\equiv\\{\tilde{S}\in\mathcal{S}:\tilde{S}_{i}=S_{i},~{}~{}0\leq
i\leq j\\},$

the notation $(S,j)$, henceforth referred as a node, will be used as a
shorthand for $\mathcal{S}_{(S,j)}$.

Notice $\mathcal{S}_{(S,0)}=\mathcal{S}$ and, if
$\tilde{S}\in\mathcal{S}_{(S,j)}$, then
$\mathcal{S}_{(\tilde{S},j)}=\mathcal{S}_{(S,j)}$. Moreover for $j\leq k$ it
follows that $\mathcal{S}_{(S,k)}\subset\mathcal{S}_{(S,j)}$. On the other
hand, for any fixed pair $j<k$, $\mathcal{S}_{(S,j)}$ is the disjoint union of
$\mathcal{S}_{(\tilde{S},k)}$ with $\tilde{S}\in\mathcal{S}_{(S,j)}$, since if
$\hat{S}\notin\mathcal{S}_{(\tilde{S},k)}$ then $\mathcal{S}_{(\tilde{S},k)}$
and $\mathcal{S}_{(\hat{S},k)}$ are disjoint.

Local properties are relative to a given node. The classification of distinct
nodes is presented in the following definition:

######  Definition 2.2 (Types of nodes).

Given a trajectory space $\mathcal{S}$ and a node $(S,j)$:

  * •

$(S,j)$ is called an _up-down node_ if

$\sup_{\tilde{S}\in\mathcal{S}_{(S,j)}}~{}~{}(\tilde{S}_{j+1}-S_{j})>0\quad\mbox{and}\quad\inf_{\tilde{S}\in\mathcal{S}_{(S,j)}}~{}~{}(\tilde{S}_{j+1}-S_{j})<0.$
(1)

  * •

$(S,j)$ is called a _flat node_ if

$\sup_{\tilde{S}\in\mathcal{S}_{(S,j)}}~{}~{}(\tilde{S}_{j+1}-S_{j})=0=\inf_{\tilde{S}\in\mathcal{S}_{(S,j)}}~{}~{}(\tilde{S}_{j+1}-S_{j}).$
(2)

$(S,j)$ is called an arbitrage-free node if (1) or (2) hold, otherwise it is
called an _arbitrage node_. An arbitrage node $(S,j)$ is said to be of _type
I_ , if there exists $\hat{S}\in\mathcal{S}_{(S,j)}$ such that
$\hat{S}_{j+1}=S_{j}$; otherwise it is said to be of _type II_.

In practice, the coordinates $S_{i}$ are multidimensional in order to allow
for multiple sources of uncertainty. For simplicity we restrict to
$S_{i}\in\mathbb{R}$, one can also extend the framework to allow for several
coordinates $S_{i}^{k}$ (see [6] and [10]).

Besides the set $\mathcal{S}$, the other basic component are the _portfolios_
defined as follows.

######  Definition 2.3 (Conditional portfolio set).

For any fixed $S\in\mathcal{S}$ and $j\geq 0$, $\mathcal{H}_{(S,j)}$ will be
the set of all sequences of functions $H=(H_{i})_{i\geq j}$, where
$H_{i}:\mathcal{S}_{(S,j)}\rightarrow\mathbb{R}$ are non-anticipative in the
sense: for all $\tilde{S},\hat{S}\in\mathcal{S}_{(S,j)}$ such that
$\tilde{S}_{k}=\hat{S}_{k}$ for $j\leq k\leq i$, then
$H_{i}(\tilde{S})=H_{i}(\hat{S})$ (i.e.
$H_{i}(\tilde{S})=H_{i}(\tilde{S}_{0},\ldots,\tilde{S}_{i})$).

Again, we introduce the shorthand notation $\mathcal{H}=\mathcal{H}_{(S,0)}$.

Observe that since there are not restrictions to the conditional portfolio
sets (i.e. portfolios acting on $\mathcal{S}_{(S,j)}$), then
$\mathcal{H}_{(\tilde{S},j)}=\mathcal{H}_{(S,j)}$ whenever
$\tilde{S}_{i}=S_{i}$, $0\leq i\leq j$. Alternatively, one can start with the
‘global’ portfolios in $\mathcal{H}$ and define the sets $\mathcal{H}_{(S,j)}$
by restrictions to $\mathcal{S}_{(S,j)}$. Therefore, $H\in\mathcal{H}_{(S,j)}$
may be referred to as a _conditional portfolio_.

######  Remark 2.4.

Note that we do not impose any measurability condition on the functions
$H_{i}:\mathbb{R}^{i+1}\rightarrow\mathbb{R}$ in the representation
$H_{i}(\tilde{S})=H_{i}(\tilde{S}_{0},\ldots\tilde{S}_{i})$ of a portfolio
position. The main reason is that we will work with a subadditive outer
integral operator instead of a linear integral operator. On the one hand, this
can be viewed in analogy to the use of the outer expectation operator in
probability and statistics (see, e.g., [22]), which does not require any
measurability properties of the integrands. On the other hand, this is in line
with the protocols used in the discrete time game-theoretic approach of Shafer
and Vovk [19], where no measurability conditions are imposed on the functions
announced by the skeptic.

For a node $(S,j)$, $H\in\mathcal{H}_{(S,j)}$, $V\in\mathbb{R}$ and $n\geq j$
we define $\Pi_{j,n}^{V,H}:\mathcal{S}_{(S,j)}\rightarrow\mathbb{R}$, as:

$\Pi_{j,n}^{V,H}(\tilde{S})\equiv
V+\sum_{i=j}^{n-1}H_{i}(\tilde{S})~{}\Delta_{i}\tilde{S},\quad\mbox{where}\;\>\Delta_{i}\tilde{S}=\tilde{S}_{i+1}-\tilde{S}_{i},~{}~{}i\geq
j,~{}~{}\tilde{S}\in\mathcal{S}_{(S,j)}.$

Notice that $V$ is a constant on $\mathcal{S}_{(S,j)}$ and so its value could
change with $S$ i.e. $V=V(S)$. In the sequel, being $\mathcal{A}$ a set of
real valued functions, $\mathcal{A}^{+}$ will denote the set of its non-
negative elements.

######  Definition 2.5 (Elementary vector spaces).

For a fixed node $(S,j)$ set

$\mathcal{E}_{(S,j)}=\\{f=\Pi_{j,n_{f}}^{V,H}:H\in\mathcal{H}_{(S,j)},~{}~{}V\in\mathbb{R}~{}~{}~{}\mbox{and}~{}~{}~{}n_{f}\in\mathbb{N}\\}.$

Observe that $\mathcal{E}_{(S,j)}$ is a real vector space. Its elements are
called _elementary functions_. Let also define

$\mathcal{E}_{j}=\\{f:\mathcal{S}\rightarrow\mathbb{R}:f|_{\mathcal{S}_{(S,j)}}\in\mathcal{E}_{(S,j)}\;\;\forall
S\in\mathcal{S}\\}.$

###  2.2. Fundamental Operators and Almost Everywhere Notions

Let $Q$ denote the set of all functions from $\mathcal{S}$ to
$[-\infty,\infty]$ and $P\subseteq Q$ denotes the set of non-negative
functions. The following conventions are in effect: $0~{}\infty=0$,
$\infty+(-\infty)=\infty$, $u-v\equiv
u+(-v)~{}\forall~{}u,v\in[-\infty,\infty]$, and $\inf\emptyset=\infty$ (unless
indicated otherwise).

We say that $f\in Q$ has _maturity_ $n_{f}\in\mathbb{N}$, if
$f(S)=f(\tilde{S})$ for every $S\in\mathcal{S}$ and
$\tilde{S}\in\mathcal{S}_{(S,n_{f})}$, i.e. if $f$ depends on $S$ only through
the first $n_{f}+1$ coordinates $S_{0},\ldots,S_{n_{f}}$. In this case, we
sometimes write $f(S_{0},\ldots,S_{n_{f}})$ in place of $f(S)$. If $f$ has
maturity $n_{f}$ for some $n_{f}\in\mathbb{N}$, we will speak of a function
$f$ with finite maturity.

We define next the operator
$\overline{I}_{j}:P\rightarrow\mathcal{E}^{+}_{j}$, which is a conditional
extension of the operator $\overline{I}$ defined in [9] and it is used to
define null sets.

######  Definition 2.6.

For a given node $(S,j)$ and a general $f\in P$ define

$\overline{I}_{j}f(S)\equiv\inf\left\\{\sum_{m\geq
1}V^{m}:~{}~{}f\leq\sum_{m\geq
1}\liminf_{n\to\infty}~{}~{}\Pi_{j,n}^{V^{m},H^{m}}\;\;\;\mbox{on}\;\;\mathcal{S}_{(S,j)},\;\Pi_{j,n}^{V^{m},H^{m}}\in\mathcal{E}_{(S,j)}^{+}~{}~{}\forall~{}~{}n\geq
j\right\\}.$

We will use the notation $\overline{I}f\equiv\overline{I}_{0}f$. We also set,
for a general $f\in Q$:

$||f||_{j}(S)\equiv\overline{I}_{j}|f|(S)~{}~{}\mbox{and}~{}~{}||f||\equiv||f||_{0}(S).$

Notice that $\overline{I}_{j}f(S)=\overline{I}_{j}f(S_{0},\ldots,S_{j})$, i.e.
$\overline{I}_{j}f(\cdot)$ is constant on $\mathcal{S}_{(S,j)}$. Moreover
$\sum_{m\geq 1}V^{m}\geq 0$ hence, $\overline{I}_{j}f\geq 0$, so
$||0||_{j}=0$. $||\cdot||_{j}(S)$ will be called a conditional norm.

Next we introduce the notions of _conditional null set_ and the _conditional
$a.e.$ property_.

######  Definition 2.7 (Conditional a.e. notions).

Given a node $(S,j)$, a function $g\in Q$ is a _conditionally null function
at_ $(S,j)$ if:

$\|g\|_{j}(S)=0.$

A subset $E\subset\mathcal{S}$ is a _conditionally null set at $(S,j)$_ if
$\|\mathbf{1}_{E}\|_{j}(S)=0$. A property is said to hold conditionally a.e.
at $(S,j)$ (or equivalently: the property holds “a.e. on
$\mathcal{S}_{(S,j)}$”) if the subset of $\mathcal{S}_{(S,j)}$ where it does
not hold is a conditionally null set at $(S,j)$. In particular, the latter
definition applies to $g=f$ a.e. on $\mathcal{S}_{(S,j)}$, which also will be
noted with $g\doteq f$ when $j=0$.

Notice that when $j=0$, the previous notions do not depend on $S$ and we apply
the abbreviation “a.e.” for “a.e. at $(S,0)$”. Moreover, $E\subset\mathcal{S}$
is called a _null set_ and $g$ is called a _null function_ , if
$\|\mathbf{1}_{E}\|=0$ and $\|g\|=0$, respectively.

The next results, from [9], gives properties of null functions and null sets
that are widely used.

######  Proposition 2.8.

[9, Proposition 1] $\overline{I}$ is isotone, positive homogeneous, countable
subadditive and $\overline{I}(1_{\mathcal{S}})\leq 1$.

######  Proposition 2.9.

[9, Proposition 2] Consider $f,g:\mathcal{S}\rightarrow[-\infty,\infty]$, then

  1. (1)

$\|g\|=0$ iff $g=0$ a.e.

  2. (2)

The countable union of null sets is a null set.

All appearing equalities and inequalities are valid for all points in the
spaces where the functions are defined unless qualified by an explicit a.e.
The notation $f\dot{=}g$ is also used for the equality being valid only a.e.

We introduce next the operator
$\overline{\sigma}_{j}:Q\rightarrow\mathcal{E}_{j}$, which we will call a
_conditional superhedging outer integral_ (or conditional outer integral); it
is the main tool to define the notion of _trajectorial supermartingales_ , the
main object of study in our paper.

######  Definition 2.10 (Conditional Outer Integral).

For a node $(S,j)$ and a general $f\in Q$,

$\overline{\sigma}_{j}f(S)\equiv\inf\left\\{\sum_{m\geq
0}V^{m}:~{}~{}f\leq\sum_{m\geq
0}f_{m}\;\;\mbox{on}\;\mathcal{S}_{(S,j)}\right\\},$

where

$f_{0}=\Pi_{j,n_{0}}^{V^{0},H^{0}}\\!\\!\in\mathcal{E}_{(S,j)};\;\mbox{for}\;m\geq
1,\;f_{m}=\liminf_{n\to\infty}~{}~{}\Pi_{j,n}^{V^{m},H^{m}},~{}~{}\mbox{and}\;\;\Pi_{j,n}^{V^{m},H^{m}}\\!\\!\in\mathcal{E}_{(S,j)}^{+}~{}~{}\forall~{}~{}~{}n\geq
j.$

Define also $\underline{\sigma}_{j}f(S)\equiv-\overline{\sigma}_{j}(-f)(S)$.
We will also set $\overline{\sigma}f\equiv\overline{\sigma}_{0}f$.

In some cases we may use the notation $\overline{\sigma}_{(S,j)}f$ to make
clear that the quantity $\overline{\sigma}_{(S,j)}f$ keeps $(S,j)$ fixed. More
common and useful is our reliance on the defining notation
$\overline{\sigma}_{j}f(S)$ treating $\overline{\sigma}_{j}f$ as a function on
$\mathcal{S}$. Note that the initial endowments $V^{m}$ at node $(S,j)$ may
depend on $S$ through $(S_{0},\ldots,S_{j})$ in all appearances.

######  Remark 2.11.

Note that
$\overline{\sigma}_{j}f(S)=\overline{\sigma}_{j}f(S_{0},\ldots,S_{j})$. Also,
$f_{0}$ can also be written in a similar form as the $f_{m}$, $m\geq 1$ by
mean of $f_{0}=\liminf_{n\to\infty}~{}~{}\Pi_{j,n}^{V^{0},H^{0}}$ with
$H_{i}^{0}\equiv 0\,$ for $\,i\geq n_{0}$.

Here we indicate the intuitions behind the definition of the operator
$\overline{\sigma}_{j}f$ (with analogous explanations for
$\overline{I}_{j}f$). The main simple portfolio superhedging $f$ is given by
$f_{0}$, the role of the idealization of an infinite number of nonnegative
portfolios $\sum_{m\geq 1}f_{m}$ is used to detect general type II arbitrage
nodes as null sets. This is in close analogy to the use of elementary regions
to define the area of non-elementary regions of the plane (for example). In
our case, elementary regions are replaced by simple portfolios, a class closed
under linear combinations (but not being a vector lattice) and playing the
role of the simple functions in Lebesgue’s theory of integration. The quantity
$\overline{\sigma}_{j}f(S)$ is then interpreted as a conditional outer
integral (we are conditioning on $(S,j)$ and hence restricting the future to
$\mathcal{S}_{(S,j)}$), a functional version of
Carath$\acute{\mbox{e}}$odory’s outer measure. Another idealization, which is
used in model free finance ([18], [23]), is the presence of $\liminf$ in the
functions $f_{m}$, $m\geq 1$, which should only play a meaningful role in
situations where an arbitrary large number of portfolio rebalances is
warranted. In our results such idealization is used to detect further null
sets that appear ‘at infinite time’, for example, in Doob’s pointwise
convergence theorem. In particular, financial positions are considered null if
they are cheap to superhedge by means of idealized portfolios allowing an
unbounded number of transactions. We note that the mentioned idealizations are
in line with the modern theory of integration where countable number of
operations are allowed. As an alternative to this approach one could study the
trajectorial analogue to the modern theory of integration such as the finitely
additive theory developed in [7].

###  2.3. Property $(L)$

The property $(L_{(S,j)})$, introduced in the following definition,
generalizes a non-conditional version from [14] and will be called
(conditional) _continuity from below_. Property $(L_{(S,j)})$ will be key to
obtain several of the results in the paper, in particular it will imply that
$\overline{\sigma}_{j}$ preserves the property of non-negativity.

######  Definition 2.12 (Property $(L_{(S,j)})$).

For a fixed node $(S,j)$,
$~{}~{}~{}~{}f=\Pi^{V,H}_{j,n_{f}}\in\mathcal{E}_{(S,j)}$ and
$f_{m}=\liminf_{n\rightarrow\infty}\Pi^{V^{m},H^{m}}_{j,n}$ with
$\Pi^{V^{m},H^{m}}_{j,n}\in\mathcal{E}_{(S,j)}^{+}~{}\mbox{for all}~{}n\geq
j~{}\mbox{and}~{}m\geq 1$, define property $(L_{(S,j)})$ by

$(L_{(S,j)}):\quad f\leq\sum_{m\geq
1}f_{m}\;\;\mbox{on}\;\mathcal{S}_{(S,j)}\implies V\leq\sum_{m\geq 1}V^{m}.$

If $(L_{(S,j)})$ holds for $S\in\mathcal{S}\;\;\overline{I}-a.e.$, it will be
written $(L_{j})$ holds $a.e.$ (see also Definition 2.15).

######  Remark 2.13.

Property $(L_{(S,j)})$ is equivalent to

$0\leq\sum_{m\geq 0}f_{m}\;\;\mbox{on}\;\mathcal{S}_{(S,j)}\implies
0\leq\sum_{m\geq 0}V^{m}.$

where the $f_{m},~{}m\geq 1$, are as in Definition 2.12 and
$f_{0}\in\mathcal{E}_{(S,j)}$.

Since $(L_{(S,0)})$ does not depend on $S$, it will be denoted by $(L)$. If
$(L_{(S,j)})$ holds, it follows that $\overline{\sigma}_{j}f(S)=V$ for
$f=\Pi^{V,H}_{j,n}\in\mathcal{E}_{(S,j)}$ (see Proposition B.3). It will be
proven in Proposition B.3 of Appendix B that $(L_{(S,j)})$ is equivalent to
several useful alternative statements. In particular, property $(L_{(S,j)})$
is equivalent to $0\leq\overline{\sigma}_{j}0(S)$ (and so to
$0=\overline{\sigma}_{j}0(S)$) which in turn implies
$\underline{\sigma}_{j}f(S)\leq\overline{\sigma}_{j}f(S)$ for any
$f:\mathcal{S}\rightarrow[-\infty,\infty]$ (i.e. $f\in Q$). $(L_{(S,j)})$ will
be established under appropriate conditions in Section 7 and Appendix C.

######  Definition 2.14 (Conditional Integrable Functions).

$f\in Q$ is called a _conditional integrable_ function at $j$ if it satisfies:

$\underline{\sigma}_{j}f=\overline{\sigma}_{j}f,~{}~{}\overline{I}-a.e.$

In particular, $f\in Q$ is integrable if
$\underline{\sigma}_{0}f=\overline{\sigma}_{0}f$.

Integrable functions will play a side role in our work as we will make clear
at due time.

Note that the conditional outer integral $\overline{\sigma}_{j}f(S)$ and the
conditional inner integral $\underline{\sigma}_{j}f(S)$ are defined at any
node $(S,j)$ and for any function $f\in Q$. However, it is possible to see
that, $\overline{\sigma}_{j}f(S)=-\infty$ and so
$\underline{\sigma}_{j}f(S)=+\infty$ for any bounded function $f\in Q$, if
$(L_{(S,j)})$ fails at node $(S,j)$. Therefore, the assumption in Definition
2.15 below will be explicitly required for several results.

######  Definition 2.15 (Assumption $(L)-a.e.$).

The following two properties will be referred as the assumption $(L)$-a.e.:

  * i)

$(L)$ (i.e. $(L_{(S,0)}$)) holds,

  * ii)

$\mathcal{N}^{(L)}\equiv\\{S\in\mathcal{S}:\exists j\geq
0\;s.t.\;(L_{(S,j)})\;\mbox{fails}\\}$ (3)

is a null set (in particular, $(L_{j})$ holds a.e. for every $j$).

Sufficient conditions for this assumption and examples will be presented in
Section 7 and Appendix C.

######  Remark 2.16.

a) If (L) fails, then the set $\mathcal{N}^{(L)}$ in (3) equals the whole
trajectory set $\mathcal{S}$. Therefore, in order to avoid trivialities, we
include property (L) in the definition of the assumption $(L)$-a.e.
b) The continuity property $(L)$ implies that $\overline{I}({\bf
1}_{\mathcal{S}})=1$ and so $1=\overline{I}({\bf
1}_{\mathcal{S}})\leq\overline{I}({\bf
1}_{\mathcal{S}\setminus\mathcal{M}})+\overline{I}({\bf
1}_{\mathcal{M}})\leq\overline{I}({\bf
1}_{\mathcal{S}\setminus\mathcal{M}})\leq 1$ where $\mathcal{M}$ is any
arbitrary null set. In particular, $\overline{I}({\bf
1}_{\mathcal{S}\setminus\mathcal{N}^{(L)}})=1$ whenever $(L)$-a.e. holds.

######  Remark 2.17.

In this remark we relate our setting, at a conceptual level, to the abstract
discrete-time game-theoretic framework which is detailed in Chapter 7 of [19].
In their setting, our nodes $(S,j)$ correspond to _situations_ $s$ and our
(conditional) outer integrals $\overline{\sigma}_{(S,j)}$ can be considered as
analogues of their _upper expected values_ $\overline{\mathbb{E}}_{s}$ in
situation $s$, although there are some (subtle, but important) differences in
the exact definition of both families of operators. Lemma 7.6 in [19] (applied
to a constant function $f\equiv c$) implies that
$\overline{\mathbb{E}}_{s}[c]=c$ for every constant $c$ and every situation
$s$. The analogue of such a condition in our setting is to ask that
$(L_{(S,j)})$ holds for every node $(S,j)$. Our assumption $(L)$-a.e. is
considerably weaker and allows, in view of Theorem 7.1, to deal with arbitrage
opportunities which arise at arbitrage nodes of type II or by trading on an
unbounded time horizon (the latter ones in the case that the model is
trajectorially incomplete, see Section 7).

##  3\. Supermartingales: Definition and Examples

Next is our definition of _trajectorial supermartingales, submartingales, and
martingales_ and of _stopping times_.

######  Definition 3.1.

Consider a sequence $(f_{j})_{j\geq 0}$ of non-anticipative functions
$f_{j}:\mathcal{S}\rightarrow[-\infty,\infty]$, $j\geq 0$. We say,

$(f_{j})$ is a _supermartingale_ if

$\overline{\sigma}_{j}f_{j+1}\leq f_{j}~{}~{}a.e.\;\;0\leq j<\infty,$ (4)

$(f_{j})$ is a _submartingale_ if

$f_{j}\leq\underline{\sigma}_{j}f_{j+1}~{}~{}a.e.\;\;0\leq j<\infty,$ (5)

$(f_{j})$ is a _martingale_ if

$\underline{\sigma}_{j}f_{j+1}=\overline{\sigma}_{j}f_{j+1}=f_{j}~{}~{}a.e.\;\;0\leq
j<\infty.$

######  Remark 3.2.

Notice that if $(f_{j})$ is a martingale, then according to Definition 2.14,
$f_{j+1}$ is conditionally integrable at $j$ for any $j\geq 0$. Moreover, if
$(f_{j})$ is a sub- and a supermartingale, then, under Assumption $(L)$-a.e.,
$(f_{j})$ is a martingale, since by Proposition B.3 in Appendix B
$\underline{\sigma}_{j}f\leq\overline{\sigma}_{j}f$ holds $a.e$ for any $f\in
Q$.

######  Definition 3.3 (Stopping Time, as per Definition 8 in [9]).

Given a trajectory space $\mathcal{S}$, a _trajectory based stopping time_ (or
_stopping time_ for short) is a function
$\tau:\mathcal{S}\rightarrow\mathbb{N}\cup\\{\infty\\}$ such that:

$\text{for any}\quad S,S^{\prime}\in\mathcal{S}\quad\text{if}\quad
S_{k}=S^{\prime}_{k}\quad\text{for}\quad 0\leq
k\leq\tau(S),\quad\text{then}\quad\tau(S)=\tau(S^{\prime}).$

We next provide examples for the above definitions.

######  Example 1.

a) Suppose that $(L)$-a.e. holds. If $V\in\mathbb{R}$ and $(H_{j})_{j\geq 0}$
is a non-anticipative sequence, then

$M_{j}(S)\equiv V+\sum_{i=0}^{j-1}H_{i}(S)\Delta_{i}S,\quad
S\in\mathcal{S},\;j\geq 0,$

satisfies the martingale property $\overline{\sigma}_{j}M_{j+1}(S)=M_{j}(S)$
whenever $(L_{(S,j)})$ holds (this is so by Proposition B.3 item $(4)$). In
particular, the projection maps $(T_{j})_{j\geq 0}$, $T_{j}(S)\equiv S_{j}$
are a martingale sequence (with $H_{i}\equiv 1$ and $V=S_{0}$).

b) For any $f\in Q$, the sequence $(f_{j})_{j\geq 0}$ defined by

$f_{j}(S)\equiv\overline{\sigma}_{j}f(S),\quad S\in\mathcal{S},\;j\geq 0,$

forms a supermartingale and the sequence $(f_{j})$ defined by

$f_{j}(S)\equiv\underline{\sigma}_{j}f(S),\quad S\in\mathcal{S},\;j\geq 0,$

forms a submartingale by the tower property in Proposition 3.5 below.

c) If $(f_{j})_{j\geq 0}$ is a supermartingale and $(D_{j})_{j\geq 0}$ is a
non-anticipative sequence of nonnegative functions, then the _supermartingale
transform_ $(g_{j})_{j\geq 0}$

$g_{j}(S)\equiv f_{0}+\sum_{i=0}^{j-1}D_{i}(f_{i+1}-f_{i}),\quad
S\in\mathcal{S},\;j\geq 0,$

is again a supermartingale. This follows from subadditivity of
$\overline{\sigma}_{j}$ and the remark that:
$\overline{\sigma}_{j}g_{j+1}(S)\leq\overline{\sigma}_{j}g_{j}(S)+\overline{\sigma}_{j}(D_{j}(f_{j+1}-f_{j}))(S)\leq
g_{j}(S)+D_{j}(S)~{}\overline{\sigma}_{j}(f_{j+1}-f_{j})(S)\leq
g_{j}(S)+D_{j}(S)[\overline{\sigma}_{j}f_{j+1}(S)-f_{j}(S)]\leq g_{j}(S)$
(where we relied on Proposition B.1).

If $(f_{j})$ is a submartingale, then we call $(g_{j})$ (defined as above) a
submartingale transform, which by the duality $f_{j}\rightarrow-f_{j}$,
$\overline{\sigma}_{j}\rightarrow\underline{\sigma}_{j}$, is a submartingale.

d) If $(f_{j})_{j\geq 0}$ is a supermartingale and $\tau$ is a stopping time,
then the _stopped sequence_ $(f_{j}^{\tau})_{j\geq 0}$ defined by

$f_{j}^{\tau}(S)\equiv f_{\tau(S)\wedge j}(S),$

is a supermartingale. This is a consequence of the previous item with the
choice

$D_{i}(S)=\begin{cases}1,&\tau(S)>i\\\ 0,&\tau(S)\leq i\end{cases},\quad
S\in\mathcal{S},\;j\geq 0,$

which is non-anticipative by Lemma 3.4 below.

######  Lemma 3.4.

Let $\tau$ a stopping time and $H^{k}=(H^{k}_{i})_{i\geq 0},\;k=1,2$ sequences
of non-anticipative functions. For $S\in\mathcal{S},j\geq 0$ define the
following functions on $\mathcal{S}_{(S,j)}$:

$H^{\tau}_{i}(\tilde{S})=\left\\{\begin{array}[]{lll}H^{1}_{i}(\tilde{S})&\mbox{if}&j\leq
i<\tau(\tilde{S})\\\ H^{2}_{i}(\tilde{S})&\mbox{if}&\tau(\tilde{S})\leq
i.\end{array}\right.$

Then $H^{\tau}=(H^{\tau}_{i})_{i\geq 0}$ is a sequence of non-anticipative
functions.

###### Proof.

Let $\tilde{S}_{k}=S_{k},\;j\leq k\leq i$. If $j\leq\tau(S)\leq
i\Rightarrow\tau(\tilde{S})=\tau(S)\leq
i\;\&\;H^{\tau}_{i}(\tilde{S})=H^{2}_{i}(\tilde{S})=H^{\tau}_{i}(S)$. While,
by symmetry in previous reasoning, $i<\tau(S)\Rightarrow
i<\tau(\tilde{S})\;\&\;H^{\tau}_{i}(\tilde{S})=H^{1}_{i}(S)=H^{1}_{i}(\tilde{S})=H^{\tau}_{i}(\tilde{S})$.
∎

######  Proposition 3.5 (Tower Inequality).

Let $S$ be an arbitrary element of $\mathcal{S}$ and $j\leq k$ non-negative
integers; also let $f\in Q$. Then,

$\overline{\sigma}_{j}(\overline{\sigma}_{k}~{}f)(S)\leq\overline{\sigma}_{j}f(S).$
(6)

###### Proof.

We begin with the following inequality valid on $\mathcal{S}_{(S,j)}$:
$f\leq\Pi_{j,n_{0}}^{V^{0},H^{0}}+\sum_{m\geq
1}\liminf_{n\rightarrow\infty}\Pi_{j,n}^{V^{m},H^{m}}$ where
$\Pi_{j,n_{0}}^{V^{0},H^{0}}\in\mathcal{E}_{(S,j)}$ and
$\Pi_{j,n}^{V^{M},H^{m}}\in\mathcal{E}^{+}_{(S,j)}$ for every $n\geq j$. This
inequality implies that $\overline{\sigma}_{k}f(\tilde{S})\leq\sum_{m\geq
0}\Pi_{j,k}^{V^{m},H^{m}}(\tilde{S})$ holds for all
$\tilde{S}\in\mathcal{S}_{(S,j)}$, which in turn implies
$\overline{\sigma}_{j}(\overline{\sigma}_{k}f)(S)\leq\sum_{m\geq 0}V^{m}(S)$.
Therefore the result follows by the definition of $\overline{\sigma}_{j}$ as
an infimum. ∎

We next present sufficient conditions for the tower inequality to turn into an
equality (and so obtaining an analogous result to the classical tower
property), which in turn implies that $(\overline{\sigma}_{j}f)_{j\geq 0}$ is
a martingale sequence.

######  Corollary 3.6.

Assume $(L)$-a.e. and fix some $f\in Q$. Then,
$(\overline{\sigma}_{j}f)_{j\geq 0}$ is a martingale whenever $a)$ or $b)$
below hold:

  * a)

$f$ is conditionally integrable for every $j\geq 0$;

  * b)

$f$ is integrable and $\overline{I}=\overline{\sigma}$ on positive functions
with finite maturity.

###### Proof.

We start with the following preliminary considerations: Since $(L)$-a.e.
holds, we conclude from Proposition B.3 that
$\underline{\sigma}_{j}g\leq\overline{\sigma}_{j}g$ a.e. for every $g\in Q$
and $j\geq 0$. Applying this twice with $g=f$ and with
$g=\overline{\sigma}_{k}f$, in view of Corollary B.2, the following chain of
inequalities holds a.e.

$\underline{\sigma}_{j}[\underline{\sigma}_{k}f]\leq\underline{\sigma}_{j}[\overline{\sigma}_{k}f]\leq\overline{\sigma}_{j}[\overline{\sigma}_{k}f].$

Applying Theorem 3.5, we obtain,

$\underline{\sigma}_{j}f\leq\underline{\sigma}_{j}[\underline{\sigma}_{k}f]\leq\underline{\sigma}_{j}[\overline{\sigma}_{k}f]\leq\overline{\sigma}_{j}[\overline{\sigma}_{k}f]\leq\overline{\sigma}_{j}f,\quad
a.e.$ (7)

a) The conditional integrability assumption now turns all a.e.-inequalities in
(7) into a.e.-identities. In particular, $(\overline{\sigma}_{j}f)_{j\geq 0}$
is a martingale.

b) Let $j=0$. Then, the integrability assumption turns all inequalities in (7)
into identities. In particular, we obtain
$\overline{\sigma}[\overline{\sigma}_{k}f]=\underline{\sigma}[\underline{\sigma}_{k}f]$,
$\overline{\sigma}[\overline{\sigma}_{k}f]=\underline{\sigma}[\overline{\sigma}_{k}f]$
and also
$\underline{\sigma}[\underline{\sigma}_{k}f]=\overline{\sigma}[\underline{\sigma}_{k}f]$
(with $-f$ in place of $f$). We then have access to Corollary B.5 to compute

$\overline{\sigma}[\overline{\sigma}_{k}f-\underline{\sigma}_{k}f]=\overline{\sigma}[\overline{\sigma}_{k}f]+\overline{\sigma}[-\underline{\sigma}_{k}f]=\overline{\sigma}[\overline{\sigma}_{k}f]-\underline{\sigma}[\underline{\sigma}_{k}f]=0.$

Therefore, given that $\overline{\sigma}_{k}f-\underline{\sigma}_{k}f\geq 0$
a.e. (as per Proposition B.3) we have

$\overline{I}[(\overline{\sigma}_{k}f-\underline{\sigma}_{k}f)_{+}]=\overline{\sigma}[(\overline{\sigma}_{k}f-\underline{\sigma}_{k}f)_{+}]=\overline{\sigma}[\overline{\sigma}_{k}f-\underline{\sigma}_{k}f]=0,$

which, by Proposition 2.9 item (1), implies
$\overline{\sigma}_{k}f-\underline{\sigma}_{k}f\leq 0$ a.e. The two
inequalities together yield $\overline{\sigma}_{k}f=\underline{\sigma}_{k}f$
a.e., hence the conditional integrability of $f$ at $k$. Since $k$ is
arbitrary, b) is reduced to a). ∎

##  4\. Supermartingale decomposition

In this section we prove a supermartingale representation theorem. It can be
considered as an analogue of the uniform Doob decomposition in discrete time
(see, e.g., Theorem 7.5 in [11]) or the optional decomposition theorem in
continuous time [13], which apply to stochastic processes that are
supermartingales simultaneously under a family of probability measures.

######  Theorem 4.1 (Supermartingale decomposition).

Under Assumption $(L)$-a.e., let $(f_{j})_{j\geq 0}$ be a sequence of non-
anticipating real-valued functions. Then, the following assertions are
equivalent:

  * (i)

$(f_{j})_{j\geq 0}$ is a supermartingale.

  * (ii)

For every sequence $(\delta_{j})_{j\geq 0}$ of positive real numbers there are
sequences $(H_{j})_{j\geq 0}$ and $(A_{j})_{j\geq 0}$, of non-anticipating
real-valued functions defined on $\mathcal{S}$, such that $(A_{j})_{j\geq 0}$
is nondecreasing, $A_{0}=0$, and

$f_{i}(S)=f_{0}+\sum_{j=0}^{i-1}H_{j}(S)\Delta_{j}S-A_{i}(S)+\sum_{j=0}^{i-1}\delta_{j},$

for every $S\in\mathcal{S}\setminus N_{f}$ and $i\geq 0$. Here $N_{f}$ is an
$\overline{I}$-null set independent of $(\delta_{j})_{j\geq 0}$.

######  Remark 4.2.

Theorem 4.1 shows that up to the small $\delta$-errors and null sets,
supermartingales can be decomposed into a difference of martingale (of the
special form as in Example 1-(a)) and a non-anticipative, nondecreasing
sequence.

We will illustrate the supermartingale decomposition theorem, its assumptions,
and its applicability beyond the classical probabilistic setting in a series
of examples in Section 7.

The proof of Theorem 4.1 relies on two lemmas, which we call Finite Maturity
Lemma and Aggregation Lemma.

######  Lemma 4.3 (Finite Maturity).

Suppose $f:\mathcal{S}\rightarrow\mathbb{R}$ has maturity $n_{f}$ for some
$n_{f}\in\mathbb{N}$. Let $j\leq n_{f}$ and $S^{\ast}\in\mathcal{S}$ be such
that the property $(L_{(S^{\ast},n_{f})})$ holds.

If
$f_{m}=\liminf_{n\to\infty}~{}\Pi_{j,n}^{V^{m},H^{m}},~{}~{}\Pi_{j,n}^{V^{m},H^{m}}\in\mathcal{E}^{+}_{(S^{\ast},j)}\;m\geq
1,~{}\mbox{and}~{}~{}f_{0}=~{}\Pi_{j,n_{0}}^{V^{0},H^{0}}\in\mathcal{E}_{(S^{\ast},j)}$
satisfy

$f\leq\sum\limits_{m=0}^{\infty}f_{m},\;\;~{}~{}\mbox{on}~{}~{}\mathcal{S}_{(S^{\ast},j)}.~{}~{}$
(8)

Then

$f\leq\sum_{m=0}^{\infty}\Pi_{j,n_{f}}^{V^{m},H^{m}}\;\;\mbox{on}~{}~{}\mathcal{S}_{(S^{\ast},n_{f})}~{}~{}(\mbox{each
side is constant on}~{}~{}\mathcal{S}_{(S^{\ast},n_{f})}).$ (9)

###### Proof.

Define for each $\tilde{S}\in\mathcal{S}_{(S^{\ast},n_{f})}$,

$U^{0}(S^{\ast})\equiv-f(S^{\ast})+\Pi_{j,n_{f}\wedge
n_{0}}^{V^{0},H^{0}}(S^{\ast})=-f(S^{\ast})+\Pi_{j,n_{f}}^{V^{0},H^{0}}(S^{\ast}),~{}~{}$
$g_{0}(\tilde{S})\equiv U^{0}(S^{\ast})+\sum_{i=n_{f}\wedge
n_{0}}^{n_{0}-1}H_{i}^{0}(\tilde{S})\Delta_{i}\tilde{S}=U^{0}(S^{\ast})+\sum_{i=n_{f}}^{n_{0}-1}H_{i}^{0}(\tilde{S})\Delta_{i}\tilde{S},$

where we have used the fact that $H_{i}^{0}=0$ for $i\geq n_{0}$, and for
$m\geq 1$

$U^{m}(S^{\ast})\equiv\Pi_{j,n_{f}}^{V^{m},H^{m}}(S^{\ast}),\qquad
g_{m}(\tilde{S})\equiv
U^{m}(S^{\ast})+\liminf_{n\rightarrow\infty}\sum_{i=n_{f}}^{n-1}H_{i}^{m}(\tilde{S})\Delta_{i}\tilde{S}.$

It follows that

$\Pi_{n_{f},n}^{U^{0}(S^{\ast}),H^{0}}=-f(S^{*})+\Pi_{j,n}^{V^{0},H^{0}}\in\mathcal{E}_{(S^{*},n_{f})}~{}\mbox{for
any}~{}n\geq n_{f},$

and for $m\geq 1$

$\Pi_{n_{f},n}^{U^{m}(S^{\ast}),H^{m}}=\Pi_{j,n}^{V^{m},H^{m}}\in\mathcal{E}^{+}_{(S^{*},n_{f})}~{}\mbox{for
any}~{}n\geq n_{f},$

Notice that (8) implies $0\leq\sum\limits_{m\geq 0}g_{m}$ holds on
$\mathcal{S}_{(S^{\ast},n_{f})}$ and since property $(L_{(S^{*},n_{f})})$
holds,

$0\leq\sum\limits_{m=0}^{\infty}U^{m}(S^{\ast})=-f(S^{*})+\sum\limits_{m=0}^{\infty}\Pi_{j,n_{f}}^{V^{m},H^{m}}(S^{*}),$

from where (9) holds. ∎

The following aggregation lemma is proved in [9] (Lemma 3) under the
assumption that all nodes are up-down nodes. The proof there can easily be
adapted to our more general setting (which allows for any type of node).

######  Lemma 4.4 (Aggregation Lemma).

For $j\geq 0$ fixed let, for $m\geq 1$ $~{}~{}H^{m}=(H_{i}^{m})_{i\geq j}$, be
sequences of non-anticipative functions on $\mathcal{S}$, and $V^{m}$
functions defined on $\mathcal{S}$, depending for each $S$ only on
$S_{0},...,S_{j}$.

Fix a node $(S,j)$ and assume for any $m\geq 1$, $n\geq j$, and
$\tilde{S}\in\mathcal{S}_{(S,j)}$ that:

$\Pi_{j,n}^{V^{m},H^{m}}(\tilde{S})=V^{m}(S)+\sum\limits_{i=j}^{n-1}H_{i}^{m}(\tilde{S})\Delta_{i}\tilde{S}\geq
0,~{}~{}\;\quad\mbox{and}\quad\sum\limits_{m\geq 1}V^{m}(S)<\infty.$

For every $\hat{S}\in\mathcal{S}_{(S,j)}$ and $k\geq j$ the following holds:
If $(\hat{S},k)$ is an up-down node and, for every $j\leq p<k$, $(\hat{S},p)$
is an up-down node or $\hat{S}_{p+1}=\hat{S}_{p}$, then,

$\sum_{m\geq 1}H_{k}^{m}(\hat{S})~{}\mbox{converges in $\mathbb{R}$}.$ (10)

###### Proof of Theorem 4.1.

Let

$\displaystyle\mathcal{N}^{(I)}$ $\displaystyle=$
$\displaystyle\\{S\in\mathcal{S}:\exists j\geq 0\;s.t.\;(S,j)\;\mbox{is a type
I arbitrage node and}\;S_{j+1}\neq S_{j}\\},$
$\displaystyle\mathcal{N}^{(II)}$ $\displaystyle=$
$\displaystyle\\{S\in\mathcal{S}:\exists j\geq 0\;s.t.\;(S,j)\;\mbox{is a type
II arbitrage node}\\},$

and recall that $\mathcal{N}^{(L)}$, defined in (3), is a null set by
assumption. Note that $\mathcal{N}^{(I)}$ and $\mathcal{N}^{(II)}$ are null
sets by Lemma A.3 and that $\mathcal{N}^{(II)}\subset\mathcal{N}^{(L)}$ by
Lemma A.1.
(i) $\Rightarrow$ (ii): By the supermartingale property of $(f_{j})_{j\geq 0}$
we may fix an $\overline{I}$-null set $\mathcal{N}_{f}$ such that
$\overline{\sigma}_{j}f_{j+1}(S)\leq f_{j}(S)$ for every
$S\in\mathcal{S}\setminus\mathcal{N}_{f}$ and $j\geq 0$. Let
$N_{f}\equiv\mathcal{N}^{(I)}\cup\mathcal{N}^{(L)}\cup\mathcal{N}_{f}$. We
first introduce the stopping time

$\tau^{\\#}(S)=\inf\\{k\geq 0:L_{(S,k)}\;\mbox{fails,
or}\;\overline{\sigma}_{k}f_{k+1}(S)>f_{k}(S),\;\mbox{or}$ $[(S,k-1)\;\mbox{is
a type I arbitrage node and}\;S_{k}\neq S_{k-1}]\\},$

with the convention $\inf\emptyset=+\infty$. Note that $\tau^{\\#}(S)=\infty$,
if $S\notin N_{f}$.

Now we fix some $j\geq 0$ and choose a family
$\\{S^{\lambda}\\}_{\lambda\in\Lambda_{j}}$ for some index set $\Lambda_{j}$
so that $\\{\mathcal{S}_{(S^{\lambda},j)}:\lambda\in\Lambda_{j}\\}$ is a
partition of $\mathcal{S}$ (see Definition A.2 in appendix A).

We construct now a function $H_{j}:\mathcal{S}\rightarrow\mathbb{R}$ constant
on the nodes of the partition (and, thus, non-anticipating) in the following
way. Consider an arbitrary but fixed node $(S^{\lambda},j)$ of the partition:

If $\tau^{\\#}(S^{\lambda})\leq j$, then $\mathcal{S}_{(S^{\lambda},j)}\subset
N_{f}$ and we simply let $H_{j}(S)=0$ for any
$S\in\mathcal{S}_{(S^{\lambda},j)}$.

If $\tau^{\\#}(S^{\lambda})\geq j+1$ note that, in particular,
$(L_{(S^{\lambda},j)})$ holds and
$\overline{\sigma}_{j}f_{j+1}(S^{\lambda})\leq
f_{j}(S^{\lambda})\in\mathbb{R}$. Applying the defnition of
$\overline{\sigma}_{j}$, we find $g_{m}$’s such that

$f_{j+1}\leq g_{m}\mbox{ on }\mathcal{S}_{(S^{\lambda},j)},$

where
$g_{m}=\liminf_{n\to\infty}~{}\Pi_{j,n}^{V^{m},H^{m}},~{}~{}\Pi_{j,n}^{V^{m},H^{m}}\in\mathcal{E}^{+}_{(S^{\lambda},j)}\;m\geq
1,~{}\mbox{and}~{}~{}g_{0}=~{}\Pi_{j,n_{0}}^{V^{0},H^{0}}\in\mathcal{E}_{(S^{\lambda},j)}$
and

$\sum_{m=0}^{\infty}V_{m}\leq f_{j}(S^{\lambda})+\delta_{j}.$ (11)

Notice that $H^{m}_{j}$ is constant on $\mathcal{S}_{(S^{\lambda},j)}$, and we
write $h_{m}$ for this constant value. If $(S^{\lambda},j)$ is an up-down
node, then $\sum_{m=0}^{\infty}h_{m}$ converges in $\mathbb{R}$ by the
Aggregation Lemma 4.4. We now define $H_{j}(S)\equiv H_{j}(S^{\lambda})$ for
$S\in\mathcal{S}_{(S^{\lambda},j)}$ in the following way:

  * •

If $(S^{\lambda},j)$ is an up-down node, then
$H_{j}(S^{\lambda})\equiv\sum_{m=0}^{\infty}h_{m}$,

  * •

Otherwise $H_{j}(S^{\lambda})\equiv 0$.

If $S\in(S^{\lambda},j)$ satisfies $(L_{(S,j+1)})$, then, by the Finite
Maturity Lemma 4.3 with $n_{f_{j+1}}=j+1$,

$f_{j+1}(S)\leq\sum_{m=0}^{\infty}V_{m}+h_{m}\Delta_{j}S.$ (12)

Case A: $S\in(S^{\lambda},j)$ satisfies $(L_{(S,j+1)})$ and [$(S^{\lambda},j)$
is an up-down or $S_{j+1}=S_{j}$].

Then, by (11) and (12),

$\begin{array}[]{lll}f_{j}(S)+\delta_{j}&=&f_{j}(S^{\lambda})+\delta_{j}\geq\sum\limits_{m=0}^{\infty}V_{m}\\\
\\\
&\geq&f_{j+1}(S)-\sum\limits_{m=0}^{\infty}h_{m}\Delta_{j}S=f_{j+1}(S)-H_{j}(S)\Delta_{j}S.\end{array}$

Case B: For $S\in\mathcal{S}_{(S^{\lambda},j)}$, $(L_{(S,j+1)})$ fails or
[$(S^{\lambda},j)$ is not an up-down and $S_{j+1}\neq S_{j}$]. Then,
$\tau^{\\#}(S)=j+1$.

Consequently, for every $S\in\mathcal{S}$ such that $\tau^{\\#}(S)\geq j+2$,
and, hence in particular for every $S\notin N_{f}$, we are in Case A and

$f_{j+1}(S)-f_{j}(S)\leq\delta_{j}+H_{j}(S)\Delta_{j}S.$ (13)

We now define, for $S\in\mathcal{S}\setminus N_{f}$

$\alpha_{j}(S)\equiv\delta_{j}+H_{j}(S)\Delta_{j}S-(f_{j+1}(S)-f_{j}(S))\geq
0;$

and set $\alpha_{j}(S)\equiv 0$ for for $S\in N_{f}$. Then for
$S\in\mathcal{S}\setminus N_{f}$

$f_{j+1}(S)-f_{j}(S)=\delta_{j}+H_{j}(S)\Delta_{j}S-\alpha_{j}(S).$

Summing this identity over $j$, yields, for every $i\geq 0$ and
$S\in\mathcal{S}\setminus N_{f}$,

$f_{i}(S)=f_{0}+\sum\limits_{j=0}^{i-1}H_{j}(S)\Delta_{j}S-A_{i}(S)+\sum\limits_{j=0}^{i-1}\delta_{j},$

with $A_{i}(S)\equiv\sum\limits_{j=0}^{i-1}\alpha_{j}(S)$.

$(ii)\Rightarrow(i)$: As $N_{f}$ is an $\bar{I}$-null set, we may deduce from
Proposition B.1–e), that $\bar{I}_{j}{\bf 1}_{N_{f}}=0$ $\bar{I}$-a.e. for
every $j\geq 0$. Hence, there is an $\bar{I}$-null set $\mathcal{N}_{f}$ such
that $\bar{I}_{j}{\bf 1}_{N_{f}}(S)=0$ for every
$S\in\mathcal{S}\setminus\mathcal{N}_{f}$ and $j\geq 0$.

We now fix some $j\geq 0$ and some positive integer $K$ and let
$\delta_{i}=1/K$. Condition (ii) implies

$f_{j+1}(S)\leq f_{j}(S)+1/K+H_{j}(S)(S_{j+1}-S_{j})$

for every $S\in\mathcal{S}\setminus{N}_{f}$. We define

$g_{j+1}(S)=f_{j}(S)+1/K+H_{j}(S)(S_{j+1}-S_{j})$

for every $S\in\mathcal{S}$. Then, $g_{j+1}\in\mathcal{E}_{j}$ and
$\overline{\sigma}_{j}g_{j+1}(S)\leq f_{j}(S)+1/K$ for every
$S\in\mathcal{S}$. Note that $(f_{j+1}-g_{j+1})_{+}$ can only be strictly
positive on the $\bar{I}$-null set $N_{f}$. Consequently,

$\displaystyle\overline{\sigma}_{j}f_{j+1}(S)$ $\displaystyle\leq$
$\displaystyle\overline{\sigma}_{j}g_{j+1}(S)+\bar{I}_{j}((f_{j+1}-g_{j+1})_{+})(S)$
$\displaystyle\leq$ $\displaystyle
f_{j}(S)+1/K+\sum_{m=1}^{\infty}\bar{I}_{j}({\bf
1}_{{N}_{f}})(S)=f_{j}(S)+1/K$

for every $S\in\mathcal{S}\setminus\mathcal{N}_{f}$. Passing to the limit
$K\rightarrow\infty$, yields (i). ∎

##  5\. Doob’s Pointwise Supermartingale Convergence

This section proves our version of Doob’s pointwise convergence theorem for
nonnegative supermartingales.

######  Theorem 5.1 (Supermartingale convergence).

Suppose $(L)$-a.e. holds. Let $(f_{i})_{i\geq 0}$ be a supermartingale with
values in $[0,\infty)$ and impose the following assumption on the trajectory
set:

  1. $(H.1)$

Whenever $(S,j)$ is an up-down node such that $(L_{(S,j)})$ holds and
$(L_{(S,j+1)})$ fails, then there are $S^{1},S^{0}\in\mathcal{S}_{(S,j)}$ such
that $S^{1}_{j+1}\geq S_{j+1}\geq S^{0}_{j+1}$ and $(L_{(S^{\iota},j+1)})$
holds for $\iota=0,1$.

Then, there exist a null set $\mathcal{N}_{div}$ such that
$\lim\limits_{i\to\infty}f_{i}(S)$ exists in $\mathbb{R}$ for every
$S\in\mathcal{S}\setminus\mathcal{N}_{div}$.

The proof’s strategy is to apply the supermartingale decomposition in Theorem
4.1 and to pass to the limit separately for the various terms. For the
martingale part $\sum_{j=0}^{i-1}H_{j}(S)\Delta_{j}S$, we can make use of
Theorem 2 in [9]. However, this result requires that
$\sum_{j=0}^{i-1}H_{j}(S)\Delta_{j}S$ is bounded from below by the same
constant for every $S\in\mathcal{S}$, while Theorem 4.1 in conjunction with
the nonnegativity of $f$ only implies boundedness of
$\sum_{j=0}^{i-1}H_{j}(S)\Delta_{j}S$ from below for a.e. $S\in\mathcal{S}$.
In view of the following lemma, the required boundedness condition can be
guaranteed under the additional assumption $(H.1)$. Useful sufficient
conditions for $(H.1)$ to hold are presented in Section 7, Corollary 7.2.

######  Lemma 5.2.

Under the assumptions of Theorem 5.1, fix a sequence $(\delta_{j})_{j\geq 0}$
of summable positive reals and construct the supermartingale decomposition of
$(f_{j})$ as in the proof of the implication $(i)\Rightarrow(ii)$ in Theorem
4.1. Then, for every $S\in\mathcal{S}$ and $i\geq 0$,

$f_{0}+\sum_{j=0}^{\infty}\delta_{j}+\sum_{j=0}^{i-1}H_{j}(S)\Delta_{j}S\geq
0.$

###### Proof.

We construct $(H_{j})$ as in the proof of the implication $(i)\Rightarrow(ii)$
in Theorem 4.1. With $\tau^{\\#}$ as in the proof of Theorem 4.1, we note
that: If $j\leq\tau^{\\#}(S)-2$ or if [$j=\tau^{\\#}(S)-1$ and $(S,j)$ is an
up-down node and $(L_{(S,j+1)})$ holds], we are in Case A of the proof of
Theorem 4.1 and, thus, by (13),

$f_{j+1}(S)-f_{j}(S)\leq\delta_{j}+H_{j}(S)(S_{j+1}-S_{j})$ (14)

Moreover, if $j\geq\tau^{\\#}(S)$ or [$j=\tau^{\\#}(S)-1$ and $(S,j)$ is not
an up-down node], then

$H_{j}(S)=0.$ (15)

We now distinguish two cases:
(i) $i<\tau^{\\#}(S)$.

Then $j\leq\tau^{\\#}(S)-2$ for any $j<i$, and, consequently, by (14),

$f_{0}+\sum_{j=0}^{\infty}\delta_{j}+\sum_{j=0}^{i-1}H_{j}(S)\Delta_{j}S\geq
f_{0}+\sum_{j=i}^{\infty}\delta_{j}+\sum_{j=0}^{i-1}f_{j+1}(S)-f_{j}(S)\geq
f_{i}(S)\geq 0.$

(ii) $i\geq\tau^{\\#}(S)$.

Then, by (15) and (14),

$\displaystyle
f_{0}+\sum_{j=0}^{\infty}\delta_{j}+\sum_{j=0}^{i-1}H_{j}(S)\Delta_{j}S$
$\displaystyle=$ $\displaystyle
f_{0}+\sum_{j=\tau^{\\#}(S)}^{\infty}\delta_{j}+\sum_{j=0}^{\tau^{\\#}(S)-2}\delta_{j}+H_{j}(S)(S_{j+1}-S_{j})$
$\displaystyle+\left(\delta_{\tau^{\\#}(S)-1}+H_{\tau^{\\#}(S)-1}(S)(S_{\tau^{\\#}(S)}-S_{\tau^{\\#}(S)-1})\right)$
$\displaystyle\geq$ $\displaystyle
f_{\tau^{\\#}(S)-1}(S)+\left(\delta_{\tau^{\\#}(S)-1}+H_{\tau^{\\#}(S)-1}(S)(S_{\tau^{\\#}(S)}-S_{\tau^{\\#}(S)-1})\right).$

If $(S,\tau^{\\#}(S)-1)$ is not an up-down-node, then, by (15),

$f_{0}+\sum_{j=0}^{\infty}\delta_{j}+\sum_{j=0}^{i-1}H_{j}(S)\Delta_{j}S\geq
f_{\tau^{\\#}(S)-1}(S)\geq 0.$

If $(S,\tau^{\\#}(S)-1)$ is an up-down-node and $(L_{(S,\tau^{\\#}(S))})$
holds, then, by (14),

$f_{0}+\sum_{j=0}^{\infty}\delta_{j}+\sum_{j=0}^{i-1}H_{j}(S)\Delta_{j}S\geq
f_{\tau^{\\#}(S)}(S)\geq 0.$

If $(S,\tau^{\\#}(S)-1)$ is an up-down-node and $(L_{(S,\tau^{\\#}(S))})$
fails, then, by Assumption $(H.1)$ with $j=\tau^{\\#}(S)-1$, there are
$S^{1},S^{0}\in(S,\tau^{\\#}(S)-1)$ such that $L_{(S^{\iota},\tau^{\\#}(S))}$
holds for $\iota=0,1$ and $S^{0}_{\tau^{\\#}(S)}\leq S_{\tau^{\\#}(S)}\leq
S^{1}_{\tau^{\\#}(S)}$. If $H_{\tau^{\\#}(S)-1}(S)\geq 0$, then, by invoking
(14) for $S^{1}$,

$\displaystyle
f_{0}+\sum_{j=0}^{\infty}\delta_{j}+\sum_{j=0}^{i-1}H_{j}(S)\Delta_{j}S$
$\displaystyle\geq$ $\displaystyle
f_{\tau^{\\#}(S)-1}(S^{1})+\left(\delta_{\tau^{\\#}(S)-1}+H_{\tau^{\\#}(S)-1}(S^{1})(S^{1}_{\tau^{\\#}(S)}-S^{1}_{\tau^{\\#}(S)-1})\right)\geq
f_{\tau^{\\#}(S)}(S^{1})\geq 0.$

If $H_{\tau^{\\#}(S)-1}(S)<0$, then, the same argument with $S^{0}$ in place
of $S^{1}$ yields

$f_{0}+\sum_{j=0}^{\infty}\delta_{j}+\sum_{j=0}^{i-1}H_{j}(S)\Delta_{j}S\geq
f_{\tau^{\\#}(S)}(S^{0})\geq 0.$

∎

###### Proof of Theorem 5.1.

Fix a sequence $(\delta_{j})_{j\geq 0}$ of positive reals with
$\sum_{j}\delta_{j}<\infty$. By Theorem 4.1 there are sequences
$({H}_{j})_{j\geq 0}$, $(A_{j})_{j\geq 0}$ of non-anticipating real-valued
functions such that $(A_{j})_{j\geq 0}$ is nondecreasing, $A_{0}=0$, and

$f_{i}(S)=f_{0}+\sum\limits_{j=0}^{i-1}{H}_{j}(S)\Delta_{j}S-A_{i}(S)+\sum_{j=0}^{i-1}\delta_{j},$
(16)

for every $S\in\mathcal{S}\setminus{N}_{f}$ and $i\geq 0$, with $N_{f}$ a null
set independent of $(\delta_{j})_{j\geq 0}$. Besides, with ${V}\equiv
f_{0}+\sum\limits_{j=0}^{\infty}\delta_{j}$ it follows by Lemma 5.2 that

$\Pi^{{V},{H}}_{i}(S)=f_{0}+\sum_{j=0}^{\infty}\delta_{j}+\sum_{j=0}^{i-1}{H}_{j}(S)\Delta_{j}S\geq
0,$ (17)

for every $i\geq 0$ and $S\in\mathcal{S}$.

Having in mind that there are no portfolio restrictions on $\mathcal{H}$, from
[9, Theorem 2] it follows that there exists a null set $N_{0}$ such that
$\lim\limits_{i\to\infty}\Pi^{V,{H}}_{i}(S)$ exists and is finite for any
$S\in\mathcal{S}\setminus N_{0}$. Consequently $(\Pi^{{V},{H}}_{i}(S))_{i\geq
0}$ is bounded for those $S$. Let $\mathcal{N}_{div}\equiv N_{0}\cup N_{f}$
and restrict the following arguments to
$S\in\mathcal{S}\setminus\mathcal{N}_{div}$. Therefore, from (16) and the
nonnegativity of $f$, $(A_{i}(S))_{i\geq 0}$ results bounded and since it is
nondecreasing, $\lim_{i\to\infty}A_{i}(S)$ also exists (and so it is finite).
So $\lim\limits_{i\to\infty}f_{i}(S)$ exists in $\mathbb{R}$ from (16), for
any $S\in\mathcal{S}\setminus\mathcal{N}_{div}$. ∎

##  6\. On the Relation between the Two Superhedging Operators

As another consequence of the supermartingale decomposition, we show, in this
section, that $\overline{\sigma}$ is the ‘correct’ superhedging operator in
the sense that, for bounded (from below) functions with finite maturity, it
corresponds to the minimal superhedging price within the usual class of simple
portfolios up to the null sets induced by $\bar{I}$. We also clarify the
relation between the two (conditional) superhedging operators
$\overline{I}_{j}$ and $\overline{\sigma}_{j}$.

######  Theorem 6.1.

Suppose that $(L)$-a.e. holds and that $f\in Q$ has maturity
$n_{f}\in\mathbb{N}$, is bounded from below and satsifies
$\overline{\sigma}f<\infty$. Let $0\leq j<n_{f}$. Then: For every
$\epsilon>0$, there are a null set $N_{f}$ and a non-anticipative sequence
$(H_{i})_{i=j,\ldots,n_{f}-1}$ such that for every $S\in\mathcal{S}\setminus
N_{f}$

$f(S)\leq(\overline{\sigma}_{j}f(S)+\epsilon)+\sum_{i=j}^{n_{f}-1}H_{i}(S)\Delta_{i}S.$

Conversely, if there are a $V\in Q$ with maturity $j$ and a null set
$\tilde{N}_{f}$ such that for every $S\in\mathcal{S}\setminus\tilde{N}_{f}$

$f(S)\leq V(S_{0},\ldots,S_{j})+\sum_{i=j}^{n_{f}-1}H_{i}(S)\Delta_{i}S,$

then $\overline{\sigma}_{j}f\leq V$ a.e.

In particular,

$\displaystyle\overline{\sigma}f$ $\displaystyle=\quad\inf\\{V\in\mathbb{R}|$
$\displaystyle\exists(H_{j})_{j=0,\ldots,n_{f}-1}\textnormal{ non-anticipative
such that }$ $\displaystyle V+\sum_{j=0}^{n_{f}-1}H_{j}(S)\Delta_{j}S\geq
f(S)\textnormal{ for a.e. }S\in\mathcal{S}\\}.$

The proof combines Theorem 4.1 with the following lemma, which deals with the
issue that the supermartingale $(\overline{\sigma}_{i}f)_{i\geq 0}$ may take
values $\pm\infty$.

######  Lemma 6.2.

Suppose $(L)$-a.e. If $f\in Q$ is bounded from below by some $c\in\mathbb{R}$
and satisfies $\overline{\sigma}f<\infty$, then there is a supermartingale
$(f_{i})_{i\geq 0}$ with values in $[c;+\infty)$ such that
$\overline{\sigma}_{j}f=f_{j}$ a.e. for every $j\geq 0$.

###### Proof.

As in the proof of Theorem 4.1, we consider the stopping time

$\tau^{\\#}(S)=\inf\\{k\geq 0:L_{(S,k)}\;\mbox{fails, or}\;[(S,k-1)\;\mbox{is
a type I arbitrage node and}\;S_{k}\neq S_{k-1}]\\}.$

Define

$f_{i}(S)=\overline{\sigma}_{i}f(S){\bf 1}_{\\{i<\tau^{\\#}(S)\\}}+c{\bf
1}_{\\{i\geq\tau^{\\#}(S)\\}},\quad i\geq 0,\;S\in\mathcal{S},$

and note that $(f_{i})_{i\geq 0}$ is non-anticipative by Lemma 3.4. Recall
that $\\{S\in\mathcal{S}|\;\tau(S)<\infty\\}$ is a null set by Lemma A.3 and
by assumption $(L)$-a.e. In view of Example 1–b), we conclude that $(f_{j})$
is a supermartingale and $\overline{\sigma}_{j}f=f_{j}$ a.e. for every $j\geq
0$.

For the lower bound, note that by Proposition B.3,

$f_{i}(S)=\overline{\sigma}_{i}f(S)\geq\overline{\sigma}_{i}(c)(S)=c,$

whenever $i<\tau^{\\#}(S)$. We finally need to verify that $f_{i}(S)<\infty$
for every $i\geq 0$ and $S\in\mathcal{S}$. Since $\overline{\sigma}f<\infty$,
we find $\Pi_{0,n_{0}}^{V^{0},H^{0}}\\!\\!\in\mathcal{E}_{0}$ and, for every
$m\in\mathbb{N}$, $\Pi_{0,\cdot}^{V^{m},H^{m}}$ such that
$\Pi_{0,n}^{V^{m},H^{m}}\\!\\!\in\mathcal{E}^{+}_{0}$ for every $n\geq 0$,
$\sum_{m=1}^{\infty}V^{m}<\infty$ and

$f(S)\leq\sum_{m=0}^{\infty}\liminf_{n\rightarrow\infty}\Pi_{0,n}^{V^{m},H^{m}}(S),\quad
S\in\mathcal{S}.$

We now fix a node $(S,i)$. The previous inequality implies

$f(\tilde{S})\leq\sum_{m=0}^{\infty}\liminf_{n\rightarrow\infty}\Pi_{i,n}^{\tilde{V}(S),\tilde{H}^{m}}(\tilde{S}),\quad\tilde{S}\in\mathcal{S}_{(S,i)}$

where $\tilde{V}(S)=\Pi_{0,i}^{V^{m},H^{m}}(S)$ and
$\tilde{H}^{m}=H^{m}_{|\mathcal{S}_{(S,i)}}$. Hence,
$\overline{\sigma}_{i}f(S)\leq\sum_{m=0}^{\infty}\Pi_{0,i}^{V^{m},H^{m}}(S)$.
We still need to check that the right-hand side is finite, if
$i<\tau^{\\#}(S)$. In this case, we may (thanks to Lemma A.1) apply the
Aggregation Lemma 4.4 to conclude that

$\sum_{m=0}^{\infty}\Pi_{0,i}^{V^{m},H^{m}}(S)=\sum_{m=0}^{\infty}V^{m}+\sum_{j=0}^{i-1}\sum_{m=0}^{\infty}(H^{m}_{j}(S)\Delta_{j}S)$

where each of the series converges in $\mathbb{R}$. ∎

###### Proof of Theorem 6.1.

Fix some $\epsilon>0$ and choose a sequence $(\delta_{i})$ of positive reals
such that $\sum_{i=j}^{n_{f}-1}\delta_{j}\leq\epsilon$. We fix a
supermartingale $(f_{i})_{i\geq 0}$ as in Lemma 6.2 and note that, thanks to
Lemma B.4 and assumption $(L)$-a.e., $f_{i}=\overline{\sigma}_{i}f=f$ a.e. for
$i\geq n_{f}$. Hence, we may apply Theorem 4.1 to $(f_{i})$ in order to
construct a non-anticipative sequence $(H_{i})_{i=0,\ldots,n_{f}-1}$ such that

$f(S)=f_{n_{f}}(S)\leq\left(\overline{\sigma}_{j}f+\epsilon\right)+\sum_{i=j}^{n_{f}-1}H_{i}(S)\Delta_{i}S$

for every $S\in\mathcal{S}\setminus N_{f}$, where $N_{f}$ is a null set.

For the converse inequality, assume that there are $V\in Q$ with maturity $j$,
$H=(H_{i})_{i=j,\ldots,n_{f}-1}$ non-anticipative, and a null set
$\tilde{N}_{f}$ such that for every $S\in\mathcal{S}\setminus\tilde{N}_{f}$

$f(S)\leq V(S_{0},\ldots,S_{j})+\sum_{i=j}^{n_{f}-1}H_{i}(S)\Delta_{i}S.$

If $V(S_{0},\ldots,S_{j})=+\infty$, the inequality
$\overline{\sigma}_{j}f(S)\leq V(S)$ is trivial. Otherwise, we note that
$g=\infty{\bf 1}_{\tilde{N}_{f}}$ is a null function. Hence, by Proposition
B.1–e), there is a null set $N_{f}$ such that $\overline{I}_{j}g(S)=0$ for
every $S\in\mathcal{S}\setminus N_{f}$. Consequently, for every
$S\in\mathcal{S}\setminus N_{f}$ and $\epsilon>0$ there are sequences
$(V_{m})_{m\geq 1}$ of nonnegative reals and $(H^{m})_{m\geq 1}$ in
$\mathcal{H}_{(S,j)}$ such that
$\Pi_{j,n}^{V^{m},H^{m}}\in\mathcal{E}^{+}_{(S,j)}$ for every $n\geq j$,
$\sum_{m\geq 1}V_{m}\leq\epsilon$ and $g\leq\sum_{m\geq
1}\liminf_{n\to\infty}~{}~{}\Pi_{j,n}^{V^{m},H^{m}}$ on $\mathcal{S}_{(S,j)}$.
Let $V^{0}=V(S_{0},\ldots,S_{j})$ and define $H^{0}$ via restricting the
functions in $H$ on the conditional space $\mathcal{S}_{(S,j)}$. Then,

$f\leq\Pi_{j,n_{f}}^{V^{0},H^{0}}+\sum_{m\geq
1}\liminf_{n\to\infty}~{}~{}\Pi_{j,n}^{V^{m},H^{m}}\mbox{ on
}\mathcal{S}_{(S,j)},$

which implies $\overline{\sigma}_{j}f(S)\leq V(S_{0},\ldots,S_{j})+\epsilon$
for every $S\in\mathcal{S}\setminus N_{f}$. By passing with $\epsilon$ to
zero, we obtain $\overline{\sigma}_{j}f\leq V$ a.e. ∎

######  Remark 6.3.

Example 2 in Section 7 provides an example of a nonnegative, bounded function
$f$ with finite maturity on a trajectory set satisfying $(L)$-a.e. such that
$\overline{I}(f)>\overline{\sigma}f$. In light of the previous theorem, we may
conclude that $\bar{I}$ cannot be applied for computing superhedging prices in
general, but only serves as an auxiliary operator to determine the null sets
of the model.

The following theorem shows that inconsistencies between the two families of
superhedging operators stem from the failure of $(L_{(S,j)})$ on a null set.

######  Theorem 6.4.

The following assertions are equivalent:
(i) For every $f\in P$ with finite maturity and every node $(S,j)$,

$\overline{I}_{j}f(S)=\overline{\sigma}_{j}f(S).$

(ii) $(L_{(S,j)})$ holds at every node $(S,j)$.

###### Proof.

(i) $\Rightarrow$ (ii): Since $\overline{I}_{j}(0)(S)=0$ always holds, we
immediately obtain

$\overline{\sigma}_{j}(0)(S)=\overline{I}_{j}(0)(S)=0$

at any node $(S,j)$.
(ii) $\Rightarrow$ (i): _Step 1:_ We first consider the initial node $(S,0)$:

Noting that $\overline{\sigma}f\leq\overline{I}f$ by definition, it suffices
to show that $\overline{I}f\leq\overline{\sigma}f$, for which we may and do
assume that $\overline{\sigma}f<\infty$. We now proceed as in the first part
of the proof of Theorem 6.1, but note that the supermartingale $(f_{i})_{i\geq
0}$ can be chosen $[0,\infty)$-valued and that Lemma 5.2 is applicable,
because $(L_{(S,j)})$ holds at every node $(S,j)$. By Theorem 4.1 and Lemma
5.2, we find a non-anticipative sequence $(H_{j})_{j=0,\ldots,n_{f}-1}$ such
that

$\left(\overline{\sigma}f+\epsilon\right)+\sum_{i=0}^{\min\\{j,n_{f}-1\\}}H_{i}(S)\Delta_{i}S\geq
0$

for every $S\in\mathcal{S}$ and

$f(S)\leq\left(\overline{\sigma}f+\epsilon\right)+\sum_{i=0}^{n_{f}-1}H_{i}(S)\Delta_{i}S$

for every $S\in\mathcal{S}\setminus N_{f}$, where $N_{f}$ is a null set. Let
$V^{0}=\overline{\sigma}f+\epsilon$ and define $H^{0}$ via $H^{0}_{j}=H_{j}$
for $j\leq n_{f}-1$ and $H^{0}_{j}=0$ for $j\geq n_{f}$. Then
$\Pi_{0,j}^{V^{0},H^{0}}\in\mathcal{E}_{0}^{+}$ for every $j\geq 0$. Dealing
with the null set $N_{f}$ as in the second part of the proof of Theorem 6.1,
we conclude that $\overline{I}f\leq\overline{\sigma}f+2\epsilon$. Letting
$\epsilon$ tend to zero, the proof of Step 1 is complete.
_Step 2:_ We now consider a generic node $(\tilde{S},\tilde{j})$:

Define the auxiliary trajectory set

$\widetilde{\mathcal{S}}=\\{(S_{\tilde{j}+i})_{i\geq
0}|S\in\mathcal{S}_{(\tilde{S},\tilde{j})}\\}.$

Then, the $\overline{\sigma}$-operator and the $\overline{I}$-operator for
$\widetilde{\mathcal{S}}$ at time 0 coincide with
$\overline{\sigma}_{\tilde{j}}(\cdot)(\tilde{S})$ and
$\overline{I}_{\tilde{j}}(\cdot)(\tilde{S})$. Moreover, each node $(S,j)$ in
$\widetilde{\mathcal{S}}$ coincides with the node
$((\tilde{S}_{0},\ldots,\tilde{S}_{\tilde{j}-1},S_{0},\ldots),$ $\tilde{j}+j)$
in $\mathcal{S}$. Hence, every node $(S,j)$ in $\widetilde{\mathcal{S}}$
satisfies $(L_{(S,j)})$. These observations reduce the case of a generic node
to the case of an initial node. ∎

##  7\. Property $(L)$-a.e. and Some Examples

In this section, we provide some easy-to-check sufficient conditions for the
Assumption $(L)$-a.e., which is crucial for the main results of this paper,
and for the additional assumption (H.1) required in the Supermartingale
Convergence Theorem 5.1. We will also present some examples.

We first recall the notion of trajectorial completeness from [9]. Given a
sequence $(S^{n})_{n\geq 0}\subseteq\mathcal{S}$ satisfying

$~{}~{}~{}S^{n}_{i}=S^{n+1}_{i}\;\;0\leq i\leq
n,~{}~{}\forall~{}n,~{}~{}~{}~{}~{}~{}~{}~{}$

define

$\overline{S}=(\overline{S}_{i})_{i\geq
0}~{}~{}\mbox{by}~{}~{}~{}\overline{S}_{i}\equiv S^{i}_{i}.~{}~{}~{}\mbox{We
will use the notation}~{}~{}\overline{S}=\lim_{n\rightarrow\infty}S^{n}.$

Notice that

$\overline{S}_{i}=S^{n}_{i},~{}~{}~{}0\leq i\leq n,~{}~{}\forall~{}n\geq 0.$
(18)

Let $\overline{\mathcal{S}}$ be the set of such $\overline{S}$. Then, clearly,
$\mathcal{S}\subseteq\overline{\mathcal{S}}$ given that for $\tilde{S}$ we can
take $\tilde{S}^{n}=\tilde{S}$ for all $n\geq 0$. We say that $\mathcal{S}$ is
trajectorially complete, if $\mathcal{S}=\overline{\mathcal{S}}$.

As argued in Proposition 13 of [9], $\overline{\mathcal{S}}$ is always
trajectorially complete. Moreover, the completion process, i.e. passing from
$\mathcal{S}$ to $\overline{\mathcal{S}}$ does not alter the type of the nodes
(being up-down, no arbitrage, etc.), but it can change a null node (i.e., a
node which constitutes a null set) into a non-null node.

The proof of the following theorem is similar to the arguments leading to
Corollary 4 in [9], where $\mathcal{S}$ is, however, assumed to have up-down
nodes only. We provide the proof in Appendix C, in which we also discuss more
general sufficient conditions for $(L)$-a.e.

######  Theorem 7.1 (Sufficient conditions for $(L)$-a.e.).

Suppose $\mathcal{S}$ is trajectorially complete and the following condition
holds:

  * (H.2)

If $(S,j)$ is an arbitrage node of type II, then $j\geq 1$, $(S,j-1)$ is an
up-down node and for every $\epsilon>0$ there are
$S^{\epsilon,1},S^{\epsilon,2}\in\mathcal{S}_{(S,j-1)}$ such that

$S^{\epsilon,1}_{j}-S_{j-1}\geq-\epsilon,\quad
S^{\epsilon,2}_{j}-S_{j-1}\leq\epsilon$

and such that $(S^{\epsilon,1},j)$, $(S^{\epsilon,2},j)$ are not type II
arbitrage nodes.

Then, $(L)$-a.e. holds. More, precisely, $(L_{(S,j)})$ fails at a node
$(S,j)$, if and only if $(S,j)$ is an arbitrage node of type II.

Whenever the trajectory set $\mathcal{S}$ is trajectorially complete and has
no arbitrage nodes of type II, it follows that Theorem 7.1 implies that
$(L_{(S,j)})$ holds at any node $(S,j)$ – and then condition (H.1), which is
required in the supermartingale convergence theorem, is trivially satisfied.
The assumption of trajectorial completeness and the validity of $(L_{(S,j)})$
for all nodes $(S,j)$ is an implicit assumption in [19] (see also Remark
2.17). We weaken completeness in Theorem C.2 by means of hypothesis (H.5) and
illustrated in Remark 7.4.

The following corollary provides sufficient conditions for $(H.1)$ in the
presence of arbitrage nodes of type II.

######  Corollary 7.2.

Suppose $\mathcal{S}$ is trajectorially complete and the following condition
holds:

  * (H.3)

If $(S,j)$ is an arbitrage node of type II, then $j\geq 1$, $(S,j-1)$ is an
up-down node and there are $S^{1},S^{2}\in\mathcal{S}_{(S,j-1)}$ such that

$S^{1}_{j}>S_{j}>S^{2}_{j}$

and such that $(S^{1},j)$, $(S^{2},j)$ are not type II arbitrage nodes.

Then, $(L)$-a.e. and condition (H.1) in Theorem 5.1 hold.

###### Proof.

We first show that (H.3) implies condition (H.2) in Theorem 7.1. Suppose
$(S,j)$ is a type II arbitrage node. Then, by (H.3), $j\geq 1$ and $(S,j-1)$
is an up-down node. Consequently, there is a $\tilde{S}\in(S,j-1)$ such that
$\tilde{S}_{j}>\tilde{S}_{j-1}=S_{j-1}$. Then, for every $\epsilon>0$

$\tilde{S}_{j}-S_{j-1}>0>-\epsilon;$

and we may take $S^{\epsilon,1}=\tilde{S}$ in (H.2), provided $(\tilde{S},j)$
is not an arbitrage node of type II. Otherwise, we may apply (H.3) with
$\tilde{S}$ in place of $S$ and find some $S^{1}\in(S,j-1)=(\tilde{S},j-1)$
such that $S^{1}_{j}>\tilde{S}_{j}$ and $(S^{1},j)$ is not an arbitrage node
of type II. Then, we may take $S^{\epsilon,1}=S^{1}$. The construction of
$S^{\epsilon,2}$ follows by a ‘symmetric’ argument.

Now that (H.2) is verified, Theorem 7.1 implies that $(L)$-a.e. holds – and
that $L_{(S,j)}$ fails, if and only if $(S,j)$ is an arbitrage node of type
II. The latter implies (H.1). Indeed, if $(S,j)$ is an up-down node and
$L_{(S,j+1)}$ fails, then $(S,j+1)$ is a type II arbitrage node, and, thus, by
(H.3) there are $S^{1},S^{2}\in\mathcal{S}_{(S,j)}$ such that

$S^{1}_{j+1}>S_{j+1}>S^{2}_{j+1}$

and such that $L_{(S^{1},j+1)}$, $L_{(S^{2},j+1)}$ hold, because $(S^{1},j)$,
$(S^{2},j)$ are not type II arbitrage nodes. ∎

The following example illustrates that condition $(L)$-a.e. may hold in the
absence of martingale measures and that that the $\delta$-sequence in the
supermartingale decomposition (Theorem 4.1) cannot be dispensed with. At the
same time, the example shows that the operators $\overline{\sigma}$ and
$\overline{I}$ do, in general, not coincide on the cone of nonnegative
functions.

######  Example 2.

Let

$\mathcal{S}=\mathcal{S}^{+}\cup\mathcal{S}^{-},\quad\mathcal{S}^{\pm}=\\{S^{\pm,n}|n\in\mathbb{N}\\}$

where

$S^{+,n}_{i}=\begin{cases}1,&i=0\\\ 2,&i=1\\\ 2+\frac{1}{n},&i\geq
2\end{cases},\quad\quad S^{-,n}_{i}=\begin{cases}1,&i=0\\\
1-\frac{1}{n^{2}},&i\geq 1\end{cases}.$

Then, the node $(S^{+,1},1)=(S^{+,n},1)$, $n\geq 2$, is an arbitrage node of
type II, the initial node $(S,0)$ is up-down, and all other nodes are flat.
Trajectories are displayed in the following illustration:

a) We first show that $(L)$-a.e. holds, but there is no martingale measure for
this trajectory set.

Since $S^{-,n}_{1}-S^{-,n}_{0}=-1/n^{2}$ is arbitrarily close to zero for
sufficiently large $n$, we observe that (H.2) is satisfied. Trajectorial
completeness is obvious, because all trajectories stay constant after time
$j=2$. Hence, by Theorem 7.1, this trajectory set satisfies $(L)$-a.e. Since
all trajectories $S^{+,n}$ pass through the arbitrage node $(S^{+,1},1)$ of
type II, $\mathcal{S}^{+}$ is a null set by Lemma A.3 and, hence,
$\bar{I}({\bf 1_{\mathcal{S}^{-}}})=1$ by Remark 2.16.

It is also clear that there is no probability measure $Q$ on the power set of
$\mathcal{S}$, which turns the coordinate process
$T_{j}:\mathcal{S}\rightarrow\mathbb{R}$, $S\mapsto S_{j}$ into a martingale.
Otherwise $Q(\mathcal{S}^{+})=0$ (because any martingale measure assigns
probability zero to type II arbitrage nodes) and then $T_{1}<T_{0}$ $Q$-almost
surely – a contradiction.

Consequently, by Example 1–a), the coordinate process $(T_{j})_{j\geq 0}$ is
an example of a trajectorial martingale, which fails to be a martingale in the
classical probabilistic setup.
b) We next construct a supermartingale, for which a decomposition as in
Theorem 4.1–(ii) is not possible, if we let $\delta_{j}\equiv 0$ (and, hence,
the small $\delta$-errors cannot be avoided in the formulation of the
supermartingale decomposition theorem).

To this end, we define $f_{j}:\mathcal{S}\rightarrow\mathbb{R}$ via

$f(S)=\begin{cases}0,&S\in\mathcal{S}^{+}\\\
\frac{1}{n},&S=S^{-,n}\end{cases},\quad\quad f_{j}(S)=\begin{cases}0,&j=0\\\
f(S),&j\geq 1,\end{cases}$

and consider the sequence $(f_{j})_{j\geq 0}$. Since all of its ‘paths’
$j\mapsto f_{j}(S)$ are nondecreasing, $(f_{j})$ obviously is a submartingale.
We claim that $(f_{j})_{j\geq 0}$ is a also supermartingale (despite of the
nondecreasing paths) and apply Theorem 4.1 to verify this. Given a sequence
$(\delta_{j})_{j\geq 0}$ of positive reals, let

$H_{0}\equiv-\lceil{\delta_{0}}^{-1}\rceil,\quad H_{j}\equiv 0,\;j\geq 0,$

and note that, for every $n\in\mathbb{N}$,

$\delta_{0}+H_{0}(S^{-,n}_{1}-S^{-,n}_{0})=\delta_{0}+\lceil{\delta_{0}}^{-1}\rceil\;\frac{1}{n^{2}}\geq\frac{1}{n}=f_{1}(S^{-,n}),$

by considering the cases $n\leq\lceil{\delta_{0}}^{-1}\rceil$ and
$n>\lceil{\delta_{0}}^{-1}\rceil$ separately. Hence, we may define a
nondecreasing, nonanticipating sequence $(A_{j})$ via $A_{0}\equiv 0$ and

$A_{j}(S)-A_{j-1}(S)=\begin{cases}\delta_{0}+\lceil{\delta_{0}}^{-1}\rceil\cdot\frac{1}{n^{2}}-\frac{1}{n},&j=1\textnormal{
and }S=S^{-,n}\\\ \delta_{j-1},&j\geq 2\textnormal{ or
}S\in\mathcal{S}^{+}.\end{cases}$

It is then straightforward to check that, for every $i\geq 0$ and
$S\in\mathcal{S}^{-}$,

$f_{i}(S)=f_{0}+\sum_{j=0}^{i-1}H_{j}(S)\Delta_{j}S-A_{i}(S)+\sum_{j=0}^{i-1}\delta_{j}.$

Hence, $(f_{j})$ is a supermartingale by Theorem 4.1. Note that, in view of
Remark 3.2, $(f_{j})_{j\geq 0}$ is a martingale.

We finally prove that a decomposition as in Theorem 4.1–(ii) is not possible
for this (super-)martingale $(f_{j})$, if we let $\delta_{j}\equiv 0$. We
first show that for every $H_{0}\in\mathbb{R}$ there is an
$n_{0}\in\mathbb{N}$ such that for every $n\geq n_{0}$

$f_{1}(S^{-,n})>f_{0}+H_{0}(S^{-,n}_{1}-S^{-,n}_{0}).$

By inserting the definition of $f_{1}$ and $f_{0}$, this inequality is
equivalent to

$\frac{1}{n}\left(1-\frac{H_{0}}{n}\right)>0,$

which trivially holds for sufficiently large $n$. Hence, no matter of the
choice of $(H_{j})$ and $(A_{j})$, a decomposition as in Theorem 4.1–(ii) with
$\delta_{0}=0$ will necessarily fail at $i=1$ on the set $\mathcal{S}^{-,\geq
n_{0}}=\\{S^{-,n}|n\geq n_{0}\\}$ for some $n_{0}\in\mathbb{N}$. It, thus,
remains to show that $\bar{I}({\bf 1}_{\mathcal{S}^{-,\geq n_{0}}})>0$ for
every $n_{0}\in\mathbb{N}$. To this end, assume that for every
$S\in\mathcal{S}$,

${\bf 1}_{\mathcal{S}^{-,\geq n_{0}}}(S)\leq\sum_{m\geq
1}\liminf_{k\to\infty}~{}~{}\Pi_{0,k}^{V^{m},H^{m}}(S)$

where $\Pi_{0,k}^{V^{m},H^{m}}\in\mathcal{E}_{0}^{+}$ for all $k\geq 0$ and
$\sum_{m\geq 1}V_{m}<\infty$. Since the trajectories in $\mathcal{S}^{-}$ are
constant after time $k=1$, the previous inequality and the Aggregation Lemma
4.4 imply

$1\leq\sum_{m\geq 1}\Pi_{0,1}^{V^{m},H^{m}}(S^{-,n})=\sum_{m\geq
1}(V^{m}-H^{m}_{0}\frac{1}{n^{2}})=\sum_{m\geq
1}V_{m}-\frac{1}{n^{2}}\sum_{m\geq 1}H^{m}_{0}$

for every $n\geq n_{0}$. Passing with $n$ to infinity, we observe that
$\sum_{m\geq 1}V_{m}\geq 1$ and, hence, $\bar{I}({\bf 1}_{\mathcal{S}^{-,\geq
n_{0}}})=1$ for every $n_{0}\in\mathbb{N}$.

The foregoing also shows that $(f_{j})_{j\geq 0}$ is an example of a
martingale, which is not a.e. equal to a ‘simple’ martingale of the form
discussed in Example 1–a).
c) We finally show that for $f$ (defined in b)):
$\overline{\sigma}f=0<1/2=\overline{I}f$.

Since $(f_{j})$ is a martingale, we obtain
$0=f_{0}=\overline{\sigma}f_{1}=\overline{\sigma}f$. For the claim concerning
$\overline{I}$, it clearly suffices to show: If $\overline{I}f<\epsilon$ for
some $\epsilon>0$, then $\epsilon\geq 1/2$. If the hypothesis is satisfied,
then, by the Aggregation Lemma 4.4 at time 0, there is a constant $H_{0}$ such
that $\epsilon+H_{0}(S_{1}-S_{0})\geq 0$ for every $S\in\mathcal{S}$ and
$\epsilon+H_{0}(S_{1}-S_{0})\geq f(S)$, whenever $S=S^{-,n}$ (because the
trajectories in the down-branch stay constant after time 1). If we apply these
inequalities to the trajectories $S^{-,1}$ and $S^{+,1}$, we obtain $\epsilon-
H_{0}\geq 1$ and $\epsilon+H_{0}\geq 0$. Hence, $\epsilon\geq 1/2$.

######  Remark 7.3.

Consider the following variant of Example 2. Let
$\mathcal{S}=\mathcal{S}^{+}\cup\\{S^{0},S^{-}\\}$, where the up-branch
$\mathcal{S}^{+}$ is as in Example 2, $S^{0}\equiv 1$, $S^{-}_{0}=1$ and
$S^{-}_{j}=0$ for $j\geq 1$. Adapting the arguments in Example 2, one can
check that: 1) The point mass $Q$ on $S^{0}$ is the unique martingale measure
of this model; 2) $(L)$-a.e. holds; 3) $\bar{I}({\bf 1}_{\\{S^{0}\\}})=1$ and
$\bar{I}({\bf 1}_{\\{S^{-}\\}})=1/2$.

Hence $\\{S^{-}\\}$ is a null set for $Q$, but not w.r.t. $\bar{I}$. In such a
situation our supermartingale decomposition (Theorem 4.1) holds on a larger
set than the (uniform) Doob decomposition in the classical theory.

######  Remark 7.4.

We may also consider the following variant of Example 2, replacing the ‘sure’
arbitrage at the type II arbitrage node $(S^{+},1)$ by a ‘sure’ arbitrage
opportunity by trading up to unbounded time. To this end, we replace the up-
branch of the model by
$\tilde{\mathcal{S}}^{+}=\\{\tilde{S}^{+,n}|n\in\mathbb{N}\\}$, where now

$\tilde{S}^{+,n}_{i}=\begin{cases}1,&i=0\\\ 2,&i=1<n+1\\\ 4,&i\geq
n+1\end{cases},$

and consider the trajectory set $\tilde{\mathcal{S}}^{+}\cup\mathcal{S}^{-}$,
with the lower branch $\mathcal{S}^{-}$ defined as in Example 2. This modified
trajectory set fails to be trajectorial complete, since
$(1,2,2,2,\ldots)\notin\tilde{\mathcal{S}}^{+}\cup\mathcal{S}^{-}$, but
satisfies $(L)$-a.e. by Corollary C.3. Note that all potential losses at the
node $(\tilde{S}^{+,1},1)$ by trading between time $0$ and $1$ can be
recuperated by buying the stock (at all times $i\geq 1$) and waiting until the
stock price eventually increases to 4. All conclusions of Example 2 are easily
seen to remain valid for this variant of the trajectory set. Note, however,
that the completion of this trajectory set, i.e. adding the trajectory
$(1,2,2,2,\ldots)$, drastically changes the model. Recuperating losses after
time 1, is not possible anymore in the completed model, because the stock
price may continue constantly after time 1. For this reason, the completed up-
branch is not a null set anymore and the completed model has infinitely many
martingale measures (all of which assigning positive probability on the ‘new’
trajectory $(1,2,2,2,\ldots)$).

We finally present an example, in which $(L)$-a.e. fails, and demonstrate the
importance of this assumption for our results.

######  Example 3.

Let $\mathcal{S}=\\{S^{+,-},S^{0},S^{-}\\}\cup\\{S^{+,n}|\,n\in\mathbb{N}\\}$,
where $S_{0}=1$ for every $S\in\mathcal{S}$,

$S_{1}=\begin{cases}1,&S=S^{0}\\\
2,&S\in\\{S^{+,n},S^{+,-}|\,n\in\mathbb{N}\\}\\\ 0,&S=S^{-}\end{cases},\quad
S_{2}=\begin{cases}S_{1},&S\in\\{S^{0},S^{-}\\}\\\
3,&S\in\\{S^{+,n}|\,n\in\mathbb{N}\\}\\\ 3/2,&S=S^{+,-}\end{cases}$

and

$S_{j}=\begin{cases}S_{2},&S\in\\{S^{0},S^{-},S^{+,-}\\}\\\
3+1/n,&S=S^{+,n}\end{cases},\quad j\geq 3.$

This trajectory space is illustrated in the following figure:

Note that $(L)$ holds, because the constant trajectory $S^{0}$ is included in
the trajectory set.
a) We first show that $(L_{(S^{+,-},1)})$ fails and $\bar{I}({\bf
1}_{\\{S^{+,-}\\}})\geq 1/6$, and, thus, condition $(L)$-a.e. is violated.

Suppose $f:\mathcal{S}_{(S^{+,-},1)}\rightarrow\mathbb{R}$. Define, for
arbitrary, but fixed $k\in\mathbb{N}$,

$g_{0}(S)=-k-2(f(S^{+,-})+k)(S_{2}-S_{1}),\quad g_{m}(S)=S_{3}-S_{2},\quad
S\in\mathcal{S}_{(S^{+,-},1)},\;m\geq 1.$

Then, $g_{0}\in\mathcal{E}_{(S^{+,-},1)}$,
$g_{m}\in\mathcal{E}^{+}_{(S^{+,-},1)}$ for $m\geq 1$, and, on
$\mathcal{S}_{(S^{+,-},1)}$,

$\sum_{m=0}^{\infty}g_{m}=f(S^{+,-})+\infty\,{\bf
1}_{\\{S^{+,n}|\,n\in\mathbb{N}\\}}\geq f.$

Since $k$ was arbitrary, we obtain $\overline{\sigma}_{1}f(S^{+,-})=-\infty$.
By choosing $f\equiv 0$, we observe that $(L_{(S^{+,-},1)})$ fails.

We next show that $\bar{I}({\bf 1}_{\\{S^{+,-}\\}})\geq 1/6$. To this end,
assume that for every $S\in\mathcal{S}$,

${\bf 1}_{\\{S^{+,-}\\}}(S)\leq\sum_{m\geq
1}\liminf_{k\to\infty}~{}~{}\Pi_{0,k}^{V^{m},H^{m}}(S)$

where $\Pi_{0,k}^{V^{m},H^{m}}\in\mathcal{E}_{0}^{+}$ for all $k\geq 0$ and
$\sum_{m\geq 1}V_{m}<\infty$. Since $S^{+,-}$ stays constant after time 1 and
the $\Pi_{0,k}^{V^{m},H^{m}}$’s are nonnegative at any time, we obtain

${\bf 1}_{\\{S^{+,-}\\}}(S)\leq\sum_{m\geq 1}\Pi_{0,k}^{V^{m},H^{m}}(S)$ (19)

for every $k\geq 1$ and $S\in\mathcal{S}.$ We now write $v=\sum_{m\geq
1}V_{m}$, $a=\sum_{m\geq 1}H^{m}_{0}$ and $b=\sum_{m\geq 1}H^{m}_{1}(S^{+,-})$
and note that the series defining $a$ and $b$ converge in $\mathbb{R}$ by the
Aggregation Lemma 4.4. Inserting the pairs $S=S^{+,-}$, $k=1$; $S=S^{+,1}$,
$k=2$; $S=S^{-}$, $k=1$ into (19) yields

$(I):\,v+a-b/2\geq 1,\quad(II):\,v+a+b\geq 0,\quad(III):\,v-a\geq 0.$

Considering $2(I)+(II)+3(III)$, we observe that $v\geq 1/6$ and, hence,
$\bar{I}({\bf 1}_{\\{S^{+,-}\\}})\geq 1/6$. Note that by similar, but easier
arguments $\bar{I}({\bf 1}_{\\{S^{-}\\}})\geq 1/2$ and $\bar{I}({\bf
1}_{\\{S^{0}\\}})=1$.
b) We now define the sequence $(f_{j})_{j\geq 0}$ via $f_{0}=1$ and
$f_{j}\equiv f_{\infty}={\bf 1}_{\\{S^{0}\\}}+2\;{\bf
1}_{\mathcal{S}\setminus\\{S^{0}\\}}$ for $j\geq 1$. We claim that
$(f_{j})_{j\geq 0}$ is a supermartingale, which does not have a decomposition
as in Theorem 4.1–(ii) for sequences $(\delta_{j})$ with $0<\delta_{0}<1$. In
particular, the assumption $(L)$-a.e. cannot be dropped in the latter theorem.

In our considerations, we may ignore the null set
$\\{S^{+,n}|\,n\in\mathbb{N}\\}$ of those trajectories, which pass through the
arbitrage node $(S^{+,1},2)$ of type II. As the computation of
$\overline{\sigma}_{j}$ is trivial after trajectories have become constant, we
get

$\overline{\sigma}_{j}f_{j+1}(S)=\overline{\sigma}_{j}f_{\infty}(S)=f_{\infty}(S)=f_{j}(S)$

for $S\in\\{S^{0},S^{-}\\}$ and $j\geq 1$ and for $S=S^{+,-}$ and $j\geq 2$.
Moreover, by a),

$\overline{\sigma}_{1}f_{2}(S^{+,-})=-\infty\leq f_{1}(S^{+,-}).$

For the supermartingale property at the initial node, consider

$g_{0}(S)=1-(S_{2}-S_{1})-4(S_{2}-S_{1}),\quad g_{m}(S)=S_{3}-S_{2},\quad
S\in\mathcal{S},\;m\geq 1.$

Then, $g_{0}\in\mathcal{E}$, $g_{m}\in\mathcal{E}^{+}$ for $m\geq 1$, and, on
$\mathcal{S}$,

$\sum_{m\geq 0}g_{m}=f_{1}\,{\bf 1}_{\\{S^{0},S^{-},S^{+,-}\\}}+\infty\,{\bf
1}_{\\{S^{+,n}|\,n\in\mathbb{N}\\}}\geq f_{1}.$

Hence $\overline{\sigma}f_{1}\leq 1$. (Since the trajectory $S^{0}$ is
constant, we obviously also obtain $\overline{\sigma}f_{1}\geq
f_{1}(S^{0})=1$, i.e., $\overline{\sigma}f_{1}=1$.)

We now fix a sequence $(\delta_{j})_{j\geq 0}$ of positive reals and assume
existence of a representation for $(f_{j})$ as in Theorem 4.1–(ii). Then, at
time $i=1$,

$f_{1}(S)\leq(1+\delta_{0})+H_{0}(S_{1}-S_{0})$

for $S\in\\{S^{0},S^{-},S^{+,-}\\}$ (where $H_{0}$ is a constant), because
none of the singletons $\\{S\\}$, $S\in\\{S^{0},S^{-},S^{+,-}\\}$, is a null
set by a). This leads to the three inequalities

$1\leq(1+\delta_{0}),\quad 2\leq(1+\delta_{0})-H_{0},\quad
2\leq(1+\delta_{0})+H_{0},$

which imply $\delta_{0}\geq 1$.

##  Appendix A Partitions, Arbitrage Nodes, and Null Sets

The following Lemma shows that property $(L_{(S,j)})$ fails at type II
arbitrage nodes $(S,j)$.

######  Lemma A.1.

Given a trajectory set $\mathcal{S}$ consider a node $(S,j)$, $j\geq 0$, then:
If $(S,j)$ is a type II arbitrage node, then

$\overline{\sigma}_{j}f(S)=-\infty~{}\mbox{for any}~{}~{}f\in Q.$

###### Proof.

We may consider the case when $\tilde{S}_{j+1}>S_{j}$ for all
$\tilde{S}\in\mathcal{S}_{(S,j)}$. Take then, for all $m\geq 1$:
$H^{m}_{j}(\tilde{S})=1$ and $H^{m}_{i}(\tilde{S})=0$ for all $i>j$,
$V^{m}=0$. Also, $H^{0}_{i}=0$ for all $i\geq j$; then, for any
$V^{0}\in\mathbb{R}$:

$f(\tilde{S})\leq V^{0}+\infty=V^{0}+\sum_{m\geq
1}H^{m}_{j}(\tilde{S})~{}\Delta_{j}\tilde{S}~{}~{}\hskip 7.22743pt\mbox{holds
for any}~{}~{}\tilde{S}\in\mathcal{S}_{(S,j)}~{}~{}\mbox{and}~{}~{}f\in Q.$

Thus, the claim follows. ∎

In upcoming results it will be proved that trajectories passing through
arbitrage nodes of type II form a null set. Define for $j\geq 0$,

$N^{\circ}_{j}\equiv\\{S\in\mathcal{S}:(S,j)\;\mbox{is an arbitrage node,
and}\;\Delta_{j}S\neq 0\\},\quad$ $N_{k}\equiv\bigcup\limits_{j\geq
k}N^{\circ}_{j},~{}\mbox{for}~{}~{}k\geq
0,~{}~{}\mbox{and}~{}~{}~{}~{}N(S,j)\equiv
N_{j}\cap\mathcal{S}_{(S,j)}~{}\mbox{for}~{}j\geq 0.$ (20)

Notice that

$N_{0}=\mathcal{N}:=\mathcal{N}^{(I)}\cup\mathcal{N}^{(II)}=\\{S\in\mathcal{S}:\exists~{}~{}j\geq
0~{}~{}~{}~{}~{}\mbox{s.t.}\;(S,j)\;\mbox{is an arbitrage node
and}\;S_{j+1}\neq S_{j}\\}$

where the sets $\mathcal{N}^{(I)},\mathcal{N}^{(II)}$ where introduced in the
proof of Theorem 4.1.

It will be shown that $\mathcal{N}$ is a null set. Whenever
$S\notin\mathcal{N}$ it follows that $S\notin N^{\circ}_{j}$ for any $j\geq
0$, therefore such node $(S,j)$ is: flat, or up-down, or type I arbitrage node
with $S_{j+1}=S_{j}$. On the other hand if $(S,j)$ is a type II arbitrage node
then $\mathcal{S}_{(S,j)}\subset N^{\circ}_{j}$. Moreover, it can be that
$S\in N^{\circ}_{j}$, but $(S,k)$ is arbitrage free for some $k>j$.

######  Definition A.2.

Since for any $j\geq 0$, $\mathcal{S}$ is the disjoint union of
$\mathcal{S}_{(S,j)}$, let $\Lambda_{j}$ be an index set, such that for
$\lambda\in\Lambda_{j}$ there exists $S^{\lambda}\in\mathcal{S}$ such that

$\lambda\neq\lambda^{\prime}\;\Rightarrow
S^{\lambda^{\prime}}\notin\mathcal{S}_{(S^{\lambda},j)},\quad\mathcal{S}=\bigcup_{\lambda\in\Lambda_{j}}\mathcal{S}_{(S^{\lambda},j)},\quad\mbox{and}$

$\mbox{if}~{}(S^{\lambda},j)~{}\mbox{is an arbitrage node
then}\;|\Delta_{j}S^{\lambda}|>0.$

For $\Gamma\subset\Lambda_{j}$ define $H^{\Gamma}=(H_{i}^{\Gamma})_{i\geq
0},\;H_{i}^{\Gamma}:\mathcal{S}\rightarrow\mathbb{R}$ by

$H_{i}^{\Gamma}\equiv 0\;\;\mbox{if}\;\;i\neq j~{};\quad
H_{j}^{\Gamma}\equiv\mathbf{1}_{\mathcal{S}^{\Gamma}},\;\;\mbox{where}\;\;\mathcal{S}^{\Gamma}\equiv\bigcup\limits_{\lambda\in\Gamma}\mathcal{S}_{(S^{\lambda},j)}.$

$\mathbf{H}^{\Gamma}$ is non-anticipative: Let $\tilde{S}_{k}=S_{k},\;0\leq
k\leq i$. If $i\neq j$ then $H_{i}^{\Gamma}(\tilde{S})=H_{i}^{\Gamma}(S)=0$.
For $i=j$, $S\in\mathcal{S}_{(S^{\lambda},j)}$ iff
$\tilde{S}\in\mathcal{S}_{(S^{\lambda},j)}$ so
$H_{j}^{\Gamma}(\tilde{S})=H_{j}^{\Gamma}(S)$.

######  Lemma A.3.

Consider $j\geq 0$ and $0\leq k\leq j$, then $N^{\circ}_{j}$, thus also
$N_{j}$, are conditionally null sets at any $(S,k)$.

###### Proof.

Define $N^{\circ,+}_{j}\equiv\\{S\in N^{\circ}_{j}:\Delta_{j}S>0\\}$ and
$\Lambda_{j}^{+}\equiv\\{\lambda\in\Lambda_{j}:S^{\lambda}\in
N^{\circ,+}_{j}\\}$, and consider for $m\geq 1$,
$f_{m}=\Pi^{0,H^{\Lambda_{j}^{+}}}_{k,j+1}\in\mathcal{E}_{(S,k)}^{+}$ for any
$S\in\mathcal{S}$. Then

$\mathbf{1}_{N^{\circ,+}_{j}}\leq\sum_{m\geq 1}f_{m},\;\;\mbox{which implies
that}\;\|\mathbf{1}_{N^{\circ,+}_{j}}\|_{k}=0.$

In a similar way it is shown that $N^{\circ,-}_{j}\equiv\\{S\in
N^{\circ}_{j}:\Delta_{j}S<0\\}$ is a conditionally null set at any $(S,k)$,
consequently $N^{\circ}_{j}=N_{j}^{\circ,+}\cup N_{j}^{\circ,-}$ is also
conditionally null. ∎

##  Appendix B Basic Properties of Superhedging Functional
$\overline{\sigma}_{j}$

######  Proposition B.1 (Basic Properties).

The following properties hold for $f,g\in Q$

  * a)

$\overline{\sigma}_{j}f(S)\leq f(S)$ if $f$ is constant on
$\mathcal{S}_{(S,j)}$. (Implies $\overline{\sigma}_{j}0\leq 0$ and
$\underline{\sigma}_{j}f\geq f$.)

  * b)

$\overline{\sigma}_{j}f(S)\leq\overline{\sigma}_{j}g(S)$, if $f\leq g$ a.e. on
$\mathcal{S}_{(S,j)}$.

  * c)

$\overline{\sigma}_{j}[f+g]\leq\overline{\sigma}_{j}f+\overline{\sigma}_{j}g$.

  * d)

Let $f\in Q$, $g\in P$, and $g$ is constant on $\mathcal{S}_{(S,j)}$ then
$\overline{\sigma}_{j}(gf)(S)\leq g\overline{\sigma}_{j}f(S)$.

  * e)

Let $f\in P$ and $k\geq 0$ then
$0\leq\overline{I}(\overline{I}_{k}f)\leq\overline{I}f$. Therefore is $f$ is a
null function we get $\overline{I}_{k}f$ is a null function.

###### Proof.

The proofs are immediate but we do indicate the arguments for item $d)$.

Let $c\geq 0$ be a constant such that $g(\hat{S}=c$ for all
$\hat{S}\in\mathcal{S}_{(S,j)}$. If $c=0$
$\overline{\sigma}_{j}(gf)(S)=\overline{\sigma}_{j}(0)(S)\leq
g\overline{\sigma}_{j}0(S)$ (already covered by item $a)$. So assume $c>0$

Let $gf(\tilde{S})\leq\sum\limits_{m\geq
0}\liminf\limits_{n\rightarrow\infty}\Pi_{j,n}^{V^{m},H^{m}}(\tilde{S}),\;\tilde{S}\in\mathcal{S}_{(S,j)}$,
with $\Pi_{j,n_{0}}^{V^{0},H^{0}}\in\mathcal{E}_{(S,j)}$ and, for $m\geq 1$,
$\Pi_{j,n}^{V^{m},H^{m}}\in\mathcal{E}^{+}_{(S,j)}$ for all $n\geq 0$. For
each $\tilde{S}\in\mathcal{S}_{(S,j)}$ and $m\geq 0$ define

$U^{m}(\tilde{S})=\frac{V^{m}}{g(S)},\;\;\mbox{and}\;\;G^{m}_{i}(\tilde{S})=\frac{H^{m}_{i}(\tilde{S})}{g(S)},\;\mbox{for}\;i\geq
j.$

It follows that $f(\tilde{S})\leq\sum\limits_{m\geq
0}\liminf\limits_{n\rightarrow\infty}\Pi_{j,n_{m}}^{U^{m},G^{m}}(\tilde{S}),\;\tilde{S}\in\mathcal{S}_{(S,j)}$,
with $\Pi_{j,n_{0}}^{U^{0},G^{0}}\in\mathcal{E}_{(S,j)}$, and for $m\geq 1$,
$\Pi_{j,n}^{U^{m},G^{m}}\in\mathcal{E}^{+}_{(S,j)}$ for all $n\geq 0$. Thus

$\overline{\sigma}_{j}f(S)\leq\frac{\overline{\sigma}_{j}[gf](S)}{g(S)}.$

Notice that one actually obtains
$\overline{\sigma}_{j}(gf)(S)=g\overline{\sigma}_{j}f(S)$ if $g=c>0$.

We also note that the proof of item $e)$ is analogous to the one of
Proposition3.5. ∎

######  Corollary B.2.

Suppose $f,g\in Q$. If $f\leq g$ a.e., then
$\overline{\sigma}_{j}f\leq\overline{\sigma}_{j}g$ a.e.

###### Proof.

Note that, for every $S\in\mathcal{S}$,

$\overline{\sigma}_{j}f(S)\leq\overline{\sigma}_{j}(\max\\{f,g\\})(S)=\overline{\sigma}_{j}(g+(f-g)_{+})(S)\leq\overline{\sigma}_{j}g(S)+\overline{I}_{j}(f-g)_{+}(S),$

where $(\cdot)_{+}$ denotes the positive part. As $(f-g)_{+}$ is a null
function, we conclude by Proposition B.1 that $\overline{I}_{j}(f-g)_{+}$ is a
null function. Hence, $\overline{\sigma}_{j}f\leq\overline{\sigma}_{j}g$ a.e.
∎

Here are some equivalent properties for $(L_{(S,j)})$.

######  Proposition B.3.

For a fixed node $(S,j)$, the following items are equivalent.

  1. (1)

$\overline{\sigma}_{j}0(S)=0$.

  2. (2)

$\underline{\sigma}_{j}f(S)\leq\overline{\sigma}_{j}f(S)$ for any $f\in Q$.

  3. (3)

Property $(L_{(S,j)})$.

  4. (4)

$\underline{\sigma}_{j}f(S)=V(S)=\overline{\sigma}_{j}f(S)$ for
$f=\Pi^{V,H}_{j,n_{f}}\in\mathcal{E}_{j}$.

###### Proof.

From (1) and item c) of Proposition B.1 it follows (2), since
$0\leq\overline{\sigma}_{j}f(S)+\overline{\sigma}_{j}[-f](S)$. Assumed (2), it
follows that $0\leq\underline{\sigma}_{j}0(S)\leq\overline{\sigma}_{j}0(S)\leq
0$, first and last inequalities from item a) of Proposition B.1, so (1) holds.

From here onwards we let $f=\Pi^{V,H}_{j,n_{f}}$.

$(\ref{L_j})$ follows from $(\ref{sigma0=0})$ as follows. Let
$h_{m}=\liminf\limits_{n\rightarrow\infty}~{}\Pi^{V^{m},H^{m}}_{j,n},~{}~{}\Pi^{V^{m},H^{m}}_{j,n}\in\mathcal{E}_{(S,j)}^{+}~{}\forall~{}n\geq
j$ and $m\geq 1$, such that $f\leq\sum\limits_{m\geq 1}h_{m}$.

Then $0\leq-\Pi^{V,H}_{j,n_{f}}+\sum\limits_{m\geq 1}h_{m}$, thus (taking
$f_{0}\equiv-\Pi^{V,H}_{j,n_{f}}$, and $f_{m}=h_{m}$ for $m\geq 1$) by
Definition 2.10, $0=\overline{\sigma}_{j}(0)\leq-V+\sum\limits_{m\geq
1}V^{m}$, which leads to $V\leq\sum\limits_{m\geq 1}V^{m}$ as required.

Assumed $(\ref{L_j})$, let
$f_{0}=\Pi^{V^{0},H^{0}}_{j,n_{0}}\in\mathcal{E}_{(S,j)}$ and
$f_{m}=\liminf\limits_{n\rightarrow\infty}~{}\Pi^{V^{m},H^{m}}_{j,n},~{}~{}\Pi^{V^{m},H^{m}}_{j,n}\in\mathcal{E}_{(S,j)}^{+}\\\
~{}\forall~{}n\geq j$ and $m\geq 1$, such that $f\leq\sum\limits_{m\geq
0}f_{m}$. Then $f-f_{0}\leq\sum\limits_{m\geq 1}f_{m}$ with
$f-f_{0}\in\mathcal{E}_{(S,j)}$, so

$V(S)-V_{0}\leq\sum\limits_{m\geq 0}V^{m},$

and

$V(S)\leq\overline{\sigma}_{j}f(S).$

Since by Definition 2.10, $\overline{\sigma}_{j}f(S)\leq V(S)$, (4) holds,
having in mind that
$\underline{\sigma}_{j}f(S)=-\overline{\sigma}_{j}[-f](S)=V(S)$.

Finally, it is clear that (1) follows from (4). ∎

######  Lemma B.4.

Let $f\in Q$, $(S,j)$ a fixed node and $k\geq j$. If $f$ is constant on
$\mathcal{S}_{(S,j)}$ then $\overline{\sigma}_{k}f(S)\leq
f(S)\leq\underline{\sigma}_{k}f(S)$. Moreover once $(L_{(S,k)})$ holds then

  1. (1)

If $f$ is constant on $\mathcal{S}_{(S,j)}$ then
$\underline{\sigma}_{k}f(S)=f(S)=\overline{\sigma}_{k}f(S)$.

  2. (2)

For a general $f\in Q$, $\overline{\sigma}_{j}f$ is constant on
$\mathcal{S}_{(S,j)}$, hence:

$\overline{\sigma}_{k}[\overline{\sigma}_{j}f](S)=\overline{\sigma}_{j}f(S)=\underline{\sigma}_{k}[\overline{\sigma}_{j}f](S)\quad\mbox{and}\quad\overline{\sigma}_{k}[\underline{\sigma}_{j}f](S)=\underline{\sigma}_{j}f(S)=\underline{\sigma}_{k}[\underline{\sigma}_{j}f](S).$

###### Proof.

If $f$ is constant on $\mathcal{S}_{(S,j)}$, it is also constant on
$\mathcal{S}_{(S,k)}\subset\mathcal{S}_{(S,j)}$, then
$f\in\mathcal{E}_{(S,k)}$, so by Definition 2.10,
$\overline{\sigma}_{k}f(S)\leq f(S)$.

If, furthermore, also $(L_{(S,k)})$ holds, we have
$\underline{\sigma}_{k}f(S)=I_{k}f(S)=f(S)=\overline{\sigma}_{k}f(S)$ by
Proposition B.3 item (4). ∎

######  Corollary B.5.

Let $f,g\in Q$ and consider a fixed $S\in\mathcal{S}$. If for $j\geq 0$
property $(L_{(S,j)})$ holds and
$\overline{\sigma}_{j}f(S)-\underline{\sigma}_{j}f(S)=0=\overline{\sigma}_{j}g(S)-\underline{\sigma}_{j}g(S)$,
then all the involved quantities are finite and

$(a)\quad\overline{\sigma}_{j}(f+g)(S)=\overline{\sigma}_{j}f(S)+\overline{\sigma}_{j}g(S)=\underline{\sigma}_{j}f(S)+\underline{\sigma}_{j}g(S)=\underline{\sigma}_{j}(f+g)(S).$
$(b)\quad\overline{\sigma}_{j}(cf)(S)=c\overline{\sigma}_{j}f(S)=c\underline{\sigma}_{j}f(S)=\underline{\sigma}_{j}(cf)(S)\quad\forall
c\in\mathbb{R}.$

###### Proof.

The finiteness claims follow from our conventions in the first paragraph of
Section 2.2. We then see that the hypotheses imply that
$\overline{\sigma}_{j}f(S)=\underline{\sigma}_{j}f(S)$ and
$\overline{\sigma}_{j}g(S)=\underline{\sigma}_{j}g(S)$.

(a) holds from

$\overline{\sigma}_{j}f(S)+\overline{\sigma}_{j}g(S)=\underline{\sigma}_{j}f(S)+\underline{\sigma}_{j}g(S)\leq\underline{\sigma}_{j}[f+g](S)\leq\overline{\sigma}_{j}[f+g](S)\leq\overline{\sigma}_{j}f(S)+\overline{\sigma}_{j}g(S).$

For (b), if $c=0$ or $c=-1$ the result is clear. For $c>0$ it follows from
item $d)$ of Proposition B.1, from where, if $c<0$

$\overline{\sigma}_{j}(cf)(S)=\overline{\sigma}_{j}(-c(-f))(S)=-c\overline{\sigma}_{j}(-f)(S)=c\overline{\sigma}_{j}f(S).$

∎

##  Appendix C General Hypothesis to Establish $(L)$-a.e. and proof of Theorem
7.1

In this appendix, we state and prove a generalization of Theorem 7.1. It
relies on the following concept of ‘good nodes’.

######  Definition C.1 (Good Nodes).

A node $(S,j)$ is called _good_ , if $\mathcal{S}_{(S,j)}\neq N(S,j)$, (as
introduced in (20)). Otherwise, $(S,j)$ is said to be _bad_.

The following theorem characterizes, under suitable assumptions, the good
nodes as precisely those nodes at which the continuity from below property
holds. Note that (H.5) relaxes the assumption of trajectorial completeness.

######  Theorem C.2.

Suppose the trajectory set satisfies the following assumptions:

  * (H.4)

If $(S,j)$ is a good up-down node, then for every $\epsilon>0$ there are
$S^{\epsilon,1},S^{\epsilon,2}\in\mathcal{S}_{(S,j)}$ such that
$(S^{\epsilon,1},j+1)$ and $(S^{\epsilon,2},j+1)$ are good nodes satisfying

$S^{\epsilon,1}_{j+1}-S_{j}\geq-\epsilon,\quad
S^{\epsilon,2}_{j+1}-S_{j}\leq\epsilon$

  * (H.5)

If $(S^{n})_{n\geq 0}$ is a sequence in $\mathcal{S}$ satisfying

$~{}~{}~{}S^{n}_{i}=S^{n+1}_{i}\;\;0\leq i\leq
n,~{}~{}\forall~{}n,~{}~{}~{}~{}~{}~{}~{}~{}$

and if there is an $n_{0}\geq 0$ such that $(S^{n},n)$ is a good node for
every $n\geq n_{0}$, then $(S^{n}_{n})_{n\in\mathbb{N}_{0}}\in\mathcal{S}$
(i.e. $\overline{S}\equiv\lim_{n\rightarrow\infty}S^{n}~{}\in\mathcal{S}$).

Then, $(L_{(S,j)})$ holds, if and only if $(S,j)$ is a good node.

###### Proof.

Fix a good node $(S,j)$. Let
$~{}~{}~{}~{}f=\Pi^{V^{0},H^{0}}_{j,n_{f}}\in\mathcal{E}_{(S,j)}$ and
$f_{m}=\liminf_{n\rightarrow\infty}\Pi^{V^{m},H^{m}}_{j,n}$ with
$\Pi^{V^{m},H^{m}}_{j,n}\in\mathcal{E}_{(S,j)}^{+}~{}\mbox{for all}~{}n\geq
j~{}\mbox{and}~{}m\geq 1$ such that

$f\leq\sum_{m\geq 1}f_{m}\;\;\mbox{on}\;\mathcal{S}_{(S,j)}.$

We need to show that

$V^{0}\leq\sum_{m\geq 1}V^{m}=:V,$

(and, hence, can and will assume that $V<\infty$). Recall that

$N(S,j)=\\{\tilde{S}\in\mathcal{S}_{(S,j)}|\;(\tilde{S},k)\mbox{ is an
arbitrage node and }\tilde{S}_{k+1}\neq\tilde{S}_{k}\mbox{ for some }k\geq
j\\}.$

If $\tilde{S}\in\mathcal{S}_{(S,j)}\setminus N(S,j)$, then, for every $k\geq
j$, $(\tilde{S},k)$ is an up-down node or $\tilde{S}_{k+1}=\tilde{S}_{k}$.
Hence, by the Aggregation Lemma 4.4, for every
$\tilde{S}\in\mathcal{S}_{(S,j)}\setminus N(S,j)$ and $n\geq j$

$\sum_{m\geq
1}\Pi^{V^{m},H^{m}}_{j,n}(\tilde{S})=V+\sum_{i=j}^{n-1}H_{i}(\tilde{S})\Delta_{i}\tilde{S},$

for the non-anticipative sequence

$H_{i}(\hat{S})=\begin{cases}\sum_{m=1}^{\infty}H^{m}_{i}(\hat{S}),&\mbox{if
convergent in }\mathbb{R},\\\ 0,&\mbox{otherwise,}\end{cases}\quad i\geq
j,\;\hat{S}\in\mathcal{S}_{(S,j)}.$

Then, by Fatou’s lemma, for every $\hat{S}\in\mathcal{S}_{(S,j)}$

$f\leq
V+\liminf_{n\rightarrow\infty}\sum_{i=j}^{n-1}H_{i}(\hat{S})\Delta_{i}\hat{S}+\infty{\bf
1}_{N(S,j)}(\hat{S}),$

which we may rearrange into

$V^{0}\leq
V+\liminf_{n\rightarrow\infty}\sum_{i=j}^{n-1}\tilde{H}_{i}(\hat{S})\Delta_{i}\hat{S}+\infty{\bf
1}_{N(S,j)}(\hat{S})$

where $\tilde{H}_{i}=H_{i}-H^{0}_{i}$ for $i<n_{f}$ and $\tilde{H}_{i}=H_{i}$
otherwise.

Thus, it is enough to show the following: For every $\delta>0$ there is an
$\tilde{S}\in\mathcal{S}_{(S,j)}\setminus N(S,j)$ such that

$\liminf_{n\rightarrow\infty}\sum_{i=j}^{n-1}\tilde{H}_{i}(\tilde{S})\Delta_{i}\tilde{S}\leq\delta.$
(21)

To this end, we construct a sequence $(S^{n})$ in $\mathcal{S}$ as follows:
$S^{n}=S$ for $n\leq j$ and, inductively for $n>j$ in the following way, which
guarantees that $(S^{n},n)$ is a good node for every $n\geq j$. Assume $S^{n}$
has already been constructed for some $n\geq j$ and $(S^{n},n)$ is a good
node.

If $(S^{n},n)$ is a flat node or an arbitrage node of type I, then choose
$S^{n+1}\in\mathcal{S}_{(S^{n},n)}$ such that $S^{n+1}_{n+1}=S^{n}_{n}$. We
argue that $(S^{n+1},n+1)$ is good in the type I case (the flat case being
similar and easier). Suppose to the contrary that $(S^{n+1},n+1)$ is bad.
Then, for every
$\hat{S}\in\mathcal{S}_{(S^{n+1},n+1)}\subseteq\mathcal{S}_{(S^{n},n)}$ there
is an $i\geq n+1$ such that $(\hat{S},i)$ is an arbitrage node and
$\hat{S}_{i+1}\neq\hat{S}_{i}$ – hence, $\hat{S}\in N(S^{n},n)$. Moreover, any
$\hat{S}\in\mathcal{S}_{(S^{n},n)}\setminus\mathcal{S}_{(S^{n+1},n+1)}$
belongs to $N(S^{n},n)$ because $(S^{n},n)$ is a type I arbitrage node. Thus,
$\mathcal{S}_{(S^{n},n)}=N(S,n)$ – a contradiction.

If $(S^{n},n)$ is an up-down node, then, by (H.4), one can choose a
sufficiently small $\epsilon>0$ and $S^{n+1}\in\mathcal{S}_{(S^{n},n)}$ such
that

$\tilde{H}_{n}(S^{n})(S_{n+1}^{n+1}-S^{n}_{n})\leq|\tilde{H}_{n}(S^{n})|\epsilon\leq\delta
2^{-(n+1)}$ (22)

and $(S^{n+1},n+1)$ is a good node.

Note that $(S^{n},n)$ cannot be an arbitrage node of type II, because it is
good by the inductive hypothesis. Hence, the construction of $S^{n+1}$ is
finished.

By (H.5), $\tilde{S}=(S^{n}_{n})_{n\in\mathbb{N}_{0}}\in\mathcal{S}$. Then, by
construction, $\tilde{S}\in\mathcal{S}_{(S,j)}$ and $(\tilde{S},n)=(S^{n},n)$
is an up-down node (and then (22) holds) or
$\tilde{S}_{n+1}=S^{n+1}_{n+1}=S^{n}_{n}=\tilde{S}_{n}$, whenever $n\geq j$.
Hence, by construction, $\tilde{S}\notin N{(S,j)}$ and

$\liminf_{n\rightarrow\infty}\sum_{i=j}^{n-1}\tilde{H}_{i}(\tilde{S})\Delta_{i}\tilde{S}\leq\delta\sum_{i=j}^{\infty}2^{-(i+1)}\leq\delta,$

which establishes (21). Consequently, $(L_{(S,j)})$ holds.

Conversely, assume that $(S,j)$ is a bad node. Then, by Lemma A.3,
$\overline{I}_{j}1(S)=0$, which in turn implies
$\overline{\sigma}_{j}0(S)\leq-1$. In view of Proposition B.3, $(L_{(S,j)})$
does not hold. ∎

As an immediate consequence of Theorem C.2 and Lemma A.3, we obtain the
following result:

######  Corollary C.3.

Under the assumptions of Theorem C.2, additionally suppose that $(S,0)$ is
good. Then $(L)$-a.e. holds.

We complete this appendix with the proof of Theorem 7.1.

###### Proof of Theorem 7.1.

We first show that, under the assumptions of Theorem 7.1, $(S,j)$ is bad, if
and only if $(S,j)$ is an arbitrage node of type II. Since arbitrage nodes of
type II are always bad, we only need to show that any other nodes are good. To
this end, we fix a node $(S,j)$ which is not arbitrage of type II and
construct a trajectory $\tilde{S}\in\mathcal{S}_{(S,j)}\setminus N(S,j)$. We
set $S^{n}=S$ for $n\leq j$ and inductively construct $S^{n}$ for $j\geq n$ in
the following way, which guarantees that $(S^{n},n)$ is not a type II
arbitrage node: If $(S^{n},n)$ is flat or an arbitrage node of type I, we
choose $S^{n+1}\in\mathcal{S}_{(S^{n},n)}$ such that $S^{n+1}_{n+1}=S^{n}_{n}$
and note that $(S^{n+1},n+1)$ is not an arbitrage node of type II by (H.2). If
$(S^{n},n)$ is an up-down node, then by (H.2) again, we find some
$S^{n+1}\in\mathcal{S}_{(S^{n},n)}$ such that $(S^{n+1},n+1)$ is not an
arbitrage node of type II. Since the trajectory set is complete,
$\tilde{S}=(S^{n}_{n})_{n\in\mathbb{N}_{0}}\in\mathcal{S}$. By construction,
$\tilde{S}\in\mathcal{S}_{(S,j)}\setminus N(S,j)$.

We next argue that (H.2) implies (H.4). By the characterization of good nodes,
(H.4) can be reformulated as

  * (H.4’)

If $(S,j)$ is an up-down node, then for every $\epsilon>0$ there are
$S^{\epsilon,1},S^{\epsilon,2}\in\mathcal{S}_{(S,j)}$ such that

$S^{\epsilon,1}_{j+1}-S_{j}\geq-\epsilon,\quad
S^{\epsilon,2}_{j+1}-S_{j}\leq\epsilon$

and $(S^{\epsilon,1},j+1)$ and $(S^{\epsilon,2},j+1)$ are not arbitrage nodes
of type II.

To verify (H.4’), consider any up-down node $(S,j)$. If $(\hat{S},j+1)$ is not
a type II arbitrage node for every $\hat{S}\in\mathcal{S}_{(S,j)}$, then we
find $S^{1},S^{2}\in\mathcal{S}_{(S,j)}$ such that $S^{1}_{j+1}-S_{j}>0$ and
$S^{2}_{j+1}-S_{j}<0$. We may choose $S^{\epsilon,1}=S^{1}$ and
$S^{\epsilon,2}=S^{2}$ for every $\epsilon>0$. Otherwise, (H.2) directly
applies.

Finally, note that trajectorial completeness implies (H.5). Hence, by Theorem
C.2, $(L_{(S,j)})$ holds, if and only if $(S,j)$ is not an arbitrage node of
type II. By (H.2), the initial node $(S,0)$ is not an arbitrage node of type
II, and, thus, $(L)$ holds. Noting that the trajectories passing through type
II arbitrage nodes form a null set (by Lemma A.3), we conclude $(L)$-a.e. ∎

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