# Robust utility maximisation under proportional transaction costs for càdlàg
price processes111We thank Huy Chau for his comments on an earlier version of
the paper.

Christoph Czichowsky222Department of Mathematics, London School of Economics
and Political Science, Columbia House, Houghton Street, London WC2A 2AE, UK,
<EMAIL_ADDRESS>Raphael Huwyler333Department of Mathematical and
Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada,
<EMAIL_ADDRESS>

###### Abstract

We consider robust utility maximisation in continuous-time financial markets
with proportional transaction costs under model uncertainty. For this, we work
in the framework of Chau and Rásonyi [7], where robustness is achieved by
maximising the worst-case expected utility over a possibly uncountable class
of models that are all given on the same underlying filtered probability space
with incomplete filtration. In this setting, we give sufficient conditions for
the existence of an optimal trading strategy extending the result for utility
functions on the positive half-line of Chau and Rásonyi [7] from continuous to
general strictly positive càdlàg price processes. This allows us to provide a
positive answer to an open question pointed out in Chau and Rásonyi [7], and
shows that the embedding into a countable product space is not essential.

Key words: Utility maximisation, proportional transaction costs, model
uncertainty, incomplete filtrations.

##  1 Introduction

Maximising the expected utility from terminal wealth is a classical problem in
Mathematical Finance and Financial Economics. Recently, there has been a lot
of interest in robust utility maximisation under model uncertainty, where one
maximises the worst-case expected utility over a class of models. The
motivation for this is that the resulting trading strategies are less
sensitive to changes of the underlying model and in this sense more robust to
model misspecification.

In this paper, we consider robust utility maximisation under proportional
transaction costs in the framework of model uncertainty of Chau and Rásonyi
[7]. Here, the worst-case expected utility over a possibly uncountable class
of models, that are all given on the same underlying filtered probability
space with incomplete filtration, is maximised. In this setting, we extend the
existence result of Chau and Rásonyi [7] for utility functions on the positive
half-line from continuous to general strictly positive càdlàg price processes.
This answers an open question in Chau and Rásonyi [7]. Without frictions, a
rich amount of examples of discrete-time models for this setting of model
uncertainty has been proposed in Rásonyi and Meireles-Rodrigues [26]. Our
results cover the corresponding models under proportional transaction costs
and allow to consider both discrete- and continuous-time models in a unified
framework.

As already explained in Chau and Rásonyi [7], most of the literature typically
parameterises model uncertainty by a family $\mathbb{P}$ of probability
measures that are given on some underlying canonical probability space (see,
e.g., Biagini and Pınar [2], Lin and Riedel [21] and Neufeld and Nutz [23]).
This means that the dynamics of the risky asset is given by a fixed process
and model uncertainty is described by a family of different distributions of
this process. For diffusion models, drift uncertainty can be modelled by
considering a family of absolutely continuous measures that are dominated by a
single measure $P^{\ast}$ (see e.g., Quenez [25] and Schied [30]), while the
case of volatility uncertainty requires an uncountable family of singular
measures (see e.g., Denis and Kervarec [14]).

In contrast to the approach above, Chau and Rásonyi [7] propose a setup of
model uncertainty, where different stock price processes are considered. That
is, they suggest to work on a fixed filtered probability space, and to use a
family of stochastic processes $S^{\theta}$ indexed by $\theta$ in a non-empty
set $\Theta$ to describe model uncertainty. From a mathematical point of view,
the main advantage of this setup is that no topological or measurability
assumptions are need to be put on the set of parameters $\Theta$ representing
the different models. In contrast, the parametrisation via a family of
measures $\mathbb{P}$ incorporates technical issues such as the treatment of
null events and filtration completion (see e.g., Biagini et al. [3], Bouchard
and Nutz [4] and Nutz [24]). However, since typical examples consider an
uncountable set of models $\Theta$, one cannot complete the filtration with
respect to the null sets arising from any price process $S^{\theta}$ and has
to work with incomplete filtrations. Working with filtrations under “unusual”
conditions, that is, without the usual condition of completeness, then brings
its own challenges. However, as pointed out in Jacod [17], we may still pass
to the usual conditions from time to time, at the cost of working with
$P$-indistinguishable equivalence classes. This is in contrast to Chau and
Rásonyi [7], who still assume the filtration to be complete. Besides this
fact, we refer to the original paper of Chau and Rásonyi [7] for a more
detailed comparison between these two approaches to model uncertainty.

Within their setting of model uncertainty, Chau and Rásonyi [7] observe that,
similarly as in the case of a single model in Guasoni [16], it is more
suitable to optimise directly over trading strategies, and hence stochastic
processes, rather than over terminal wealths given by random variables as in
the classical case with only one price process. For this, they exploit that,
for continuous price processes, it is sufficient to model trading strategies
by càdlàg finite variation processes under proportional transaction costs. The
key insight is that càdlàg finite variation processes can be identified with
their values along the rational numbers and hence objects taking values in the
countable product of complete metric spaces. However, this does not avoid the
measurability problem arising from working with uncountably many models
simultaneously, if one does not assume the filtration to be complete.

While càdlàg strategies are sufficient to obtain the optimal strategies for
continuous price processes under proportional transaction costs, this is no
longer true for price processes with jumps; see Example 4.1 in Czichowsky and
Schachermayer [9]. For càdlàg price processes, it matters, whether one is
trading immediately before, just at, or immediately after a jump. Therefore,
trading strategies have to be modelled by general predictable finite variation
processes that can have left and right discontinuities and can no longer be
identified with their values along the rationals. To overcome this difficulty,
we use a version of Helly’s Theorem of Campi and Schachermayer [6]. The latter
allows us to obtain a sequence of trading strategies that converges to a
càdlàg modification of a suitable limit process at all time points except for
the discontinuity points of that modified limit process. The key is now that
we can show that the set, where the convergence can fail, is the same for all
models. Hence, it can be exhausted by countably many stopping times. The
stopping times can be obtained by either passing to the usual conditions or by
using a suitable version of the Debut Theorem for filtrations that are not
complete. This allows us to achieve the convergence at these points as well by
a diagonal sequence argument. Mathematically speaking, while Chau and Rásonyi
[7] work with the topology of $P$-a.s. convergence along all rationales on the
set of càdlàg finite variation processes, we work with the topology of
convergence in probability at all $[0,T]$-valued stopping times. Somewhat
surprisingly our result shows that the embedding into a countable product
metric space indexed by the rationals as used in Chau and Rásonyi [7] is not
essential.

We assume that our utility functions have a reasonable asymptotic elasticity
as in the classical single model framework of Kramkov and Schachermayer [19].
This allows us to obtain the optimal trading strategy by directly optimising
in the primal problem of maximising expected utility from terminal wealth, and
we do not need specific properties of the dual problem. Therefore, we only
need the existence of (locally) consistent price systems for _one_ level of
transaction costs $\lambda^{\prime}\in(0,\lambda)$ rather than for all
$\lambda^{\prime}\in(0,\lambda)$ as in Chau and Rásonyi [7]; see Remark 4.5 of
Chau and Rásonyi [7] and Lemma 4.3 below.

In the framework of model uncertainty of Chau and Rásonyi [7], a super-
replication theorem has been recently established in Chau et al. [8]. For
model uncertainty with uncountably many probability measures on the same
probability space, Bartl et al. [1] derived a duality result, in the spirit to
the one of Kramkov and Schachermayer [19], for the case of a single model, for
utility maximisation from terminal wealth without transaction costs.

The paper is organised as follows. We introduce the setting and formulate the
problem in Section 2. Our main result is stated and explained in Section 3.
The proof of the main result is covered in Section 4. For better readability,
some proofs and explanations are deferred to Appendix A.

##  2 Formulation of the problem

We consider a financial market consisting of one riskless asset and one risky
asset. The riskless asset has constant price $1$. The dynamics of the risky
asset is uncertain. For this, let $\Theta$ be a non-empty set and consider a
family of strictly positive adapted càdlàg processes $(S_{t}^{\theta})_{0\leq
t\leq T}$, for each $\theta\in\Theta$, on some underlying filtered probability
space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{0\leq t\leq T},P)$. No further
conditions are imposed on $\Theta$. By passing to the right-continuous
version, we can assume without loss of generality that the filtration
$\mathbb{F}\coloneqq(\mathcal{F}_{t})_{0\leq t\leq T}$ is right continuous.
However, the filtration does not necessarily need to be complete. We denote by
$\mathcal{F}^{P}$ the $P$-completion of $\mathcal{F}$, and accordingly
$\mathbb{F}^{P}$ the usual augmentation of $\mathbb{F}$ by adjoining all the
$P$-negligible sets to $\mathcal{F}_{0}$. In this framework, model uncertainty
is then incorporated by considering simultaneously all models
$(S_{t}^{\theta})_{0\leq t\leq T}$, for $\theta\in\Theta$, as possible
evolutions for the price process of the risky asset, so that the set $\Theta$
provides the parametrisation of model uncertainty.

In each model, we assume that the risky asset $S^{\theta}$ can be traded under
proportional transaction costs $\lambda\in(0,1)$. That is, an agent can buy
the risky asset at the higher ask price price $S^{\theta}$ but can only sell
it at the lower bid price $(1-\lambda)S^{\theta}$. The riskless asset can be
traded without transaction costs.

In the spirit of Jacod [17], we say that an
$\mathbb{R}_{+}\cup\\{\infty\\}$-valued process is _increasing_ if all
trajectories are increasing. We note that an increasing process does not
necessarily start at zero. We then define the set
$\mathcal{V}^{+}(\mathbb{F},P)$ of all equivalence classes of processes, with
respect to the relation $P$-indistinguishable444Two processes $X$ and $Y$ are
called _$P$ -indistinguishable_ if the set $\\{X\neq Y\\}$ is $P$-evanescent,
i.e., the set $N=\\{\omega\in\Omega\colon\exists
t\in\mathbb{R}_{+},\>\text{s.t.}\>(\omega,t)\in\\{X\neq Y\\}\\}$ is
$P$-negligible, that is $N\in\mathcal{F}^{P}$ and $P[N]=0$., admitting a
version $H$, which is $\mathbb{F}$-adapted, increasing and satisfies
$H_{t}<\infty$, $P$-a.s., for all $t\in\mathbb{R}_{+}$. We then define
$\mathcal{V}(\mathbb{F},P)=\mathcal{V}^{+}(\mathbb{F},P)-\mathcal{V}^{+}(\mathbb{F},P)$
as the set of differences of two elements from
$\mathcal{V}^{+}(\mathbb{F},P)$. These are exactly the equivalence classes of
$\mathbb{F}$-adapted processes, whose trajectories are $P$-a.s. of finite
variation on compacts. If $H\in\mathcal{V}(\mathbb{F},P)$, we denote the total
variation of $H$ by $|H|$. We denote the $\mathbb{F}$-predictable processes by
$\mathcal{P}(\mathbb{F})$. The set
$\mathcal{P}(\mathbb{F})\cap\mathcal{V}(\mathbb{F},P)$ consists of classes of
processes from $\mathcal{V}(\mathbb{F},P)$ with at least one
$\mathcal{P}(\mathbb{F})$-measurable version. Accordingly, we define the sets
$\mathcal{V}(\mathbb{F}^{P},P)$, $\mathcal{P}(\mathbb{F}^{P})$, etc., by
replacing $\mathbb{F}$ with $\mathbb{F}^{P}$.

Trading strategies are then modelled by $\mathbb{R}^{2}$-valued processes
$H=(H_{t}^{0},H_{t}^{1})_{0\leq t\leq T}$, where $H^{0}$ and $H^{1}$ are
elements of $\mathcal{P}(\mathbb{F})\cap\mathcal{V}(\mathbb{F},P)$. In
particular, $H^{0}$ (respectively $H^{1}$) is a processes of finite variation
that is $P$-indistinguishable from a predictable process. For each process
$H\in\mathcal{V}(\mathbb{F},P)$, there exists a unique decomposition
$H=H^{\uparrow}-H^{\downarrow}$, called _Jordan-Hahn decomposition_ , where
$H^{\uparrow}$ and $H^{\downarrow}$ are elements of
$\mathcal{V}^{+}(\mathbb{F},P)$, and such that
$|H|=H^{\uparrow}+H^{\downarrow}$. If
$H\in\mathcal{P}(\mathbb{F})\cap\mathcal{V}(\mathbb{F},P)$, then also
$H^{\uparrow},\>H^{\downarrow}\in\mathcal{P}(\mathbb{F})\cap\mathcal{V}^{+}(\mathbb{F},P)$
(see Lemma 1.35 in [17]).

Moreover, we define $\Delta H_{t}\coloneqq H_{t}-H_{t-}$, and
$\Delta_{+}H_{t}\coloneqq H_{t+}-H_{t}$, where $H_{t-}:=\lim_{s\uparrow
t}H_{s}$ and $H_{t+}:=\lim_{s\downarrow t}H_{s}$ denote the left and right
limits, respectively, and the right-continuous processes

$\displaystyle H_{t}^{d}\coloneqq\sum_{s\leq t}\Delta
H_{s},\quad\text{and}\quad H_{t}^{d,+}\coloneqq\sum_{s\leq t}\Delta_{+}H_{s}.$

Finally, we define the continuous part $H^{c}$ of $H$ by $H_{t}^{c}\coloneqq
H_{t}-H_{t}^{d}-H_{t-}^{d,+}$.

For a fixed model $\theta\in\Theta$, a strategy is called _self-financing
under transaction costs $\lambda$_, if

$\int_{s}^{t}\mathrm{d}H_{u}^{0}\leq-\int_{s}^{t}S_{u}^{\theta}\mathrm{d}H_{u}^{1,\uparrow}+\int_{s}^{t}(1-\lambda)S_{u}^{\theta}\mathrm{d}H_{u}^{1,\downarrow}$
(2.1)

for all $0\leq s<t\leq T$, where the integrals

$\displaystyle\int^{t}_{s}S_{u}dH^{1,\uparrow}_{u}$
$\displaystyle:=\int^{t}_{s}S_{u}dH_{u}^{1,\uparrow,c}+\sum_{s<u\leq
t}S_{u-}\Delta H_{u}^{1,\uparrow}+\sum_{s\leq
u<t}S_{u}\Delta_{+}H_{u}^{1,\uparrow},$
$\displaystyle\int^{t}_{s}(1-\lambda)S_{u}dH_{u}^{1,\downarrow}$
$\displaystyle:=\int^{t}_{s}(1-\lambda)S_{u}dH_{u}^{1,\downarrow,c}+\sum_{s<u\leq
t}(1-\lambda)S_{u-}\Delta H^{1,\downarrow}_{u}+\sum_{s\leq
u<t}(1-\lambda)S_{u}\Delta_{+}H^{1,\downarrow}_{u}$

can be defined pathwise by using Riemann-Stieltjes integrals. For details on
the above integrals, we refer to Section 7 in [10]. In fact, the integrals
$\int_{s}^{t}\mathrm{d}H_{u}^{0}$, $\int^{t}_{s}S_{u}dH^{1,\uparrow}_{u}$ as
well as $\int^{t}_{s}(1-\lambda)S_{u}dH_{u}^{1,\downarrow}$ represent a class
of processes up to $P$-indistinguishability, which is in line with our setup,
as $H^{0}$ and $H^{1}$ already represent an equivalence class of processes
(cf., Section 1.d in [17]). Here it is also worth noting that
$\mathcal{V}(\mathbb{F},P)=\mathcal{V}(\mathbb{F}^{P},P)$. The self-financing
condition (2.1) then states that purchases and sales of the risky asset are
accounted for in the riskless position:

$\displaystyle\mathrm{d}H_{t}^{0,c}$ $\displaystyle\leq-
S_{t}^{\theta}\mathrm{d}H_{t}^{1,\uparrow,c}+(1-\lambda)S_{t}^{\theta}\mathrm{d}H_{t}^{1,\downarrow,c},\qquad
0\leq t\leq T,$ (2.2) $\displaystyle\Delta H_{t}^{0}$ $\displaystyle\leq-
S_{t-}^{\theta}\Delta H_{t}^{1,\uparrow}+(1-\lambda)S_{t-}^{\theta}\Delta
H_{t}^{1,\downarrow},\qquad 0\leq t\leq T,$ (2.3)
$\displaystyle\Delta_{+}H_{t}^{0}$ $\displaystyle\leq-
S_{t}^{\theta}\Delta_{+}H_{t}^{1,\uparrow}+(1-\lambda)S_{t}^{\theta}\Delta_{+}H_{t}^{1,\downarrow},\qquad
0\leq t\leq T.$ (2.4)

For a fixed model $\theta\in\Theta$, a self-financing strategy $H$ is
_admissible_ under transaction costs $\lambda$, if its _liquidation value_
$V^{\mathrm{liq}}(\theta,H)$ satisfies

$V_{t}^{\mathrm{liq}}(\theta,H)\coloneqq
H_{t}^{0}+(H_{t}^{1})^{+}(1-\lambda)S_{t}^{\theta}-(H_{t}^{1})^{-}S_{t}^{\theta}\geq
0,\quad\text{a.s.},$ (2.5)

for all $t\in[0,T]$. For $x>0$ and a fixed model $\theta\in\Theta$, we denote
by $\mathcal{H}^{\theta}(x)$ the set of all self-financing, admissible trading
strategies under transaction costs $\lambda$, starting with initial endowment
$(H_{0}^{0},H_{0}^{1})=(x,0)$.

In order to get towards a model-independent setup, that is, we want to
consider self-financing trading strategies that are admissible for all models
$\theta\in\Theta$, we pass to a dominating pair $(H^{0},H^{1})$ for each
trading strategy $H\in\mathcal{H}^{\theta}(x)$ where equality holds true in
(2.1). This way we only have to specify one of the holdings, e.g., the number
of stocks $H^{1}$. For a fixed model $\theta\in\Theta$ and $x>0$, we thus
define the set

$\displaystyle\mathcal{A}^{\theta}(x)\coloneqq\big{\\{}H^{1}\in\mathcal{P}(\mathbb{F})\cap\mathcal{V}(\mathbb{F},P)\colon(H^{0},H^{1})\in\mathcal{H}^{\theta}(x),\>\mathrm{d}H_{t}^{0}=-S_{t}^{\theta}\mathrm{d}H_{t}^{1,\uparrow}+(1-\lambda)S_{t}^{\theta}\mathrm{d}H_{t}^{1,\downarrow}\big{\\}}.$

This is in line with the set of admissible trading strategies considered in
the case of one single model, that is, $\Theta=\\{\theta\\}$, as discussed in
[9]. We will also refer to this as the _non-robust_ case. Moreover, by letting
$\mathcal{A}(x)\coloneqq\bigcap_{\theta\in\Theta}\mathcal{A}^{\theta}(x)$, we
obtain the analogue of the set of model-independent admissible trading
strategies given in [7].

Note that $H^{1}\in\mathcal{A}(x)$ does no longer depend on $\theta$. However,
the holdings in the bond $H^{0}$ still depend on $\theta$. We will use the
notation $H^{0,\theta}$ to indicate this dependence. In particular, we define
$H^{0,\theta}\coloneqq
x+H_{t}^{0,\theta,\uparrow}-H_{t}^{0,\theta,\downarrow}$ with

$\displaystyle
H_{t}^{0,\theta,\downarrow}\coloneqq\int_{0}^{t}S_{u}^{\theta}\mathrm{d}H_{u}^{1,\uparrow},\quad\text{and}\quad
H_{t}^{0,\theta,\uparrow}\coloneqq\int_{0}^{t}(1-\lambda)S_{u}^{\theta}\mathrm{d}H_{u}^{1,\downarrow}.$
(2.6)

Moreover, we write $V_{t}^{\mathrm{liq}}(\theta,H^{1})$ to indicate that
$H^{0}$ in (2.5) is defined via (2.6).

Now, we have everything in place to formulate the optimisation problem. For
this, we consider an investor whose preferences are modelled by a standard
utility function555That is a strictly concave, non-decreasing and continuously
differentiable function satisfying the Inada conditions
$U^{\prime}(0)=\lim_{x\to 0}U^{\prime}(x)=\infty$ and
$U^{\prime}(\infty)=\lim_{x\to\infty}U^{\prime}(x)=0$.
$U\colon(0,\infty)\to\mathbb{R}$. For a given initial capital $x>0$, the
investor wants to maximise the expected utility of terminal wealth with
respect to the worst-case scenario of all possible models. This means that the
investor wants to find the optimal strategy $\widehat{H}^{1}\in\mathcal{A}(x)$
that maximises
$\inf_{\theta\in\Theta}\mathrm{E}\big{[}U\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,H^{1}\big{)}\big{)}\big{]}$.
The value function of this _primal optimisation problem_ is denoted by

$u(x)\coloneqq\sup_{H^{1}\in\mathcal{A}(x)}\inf_{\theta\in\Theta}\mathrm{E}\big{[}U\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,H^{1}\big{)}\big{)}\big{]}.$
(2.7)

In the sequel, we answer the question whether, and under which assumptions,
the robust primal problem (2.7) admits a solution.

##  3 Main result

In the frictionless case, the Fundamental Theorem of Asset Pricing states that
the no-arbitrage condition is equivalent to the property that the price
process admits an equivalent local martingale measure (cf. Delbaen and
Schachermayer [11]). In the setting of transaction costs, the notion of
_consistent price systems_ plays a role analogous to the notion of equivalent
martingale measures in the frictionless case.

######  Definition 3.1.

Fix $0<\lambda<1$ and $\theta\in\Theta$. A strictly positive adapted càdlàg
process $S^{\theta}$ satisfies the condition $(\mathrm{CPS}^{\lambda})$ of
having a _$\lambda$ -consistent price system_, if there exists a pair of
processes $Z^{\theta}=(Z_{t}^{0,\theta},Z_{t}^{1,\theta})_{0\leq t\leq T}$,
consisting of a density process $Z^{0,\theta}=(Z_{t}^{0,\theta})_{0\leq t\leq
T}$ of an equivalent local martingale measure $Q^{\theta}\approx P$ for a
price process $\widetilde{S}^{\theta}=(\widetilde{S}^{\theta}_{t})_{0\leq
t\leq T}$ evolving in the bid-ask spread $[(1-\lambda)S^{\theta},S^{\theta}]$,
and $Z^{1,\theta}=Z^{0,\theta}\widetilde{S}^{\theta}$. In particular,
$\widetilde{S}^{\theta}$ satisfies

$\displaystyle(1-\lambda)S_{t}^{\theta}\leq\widetilde{S}_{t}^{\theta}\leq
S_{t}^{\theta},\qquad 0\leq t\leq T.$ (3.1)

We further say that $S^{\theta}$ satisfies $(\mathrm{CPS}^{\lambda})$
_locally_ , if there exists a strictly positive stochastic process
$Z^{\theta}=(Z^{0,\theta},Z^{1,\theta})$ and a localising sequence
$(\tau_{n})_{n\geq 0}$ of stopping times, such that $(Z^{\theta})^{\tau_{n}}$
is a $\lambda$-consistent prices system for the stopped process
$(\widetilde{S}^{\theta})^{\tau_{n}}$ for each $n\geq 0$. We denote the space
of all such processes by $\mathcal{Z}^{\theta}$ and
$\mathcal{Z}_{\mathrm{loc}}^{\theta}$, respectively.

We impose the existence of consistent price systems for every model
$\theta\in\Theta$.

######  Assumption 3.2.

For each $\theta\in\Theta$ and for some $0<\lambda^{\prime}<\lambda$, the
price process $S^{\theta}$ satisfies $(\mathrm{CPS}^{\lambda^{\prime}})$
locally.

In the non-robust setting, i.e., for a fixed model $\theta\in\Theta$ and
$x>0$, we define

$\displaystyle\mathcal{C}^{\theta}(x)\coloneqq\big{\\{}g\in
L_{+}^{0}(\Omega,\mathcal{F},P)\colon g\leq
V_{T}^{\mathrm{liq}}(\theta,H),\text{ for some
}H\in\mathcal{H}^{\theta}(x)\big{\\}}.$

This is the set of terminal positions $g$ that one can superreplicate with an
admissible trading strategy $H$ and initial endowment $x$. Since we are not
interested in an analysis of the dual problem on the level of stochastic
processes, we can define the dual variables simply on the level of random
variables by setting

$\displaystyle\mathcal{D}^{\theta}(y)\coloneqq\big{\\{}h\in
L_{+}^{0}(\Omega,\mathcal{F},P)\colon\mathrm{E}[gh]\leq y\text{ for all
$g\in\mathcal{C}^{\theta}(1)$}\big{\\}},\quad\text{for all $y>0.$}$ (3.2)

Note that
$\\{yZ^{0}_{T}\colon(Z^{0},Z^{1})\in\mathcal{Z}_{\mathrm{loc}}^{\theta}\\}\subseteq\mathcal{D}^{\theta}(y)$
so that $\mathcal{D}^{\theta}(y)\neq\emptyset$. Moreover, in the non-robust
case it is possible to identify a suitable set of stochastic processes as dual
variables, so-called _optional strong supermartingale deflators_ , that yields
$\mathcal{D}^{\theta}(y)$ as terminal values of such processes. We refer to
Czichowsky and Schachermayer [9] for the duality theory for portfolio
optimisation under transaction costs on the level of stochastic processes in
the non-robust setup.

Using the sets $\mathcal{C}^{\theta}(x)$ and $\mathcal{D}^{\theta}(y)$ allows
us to define the non-robust value functions. In particular, the primal and
dual value functions for the $\theta$-model are given by

$\displaystyle
u^{\theta}(x)\coloneqq\sup_{f\in\mathcal{C}^{\theta}(x)}\mathrm{E}[U(f)],\quad\text{and}\quad
j^{\theta}(y)\coloneqq\inf_{h\in\mathcal{D}^{\theta}(y)}\mathrm{E}[J(h)].$

For our purpose, we need the following assumption.

######  Assumption 3.3.

The asymptotic elasticity of $U$ is strictly less than one, that is,

$\displaystyle\mathrm{AE}(U)\coloneqq\limsup_{x\to\infty}\frac{xU^{\prime}(x)}{U(x)}<1,$

and for each model $\theta\in\Theta$, the primal value function
$u^{\theta}(x)$ is finite for _some_ $x>0$ and hence _all_ $x>0$ by concavity
of $u^{\theta}(x)$.

In the original paper of Chau and Rásonyi [7], instead of Assumption 3.3, they
use the assumption that $j^{\theta}(y)$, $y>0$, is finite for all models
$\theta\in\Theta$ (cf. Assumption 3.5 in [7]). However, note that Assumption
3.2 and Assumption 3.3 imply Assumption 3.5 in [7] (cf. [19] together with
[20] and [9]). Moreover, Assumption 3.2 is standard and Assumption 3.3 is
satisfied for most popular utility functions, like logarithmic and power
utility.

######  Remark 3.4.

Under Assumptions 3.2 and 3.3, we obtain by Theorem 3.2 in [9] that

$\displaystyle(u^{\theta})^{\prime}(\infty)=\lim_{x\to\infty}(u^{\theta})^{\prime}(x)=0,\quad\forall\theta\in\Theta,$

which may be restated as

$\displaystyle\lim_{x\to\infty}\frac{u^{\theta}(x)}{x}=0,\quad\forall\theta\in\Theta.$

The following theorem is the main result of this paper. It extends Theorem 3.6
in [7] for utility functions defined on the positive half-line from continuous
price processes to the general price processes with jumps.

######  Theorem 3.5.

Let $x>0$. Under Assumptions 3.2 and 3.3, the robust utility maximisation
problem (2.7) admits a solution, i.e., there is
$\widehat{H}^{1}\in\mathcal{A}(x)$ satisfying

$\displaystyle
u(x)=\inf_{\theta\in\Theta}\mathrm{E}\big{[}U\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,\widehat{H}^{1}\big{)}\big{)}\big{]}.$

When $U$ is bounded from above, the same conclusion holds assuming only that
there exists (at least) one $\theta^{\prime}\in\Theta$ for which
$(\mathrm{CPS}^{\lambda^{\prime}})$ locally holds true for some
$\lambda^{\prime}\in(0,\lambda)$.

##  4 Proof of the main theorem

To prove Theorem 3.5, we need the following four results. The first result
states that for a fixed model $\theta$, the value process with respect to a
consistent price system $(\widetilde{S}^{\theta},Q^{\theta})$ is an optional
strong supermartingale under $Q^{\theta}$. Optional strong supermartingales
have been introduced by Mertens [22] as a generalisation of the notion of a
càdlàg supermartingale. We recall the definition (cf., Definition 1 of
Appendix I in [13]).

######  Definition 4.1.

An optional process $X=(X_{t})_{t\geq 0}$ is an _optional strong
supermartingale_ , if:

  1. (1)

For every bounded stopping time $\tau$, $X_{\tau}$ is integrable.

  2. (2)

For every pair of bounded stopping times $\sigma$ and $\tau$, such that
$\sigma\leq\tau$, we have

$\displaystyle\mathrm{E}[X_{\tau}|\mathcal{F}_{\sigma}]\leq X_{\sigma},\quad
P\text{-a.s.}$

For further discussion of optional strong supermartingales we refer to
Appendix I in Dellacherie and Meyer [13].

######  Proposition 4.2.

For a fixed model $\theta\in\Theta$ with price process
$S^{\theta}=(S_{t}^{\theta})_{0\leq t\leq T}$, transaction costs
$0<\lambda<1$, and an admissible self-financing trading strategy
$H\in\mathcal{H}^{\theta}(x)$, $x>0$, as above, suppose that
$(Z^{0,\theta},Z^{0,\theta}\widetilde{S}^{\theta})\in\mathcal{Z}_{\mathrm{loc}}^{\theta}$
is a locally consistent price system under transaction costs $\lambda$. Then
the process $\widetilde{V}(\theta,H)=(\widetilde{V}_{t}(\theta,H))_{0\leq
t\leq T}$, defined by

$\displaystyle\widetilde{V}_{t}(\theta,H)\coloneqq
H_{t}^{0}+H_{t}^{1}\widetilde{S}_{t}^{\theta},\qquad 0\leq t\leq T,$

satisfies $\widetilde{V}_{t}(\theta,H)\geq V_{t}^{\mathrm{liq}}(\theta,H)$
almost surely, and $Z^{0,\theta}(H^{0}+H^{1}\widetilde{S}^{\theta})$ is an
optional strong supermartingale.

In the setting where the filtration $\mathbb{F}$ satisfies the usual
conditions, the proof of Proposition 4.2 is given in [28] (cf. Proposition 2).
In the current setup, where $\mathbb{F}$ is right continuous but not complete,
we may still follow the lines of the proof given in [28], but with special
care on some technicalities. In particular, we first obtain the result for
$\mathbb{F}^{P}$ by using the aforementioned result, and then pass to an
$\mathbb{F}$-optional version by using Lemma 7 of Appendix I in [13]. Note
that this is in line with our understanding of $\widetilde{V}_{t}(\theta,H)$
being an optional process, as $H^{0}$ and $H^{1}$ already are
$P$-indistinguishable from predictable processes. Since
$\mathbb{F}\subseteq\mathbb{F}^{P}$, Property (1) of Definition 4.1 follows
immediately, while Property (2) follows by using the tower property of
conditional expectations.

It is worth noting that the proof of Proposition 2 in [28] requires the usual
conditions, because it uses Theorem IV.117 in [12]. It is still possible
though to argue similarly as in [28], by using instead Theorem B together with
Remark E in [13] (cf., complements to chapter IV) and a corollary thereof,
instead of Theorem IV.117 in [12]. To conclude the proof, one would also use
Mertens decomposition (cf., Theorem 20 of Appendix I in [13]), as it is done
in [28]. However, the result of Proposition 4.2 would not change (see also
Remark 21 of Appendix I in [13]).

As a second result, we also need the following bipolar relation for the non-
robust models.

######  Lemma 4.3.

Fix $x,y>0$. Suppose that $S^{\theta}$ satisfies
$(\mathrm{CPS}^{\lambda^{\prime}})$ locally for some
$\lambda^{\prime}\in(0,\lambda)$. Then, a random variable $g\in L_{+}^{0}(P)$
satisfies $g\in\mathcal{C}^{\theta}(x)$ if and only if $\mathrm{E}[gh]\leq xy$
for all $h\in\mathcal{D}^{\theta}(y)$ for some $y>0$.

The third result we need extends Proposition 3.4 in [6] to a more general,
model-independent view.

######  Proposition 4.4.

Let $(H^{1,n})_{n\in\mathbb{N}}\subseteq\mathcal{A}(x)$ for $x>0$. Assume that
there is $\theta\in\Theta$ so that $S^{\theta}$ satisfies
$(\mathrm{CPS}^{\lambda^{\prime}})$ locally, for some
$0<\lambda^{\prime}<\lambda$. Then there exist processes $H^{1,\uparrow}$ and
$H^{1,\downarrow}$, both in
$\mathcal{P}(\mathbb{F})\cap\mathcal{V}^{+}(\mathbb{F},P)$, with
$H^{1}=H^{1,\uparrow}-H^{1,\downarrow}\in\mathcal{A}(x)$ and a sequence of
convex combinations
$(\widetilde{H}^{1,n,\uparrow},\widetilde{H}^{1,n,\downarrow})_{n\in\mathbb{N}}\subseteq\mathrm{conv}((H^{1,n,\uparrow},H^{1,n,\downarrow}),(H^{1,n+1,\uparrow},H^{1,n+1,\downarrow}),\dots)$,
such that $(\widetilde{H}^{1,n,\uparrow},\widetilde{H}^{1,n,\downarrow})$
converges for almost every $\omega$ for every $t\in[0,T]$ to
$(H^{1,\uparrow},H^{1,\downarrow})$, i.e.,

$\displaystyle
P\big{[}(\widetilde{H}_{t}^{1,n,\uparrow},\widetilde{H}_{t}^{1,n,\downarrow})\to(H_{t}^{1,\uparrow},H_{t}^{1,\downarrow}),\quad\forall
t\in[0,T]\big{]}=1.$

The proofs of Lemma 4.3 and Proposition 4.4 are given in Appendix A. In the
proof of Proposition 4.4 we will also need the following result. It shows that
the pointwise convergence of integrators of finite variation is sufficient for
the convergence of the integrals of a fixed càdlàg function. While this is not
true for the standard Riemann-Stieltjes integrals, it is true for our notion
of the integral that is motivated by self-financing trading in financial
markets (given below (2.1)). The reason for this is that for trades
immediately before a predictable stopping time, the price paid is the left
limit of the price process. The proof of Lemma 4.5 is also given in Appendix
A.

######  Lemma 4.5.

Let $x>0$ and consider the sequence
$(H^{1,n})_{n\in\mathbb{N}}\subseteq\mathcal{A}(x)$ with canonical
decomposition $H^{1,n}=H^{1,n,\uparrow}-H^{1,n,\downarrow}$. Moreover, assume
that $H^{1}=H^{1,\uparrow}-H^{1,\downarrow}$ is a process in
$\mathcal{P}(\mathbb{F})\cap\mathcal{V}(\mathbb{F},P)$ with
$H^{1,\uparrow},\>H^{1,\downarrow}\in\mathcal{P}(\mathbb{F})\cap\mathcal{V}^{+}(\mathbb{F},P)$,
such that

$\displaystyle P\big{[}H_{t}^{1,n,\uparrow}\to H_{t}^{1,\uparrow},\ \forall
t\in[0,T]\big{]}=1,\quad\text{and}\quad P\big{[}H_{t}^{1,n,\downarrow}\to
H_{t}^{1,\downarrow},\ \forall t\in[0,T]\big{]}=1.$ (4.1)

Then, for all $\theta\in\Theta$, the sequence
$(H^{0,\theta,n})_{n\in\mathbb{N}}$ with
$H^{0,\theta,n}=x+H^{0,\theta,n,\uparrow}-H^{0,\theta,n,\downarrow}$ defined
via (2.6), i.e.,

$\displaystyle
H_{t}^{0,\theta,n,\downarrow}=\int_{0}^{t}S_{u}^{\theta}\mathrm{d}H_{u}^{1,n,\uparrow},\quad\text{and}\quad
H_{t}^{0,\theta,n,\uparrow}=\int_{0}^{t}(1-\lambda)S_{u}^{\theta}\mathrm{d}H_{u}^{1,n,\downarrow},$
(4.2)

converges for almost every $\omega$ for every $t\in[0,T]$ to
$H^{0,\theta}=x-\int_{0}^{t}S_{u}^{\theta}\mathrm{d}H_{u}^{1,\uparrow}+\int_{0}^{t}(1-\lambda)S_{u}^{\theta}\mathrm{d}H_{u}^{1,\downarrow}$.
In particular, $H_{t}^{0,\theta,\uparrow}$ and $H_{t}^{0,\theta,\downarrow}$
are elements of $\mathcal{P}(\mathbb{F})\cap\mathcal{V}^{+}(\mathbb{F},P)$
satisfying

$\displaystyle P\big{[}H_{t}^{0,\theta,n,\uparrow}\to
H_{t}^{0,\theta,\uparrow},\ \forall t\in[0,T]\big{]}=1,\quad\text{and}\quad
P\big{[}H_{t}^{0,\theta,n,\downarrow}\to H_{t}^{0,\theta,\downarrow},\ \forall
t\in[0,T]\big{]}=1,$ (4.3)

and therefore $H^{1}\in\mathcal{A}(x)$.

We are now ready to prove our main result.

###### Proof of Theorem 3.5.

Let $(H^{1,n})_{n\in\mathbb{N}}\subseteq\mathcal{A}(x)$ be a maximising
sequence, i.e.,

$\displaystyle\inf_{\theta\in\Theta}\mathrm{E}\big{[}U\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,H^{1,n}\big{)}\big{)}\big{]}\nearrow
u(x),\quad\text{as }n\to\infty.$

By Proposition 4.4 there is a sequence
$(\widetilde{H}^{1,n})_{n\in\mathbb{N}}\subseteq\mathrm{conv}(H^{1,n},H^{1,n+1},\dots)$
and $\widehat{H}^{1}\in\mathcal{A}(x)$, such that, for almost every
$\omega\in\Omega$, we have that $\widetilde{H}^{1,n}_{t}$ converges for every
$t\in[0,T]$ to $\widehat{H}^{1}_{t}$. Since the utility function is concave,
we obtain that $(\widetilde{H}^{1,n})_{n\in\mathbb{N}}$ is also a maximising
sequence, because

$\displaystyle\inf_{\theta\in\Theta}\mathrm{E}\big{[}U\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,\widetilde{H}^{1,n}\big{)}\big{)}\big{]}\geq\inf_{\theta\in\Theta}\mathrm{E}\big{[}U\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,H^{1,n}\big{)}\big{)}\big{]}\to
u(x),\quad\text{as }n\to\infty.$

We claim that $\widehat{H}^{1}$ is an optimal solution to (2.7). For this
purpose, we denote by $U^{+}$ and $U^{-}$ the positive and negative parts of
the function $U$. From Fatou’s lemma, we deduce that

$\displaystyle\liminf_{n\to\infty}\mathrm{E}\big{[}U^{-}\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,\widetilde{H}^{1,n}\big{)}\big{)}\big{]}\geq\mathrm{E}\big{[}U^{-}\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,\widehat{H}^{1}\big{)}\big{)}\big{]},$

for each model $\theta\in\Theta$. The optimality of $\widehat{H}$ will follow
if we show that

$\displaystyle\lim_{n\to\infty}\mathrm{E}\big{[}U^{+}\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,\widetilde{H}^{1,n}\big{)}\big{)}\big{]}=\mathrm{E}\big{[}U^{+}\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,\widehat{H}^{1}\big{)}\big{)}\big{]},$
(4.4)

for each model $\theta\in\Theta$. If $U(\infty)\leq 0$, then there is nothing
to prove. So we assume that $U(\infty)>0$.

We claim that the sequence
$(U^{+}(V_{T}^{\mathrm{liq}}(\theta,\widetilde{H}^{1,n})))_{n\in\mathbb{N}}$
is uniformly integrable for each $\theta\in\Theta$. This is equivalent to the
validity of (4.4). Suppose by contradiction that the sequence is not uniformly
integrable for some $\theta$. Then, using the characterization of uniform
integrability given in [15], Theorem VI.16, and passing if necessary to a
subsequence still denoted by $(\widetilde{H}^{1,n})_{n\in\mathbb{N}}$, we can
find a constant $\alpha>0$ and disjoint sets $A_{n}\in\mathcal{F}$,
$n\in\mathbb{N}$, such that

$\displaystyle\mathrm{E}\big{[}U^{+}\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,\widetilde{H}^{1,n}\big{)}\big{)}\mathbbm{1}_{A_{n}}\big{]}\geq\alpha\quad\text{for
}n\geq 1.$

We define the sequence of random variables $(h^{n})_{n\in\mathbb{N}}$ by

$\displaystyle h^{n}\coloneqq
x_{0}+\sum_{k=1}^{n}V_{T}^{\mathrm{liq}}\big{(}\theta,\widetilde{H}^{1,k}\big{)}\mathbbm{1}_{A_{k}},$

where $x_{0}=\inf\big{\\{}x>0\colon U(x)\geq 0\big{\\}}$. It follows
immediately that

$\displaystyle\mathrm{E}[U(h^{n})]\geq\sum_{k=1}^{n}\mathrm{E}\big{[}U^{+}\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,\widetilde{H}^{1,k}\big{)}\big{)}\mathbbm{1}_{A_{k}}\big{]}\geq
n\alpha.$

On the other hand, for any $f\in\mathcal{D}^{\theta}(1)$ we have by
Proposition 4.2 that

$\displaystyle\mathrm{E}[h^{n}f]\leq
x_{0}+\sum_{k=1}^{n}\mathrm{E}\big{[}V_{T}^{\mathrm{liq}}\big{(}\theta,\widetilde{H}^{1,k}\big{)}f\big{]}\leq
x_{0}+nx.$

Hence, by Lemma 4.3, we obtain $h^{n}\in\mathcal{C}^{\theta}(x_{0}+nx)$.
Therefore, we have

$\displaystyle\limsup_{x\to\infty}\frac{u^{\theta}(x)}{x}\geq\limsup_{n\to\infty}\frac{\mathrm{E}[U(h^{n})]}{x_{0}+nx}\geq\limsup_{n\to\infty}\frac{n\alpha}{x_{0}+nx}=\frac{\alpha}{x}>0.$

By Remark 3.4, this is a contradiction to our Assumptions 3.2 and 3.3. As a
result,
$(U^{+}(V_{T}^{\mathrm{liq}}(\theta,\widetilde{H}^{1,n})))_{n\in\mathbb{N}}$
indeed is a uniformly integrable sequence for each $\theta\in\Theta$.

Since for almost every $\omega\in\Omega$, we have that
$\widetilde{H}^{1,n}_{t}\to\widehat{H}^{1}_{t}$ for every $t\in[0,T]$, we get
that $V_{T}^{\mathrm{liq}}(\theta,\widetilde{H}^{1,n})\to
V_{T}^{\mathrm{liq}}(\theta,\widehat{H}^{1})$ almost surely for each
$\theta\in\Theta$ by Lemma 4.5. Therefore, Fatou’s lemma and uniform
integrability imply

$\displaystyle\limsup_{n\to\infty}\Big{(}\inf_{\theta\in\Theta}\mathrm{E}\big{[}U\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,\widetilde{H}^{1,n}\big{)}\big{)}\big{]}\Big{)}$
$\displaystyle\leq\inf_{\theta\in\Theta}\Big{(}\limsup_{n\to\infty}\mathrm{E}\big{[}U\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,\widetilde{H}^{1,n}\big{)}\big{)}\big{]}\Big{)}$
$\displaystyle\leq\inf_{\theta\in\Theta}\mathrm{E}\big{[}U\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,\widehat{H}^{1}\big{)}\big{)}\big{]},$

which proves that $\widehat{H}^{1}$ is an optimal solution to (2.7).

In the case where $U$ is bounded from above, we use Fatou’s lemma to get

$\displaystyle\limsup_{n\to\infty}\mathrm{E}\big{[}U^{+}\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,\widetilde{H}^{1,n}\big{)}\big{)}\big{]}\leq\mathrm{E}\big{[}U^{+}\big{(}V_{T}^{\mathrm{liq}}\big{(}\theta,\widehat{H}^{1}\big{)}\big{)}\big{]}$

instead of (4.4). Hence, the existence of (at least) one
$\theta^{\prime}\in\Theta$ so that $S^{\theta^{\prime}}$ satisfies
$(\mathrm{CPS}^{\lambda^{\prime}})$ locally for some
$0<\lambda^{\prime}<\lambda$ is enough to obtain the optimal strategy and to
conclude the proof. ∎

##  Appendix A Proofs for Section 4

We begin with the proof of Lemma 4.3.

###### Proof of Lemma 4.3.

Under the assumption that $S^{\theta}=(S_{t}^{\theta})_{0\leq t\leq T}$
satisfies $(\mathrm{CPS}^{\lambda^{\prime}})$ locally for some
$0<\lambda^{\prime}<\lambda$, it follows as in the proof of Lemma A.1 of
Czichowsky and Schachermayer [9] that $\mathcal{C}^{\theta}(x)$ is a closed,
convex and solid subset of $L^{0}_{+}(P)$. As we defined
$\mathcal{D}^{\theta}(y)$ as the polar of $\mathcal{C}^{\theta}(x)$ in
$L^{0}_{+}(P)$ in (3.2), the biploar theorem in $L^{0}_{+}(P)$ of Brannath and
Schachermayer [5] (cf. Theorem 1.3) yields that $g\in L^{0}_{+}(P)$ satisfies
$\mathcal{C}^{\theta}(x)$ if and only if $\mathrm{E}[gh]\leq xy$ for all
$h\in\mathcal{D}^{\theta}(y)$ for some $y>0$. ∎

We continue with a result that was originally proven in Campi and
Schachermayer [6] in the setting of Kabanov and Safarian [18]. It provides an
a posteriori, quantitative control on the size of the total variation of
admissible trading strategies. We use a slightly adjusted version of this
result to cover model independent trading strategies. Our proof is mainly the
same as the one of Lemma 3.1 in [27] (see also [29], Lemma 4.10). We also
refer to Remark 3.2 in [27] which comes together with the following result.

######  Proposition A.1.

Let $x>0$. For some $\theta^{\prime}\in\Theta$ and for some
$0<\lambda^{\prime}<\lambda$, assume that $S^{\theta^{\prime}}$ satisfies
$(\mathrm{CPS}^{\lambda^{\prime}})$ locally. Then, for any strategy
$H^{1}\in\mathcal{A}(x)$ with canonical decomposition
$H^{1}=H^{1,\uparrow}-H^{1,\downarrow}$, the elements $H_{T}^{1,\uparrow}$ and
$H_{T}^{1,\downarrow}$ as well as their convex combinations are bounded in
$L^{0}(\Omega,\mathcal{F},P)$.

###### Proof.

Fix $0<\lambda^{\prime}<\lambda$. For some $\theta^{\prime}\in\Theta$ there is
by assumption a probability measure $Q^{\theta^{\prime}}\approx P$ and a local
$Q^{\theta^{\prime}}$-martingale
$\widetilde{S}^{\theta^{\prime}}=(\widetilde{S}_{t}^{\theta^{\prime}})_{0\leq
t\leq T}$ satisfying (3.1). By stopping, we may assume that
$\widetilde{S}^{\theta^{\prime}}$ is a true martingale.

We consider a strategy $H^{1}\in\mathcal{A}(x)$, for some $x>0$. Without loss
of generality we assume that $H_{T}^{1}=0$, i.e., that the position is
liquidated at time $T$. For each $\theta\in\Theta$, using (2.6), we obtain the
holdings of the bond
$H_{t}^{0,\theta}=H^{0,\theta,\uparrow}-H^{0,\theta,\downarrow}$ via
$\mathrm{d}H_{t}^{0,\theta,\uparrow}=(1-\lambda)S_{t}^{\theta}\mathrm{d}H_{t}^{1,\downarrow}$
and
$\mathrm{d}H_{t}^{0,\theta,\downarrow}=-S_{t}^{\theta}\mathrm{d}H_{t}^{1,\uparrow}$.
Now, working with $\theta^{\prime}$ as introduced before, we first show that

$\mathrm{E}_{Q^{\theta^{\prime}}}\big{[}H_{T}^{0,\theta^{\prime},\uparrow}\big{]}\leq\frac{x}{\lambda-\lambda^{\prime}}.$
(A.1)

For this purpose, define the process
$H^{\prime}=((H^{0,\theta^{\prime}})^{\prime},(H^{1})^{\prime})$ by

$\displaystyle
H^{\prime}_{t}=((H^{0,\theta^{\prime}})^{\prime}_{t},(H^{1})^{\prime}_{t})=\Big{(}H_{t}^{0,\theta^{\prime}}+\frac{\lambda-\lambda^{\prime}}{1-\lambda}H_{t}^{0,\theta^{\prime},\uparrow},H_{t}^{1}\Big{)},\qquad
0\leq t\leq T.$

This is a self-financing trading strategy under transaction costs
$\lambda^{\prime}$: indeed, whenever we have
$\mathrm{d}H_{t}^{0,\theta^{\prime}}>0$ such that
$\mathrm{d}H_{t}^{0,\theta^{\prime}}=\mathrm{d}H_{t}^{0,\theta^{\prime},\uparrow}$,
the agent sells some units of stock and receives
$\mathrm{d}H_{t}^{0,\theta^{\prime},\uparrow}=(1-\lambda)S_{t}^{\theta^{\prime}}\mathrm{d}H_{t}^{1,\downarrow}$
(resp.,
$(1-\lambda^{\prime})S_{t}^{\theta^{\prime}}\mathrm{d}H_{t}^{1,\downarrow}=\frac{1-\lambda^{\prime}}{1-\lambda}\mathrm{d}H_{t}^{0,\theta^{\prime},\uparrow}$)
many bonds under transaction costs $\lambda$ (resp., $\lambda^{\prime}$). The
difference between these two terms is
$\frac{\lambda-\lambda^{\prime}}{1-\lambda}\mathrm{d}H_{t}^{0,\theta^{\prime},\uparrow}$;
this is the amount by which the $\lambda^{\prime}$-agent does better than the
$\lambda$-agent. It is also clear that
$((H^{0,\theta^{\prime}})^{\prime},(H^{1})^{\prime})$ under transaction costs
$\lambda^{\prime}$ still is an $x$-admissible strategy for the model
$\theta^{\prime}$, i.e., $H^{\prime}\in\mathcal{H}^{\theta^{\prime}}(x)$.

By Proposition 4.2, the process
$\widetilde{V}(\theta^{\prime},H^{\prime})=(\widetilde{V}_{t}(\theta^{\prime},H^{\prime}))_{0\leq
t\leq T}$ defined by

$\displaystyle\widetilde{V}_{t}(\theta^{\prime},H^{\prime})=(H^{0,\theta^{\prime}})^{\prime}_{t}+(H^{1})^{\prime}_{t}\widetilde{S}_{t}^{\theta^{\prime}}=H_{t}^{0,\theta^{\prime}}+\frac{\lambda-\lambda^{\prime}}{1-\lambda}H_{t}^{0,\theta^{\prime},\uparrow}+H_{t}^{1}\widetilde{S}_{t}^{\theta^{\prime}}$

is an optional strong $Q^{\theta^{\prime}}$-supermartingale. We thus get

$\displaystyle\mathrm{E}_{Q^{\theta^{\prime}}}\big{[}\widetilde{V}_{T}\big{(}\theta^{\prime},H^{\prime}\big{)}\big{]}=\mathrm{E}_{Q^{\theta^{\prime}}}\big{[}H_{T}^{0,\theta^{\prime}}+H_{T}^{1}\widetilde{S}_{T}^{\theta^{\prime}}\big{]}+\frac{\lambda-\lambda^{\prime}}{1-\lambda}\mathrm{E}_{Q^{\theta^{\prime}}}\big{[}H_{T}^{0,\theta^{\prime},\uparrow}\big{]}\leq
0.$

By admissibility of $H^{1}$, we have
$H_{T}^{0,\theta^{\prime}}+H_{T}^{1}\widetilde{S}_{T}^{\theta^{\prime}}\geq-x$,
and thus we have shown (A.1).

To obtain control on $H_{T}^{0,\theta^{\prime},\downarrow}$ too, we note that
$H_{T}^{0,\theta^{\prime}}\geq-x$, since $H_{T}^{1}=0$. So we have
$H_{T}^{0,\theta^{\prime},\downarrow}\leq
x+H_{T}^{0,\theta^{\prime},\uparrow}$. Therefore, we obtain the following
estimate for the total variation
$H_{T}^{0,\theta^{\prime},\uparrow}+H_{T}^{0,\theta^{\prime},\downarrow}$ of
$H^{0,\theta^{\prime}}$:

$\displaystyle\mathrm{E}_{Q^{\theta^{\prime}}}\big{[}H_{T}^{0,\theta^{\prime},\uparrow}+H_{T}^{0,\theta^{\prime},\downarrow}\big{]}\leq
x\Big{(}1+\frac{2}{\lambda-\lambda^{\prime}}\Big{)}.$ (A.2)

Now we transfer the $L^{1}(Q^{\theta^{\prime}})$-estimate in (A.2) to an
$L^{0}(P)$-estimate. For $\varepsilon>0$, there exists
$\delta_{\theta^{\prime}}>0$, so that for $A\in\mathcal{F}$ with
$Q^{\theta^{\prime}}[A]<\delta_{\theta^{\prime}}$, we have
$P[A]<\frac{\varepsilon}{2}$. Letting
$C^{0,\theta^{\prime}}\coloneqq\frac{x}{\delta_{\theta^{\prime}}}\big{(}1+\frac{2}{\lambda-\lambda^{\prime}}\big{)}$
and applying Markov’s inequality to (A.2), we get

$\displaystyle
P\big{[}H_{T}^{0,\theta^{\prime},\uparrow}+H_{T}^{0,\theta^{\prime},\downarrow}\geq
C^{0,\theta^{\prime}}\big{]}<\frac{\varepsilon}{2},$ (A.3)

which is the desired $L^{0}(P)$-estimate. At this point we remark that (A.1)
implies that the convex hull of the functions
$H_{T}^{0,\theta^{\prime},\uparrow}$ is bounded in $L^{1}(Q^{\theta})$ and
(A.2) yields the same for $H_{T}^{0,\theta^{\prime},\downarrow}$. So by the
above reasoning we obtain that also the convex combinations of
$H_{T}^{0,\theta^{\prime},\uparrow}$ and
$H_{T}^{0,\theta^{\prime},\downarrow}$ remain bounded in $L^{0}(P)$.

As before, it follows from (2.6) that
$\mathrm{d}H_{t}^{0,\theta^{\prime},\downarrow}=S_{t}^{\theta^{\prime}}\mathrm{d}H_{t}^{1,\uparrow}$,
which can be rewritten as

$\mathrm{d}H_{t}^{1,\uparrow}=\frac{\mathrm{d}H_{t}^{0,\theta^{\prime},\downarrow}}{S_{t}^{\theta^{\prime}}}.$
(A.4)

By assumption, the trajectories of $S^{\theta^{\prime}}$ are strictly
positive. In fact, we even have for almost all trajectories
$(S_{t}^{\theta^{\prime}}(\omega))_{0\leq t\leq T}$, that $\inf_{0\leq t\leq
T}S_{t}^{\theta^{\prime}}(\omega)$ is strictly positive. Indeed,
$\widetilde{S}^{\theta^{\prime}}$ being a $Q^{\theta^{\prime}}$-martingale
with $\widetilde{S}_{T}^{\theta^{\prime}}>0$ almost surely satisfies that
$\inf_{0\leq t\leq T}\widetilde{S}_{t}^{\theta^{\prime}}$ is
$Q^{\theta^{\prime}}$-a.s. and therefore $P$-a.s. strictly positive. In
particular, for $\varepsilon>0$ we may find
$\delta^{\prime}_{\theta^{\prime}}>0$ such that

$P\Big{[}\inf_{0\leq t\leq
T}S_{t}^{\theta^{\prime}}<\delta^{\prime}_{\theta^{\prime}}\Big{]}<\frac{\varepsilon}{2}.$
(A.5)

Taking
$\eta_{\theta^{\prime}}\coloneqq\min(\delta_{\theta^{\prime}},\delta^{\prime}_{\theta^{\prime}})$
and letting
$C^{1,\theta^{\prime}}\coloneqq\frac{x}{\eta_{\theta^{\prime}}^{2}}\big{(}1+\frac{2}{\lambda-\lambda^{\prime}}\big{)}$,
we obtain from (A.3), (A.4) and (A.5) that

$P\Big{[}H_{T}^{1,\uparrow}\geq C^{1,\theta^{\prime}}\Big{]}\leq
P\Big{[}\inf_{0\leq t\leq
T}S_{t}^{\theta^{\prime}}<\delta^{\prime}_{\theta^{\prime}}\Big{]}+P\big{[}H_{T}^{0,\theta^{\prime},\downarrow}\geq
C^{0,\theta^{\prime}}\big{]}<\varepsilon.$ (A.6)

To control the term $H_{T}^{1,\downarrow}$, we observe that
$H_{T}^{1,\uparrow}-H_{T}^{1,\downarrow}=H_{T}^{1}=0$. Therefore, we may use
the estimate (A.6) of $H_{T}^{1,\uparrow}$ to also control
$H_{T}^{1,\downarrow}$. Moreover, we note that (A.4) also holds for convex
combinations of $H^{1,\uparrow}$. Indeed, for another strategy
$\widehat{H}^{1}\in\mathcal{A}(x)$ and $\alpha\in[0,1]$ we have

$\displaystyle
S_{t}^{\theta^{\prime}}\mathrm{d}((1-\alpha)H_{t}^{1,\uparrow}+\alpha\widehat{H}_{t}^{1,\uparrow})$
$\displaystyle=(1-\alpha)S_{t}^{\theta^{\prime}}\mathrm{d}H_{t}^{1,\uparrow}+\alpha
S_{t}^{\theta^{\prime}}\mathrm{d}\widehat{H}_{t}^{1,\uparrow}=(1-\alpha)\mathrm{d}H_{t}^{0,\theta^{\prime},\downarrow}+\alpha\mathrm{d}\widehat{H}_{t}^{0,\theta^{\prime},\downarrow},$

so that dividing by $S_{t}^{\theta^{\prime}}$ yields

$\displaystyle\mathrm{d}((1-\alpha)H_{t}^{1,\uparrow}+\alpha\widehat{H}_{t}^{1,\uparrow})$
$\displaystyle=\frac{\mathrm{d}((1-\alpha)H_{t}^{0,\theta^{\prime},\downarrow}+\alpha\widehat{H}_{t}^{0,\theta^{\prime},\downarrow})}{S_{t}^{\theta^{\prime}}}.$

Since (A.3) also holds for convex combinations of
$H_{T}^{0,\theta^{\prime},\uparrow}$ and
$H_{T}^{0,\theta^{\prime},\downarrow}$, we obtain that also the convex
combinations of $H_{T}^{1,\uparrow}$ and $H_{T}^{1,\downarrow}$ remain bounded
in $L^{0}(P)$. ∎

Next, we establish the proof of Lemma 4.5.

###### Proof of Lemma 4.5.

Let $\theta\in\Theta$ and fix an arbitrary $\varepsilon>0$. Also fix
$\omega\in\Omega$ so that $S^{\theta}(\omega)$ has càdlàg trajectories and
such that $H_{t}^{1,n,\uparrow}(\omega)\to H_{t}^{1,\uparrow}(\omega)$ for all
$t\in[0,T]$. Since the function
$S^{\theta}_{\cdot}(\omega):[0,T]\to\mathbb{R}$ is càdlàg, there can only be
finitely many times $0\leq\tau_{1}<\ldots<\tau_{k}\leq T$ such that
$\big{|}\Delta S^{\theta}_{\tau_{i}}(\omega)\big{|}\geq\varepsilon$. Indeed,
if there were infinitely many time points with jumps of size larger than
$\varepsilon$, the jump times would have a cluster point in the compact set
$[0,T]$, leading to a contradiction to the existence of right or left limits
of $S^{\theta}_{t}(\omega)$ at every $t\in[0,T]$.

Therefore, setting

$\displaystyle S_{t}^{\theta,\varepsilon}(\omega)\coloneqq
S_{t}^{\theta}(\omega)-\sum_{i=1}^{k}\Delta
S_{\tau_{i}}^{\theta}(\omega)\mathbbm{1}_{[\tau_{i},T]}(t),\quad t\in[0,T],$

gives a càdlàg function with $\big{|}\Delta
S^{\theta,\varepsilon}_{t}(\omega)\big{|}\leq\varepsilon$ for all $t\in[0,T]$.
Then, there are finitely many times
$\sigma_{0}=0<\sigma_{1}<\ldots<\sigma_{m-1}<T$ such that
$\big{|}S_{\sigma_{i+1}}^{\theta,\varepsilon}(\omega)-S_{\sigma_{i}}^{\theta,\varepsilon}(\omega)\big{|}>\varepsilon$,
because $S^{\theta,\varepsilon}_{\cdot}(\omega):[0,T]\to\mathbb{R}$ is càdlàg.
Indeed, if there were infinitely many such time points $(\sigma_{i})_{i\in I}$
for some infinite index set $I$, the times $(\sigma_{i})_{i\in I}$ would have
a cluster point in the compact set $[0,T]$ leading to a contradiction to the
existence of right or left limits of $S^{\theta,\varepsilon}_{t}(\omega)$ at
every $t\in[0,T]$. Because the jumps of $S^{\theta,\varepsilon}(\omega)$ are
bounded by $\varepsilon$, we obtain
$\big{|}S_{t}^{\theta,\varepsilon}(\omega)-S_{\sigma_{i}}^{\theta,\varepsilon}(\omega)\big{|}\leq
2\varepsilon$ for all $t\in[\sigma_{i},\sigma_{i+1}]$ for $i=0,\ldots,m-1$
with $\sigma_{m}:=T$. Therefore, the step functions
$S^{\theta,\varepsilon,m}(\omega)$ given by

$\displaystyle S_{t}^{\theta,\varepsilon,m}(\omega)$ $\displaystyle\coloneqq
S_{0}^{\theta,\varepsilon}(\omega)\mathbbm{1}_{\\{0\\}}(t)+\sum_{i=1}^{m}S_{\sigma_{i-1}}^{\theta,\varepsilon}(\omega)\mathbbm{1}_{(\sigma_{i-1},\sigma_{i}]}(t),\quad
t\in[0,T],$

satisfy
$\big{|}S_{t}^{\theta,\varepsilon}(\omega)-S_{t}^{\theta,\varepsilon,m}(\omega)\big{|}\leq
2\varepsilon$ for all $t\in[0,T]$, which implies

$\displaystyle\bigg{|}\int_{0}^{t}\big{(}S_{u}^{\theta,\varepsilon}(\omega)-S_{u}^{\theta,\varepsilon,m}(\omega)\big{)}\mathrm{d}H_{u}^{1,\uparrow,n}(\omega)\bigg{|}$
$\displaystyle\leq 2\varepsilon H_{T}^{1,\uparrow,n}(\omega),$

and

$\displaystyle\bigg{|}\int_{0}^{t}\big{(}S_{u}^{\theta,\varepsilon}(\omega)-S_{u}^{\theta,\varepsilon,m}(\omega)\big{)}\mathrm{d}H_{u}^{1,\uparrow}(\omega)\bigg{|}$
$\displaystyle\leq 2\varepsilon H_{T}^{1,\uparrow}(\omega).$

Moreover, because of our definition of the stochastic integral at left jumps
of the integrator given below the self-financing condition (2.1), we have that

$\displaystyle\int_{0}^{t}S_{u}^{\theta,\varepsilon,m}(\omega)\mathrm{d}H_{u}^{1,n,\uparrow}(\omega)$
$\displaystyle=\sum_{i=1}^{m}S_{\sigma_{i-1}}^{\theta,\varepsilon}(\omega)\big{(}H_{\sigma_{i}\wedge
t}^{1,n,\uparrow}(\omega)-H_{\sigma_{i-1}\wedge
t}^{1,n,\uparrow}(\omega)\big{)},$

as well as

$\displaystyle\int_{0}^{t}S_{u}^{\theta,\varepsilon,m}(\omega)\mathrm{d}H_{u}^{1,\uparrow}(\omega)$
$\displaystyle=\sum_{i=1}^{m}S_{\sigma_{i-1}}^{\theta,\varepsilon}(\omega)\big{(}H_{\sigma_{i}\wedge
t}^{1,\uparrow}(\omega)-H_{\sigma_{i-1}\wedge t}^{1,\uparrow}(\omega)\big{)},$

so that

$\displaystyle\int_{0}^{t}S_{u}^{\theta,\varepsilon,m}(\omega)\mathrm{d}H_{u}^{1,n,\uparrow}(\omega)\to\int_{0}^{t}S_{u}^{\theta,\varepsilon,m}(\omega)\mathrm{d}H_{u}^{1,\uparrow}(\omega),\quad
n\to\infty,$

as $H_{t}^{1,n,\uparrow}(\omega)\to H_{t}^{1,\uparrow}(\omega)$ for all
$t\in[0,T]$. In particular, for $n\in\mathbb{N}$ large enough, we have

$\displaystyle\bigg{|}\int_{0}^{t}S_{u}^{\theta,\varepsilon}(\omega)\mathrm{d}H_{u}^{1,n,\uparrow}(\omega)-\int_{0}^{t}S_{u}^{\theta,\varepsilon}(\omega)\mathrm{d}H_{u}^{1,\uparrow}(\omega)\bigg{|}$
$\displaystyle\qquad\leq\bigg{|}\int_{0}^{t}\big{(}S_{u}^{\theta,\varepsilon}(\omega)-S_{u}^{\theta,\varepsilon,m}(\omega)\big{)}\mathrm{d}H_{u}^{1,n,\uparrow}(\omega)\bigg{|}+\bigg{|}\int_{0}^{t}\big{(}S_{u}^{\theta,\varepsilon,m}(\omega)-S_{u}^{\theta,\varepsilon}(\omega)\big{)}\mathrm{d}H_{u}^{1,\uparrow}(\omega)\bigg{|}$
$\displaystyle\qquad\quad+\bigg{|}\int_{0}^{t}S_{u}^{\theta,\varepsilon,m}(\omega)\mathrm{d}H_{u}^{1,n,\uparrow}(\omega)-\int_{0}^{t}S_{u}^{\theta,\varepsilon,m}(\omega)\mathrm{d}H_{u}^{1,\uparrow}(\omega)\bigg{|}$
$\displaystyle\qquad\leq
4\varepsilon\big{(}H_{T}^{1,\uparrow}(\omega)+\varepsilon\big{)}+\varepsilon.$

On the other hand, the finite sum $\sum_{i=1}^{k}\Delta
S_{\tau_{i}}^{\theta}(\omega)\mathbbm{1}_{[\tau_{i},T]}(t)$, where the
$\tau_{i}$’s are the times where $\big{|}\Delta
S^{\theta}_{\tau_{i}}(\omega)\big{|}\geq\varepsilon$ for $i=1,\ldots,k$,
satisfies

$\displaystyle\int_{0}^{t}\bigg{(}\sum_{i=1}^{k}\Delta
S_{\tau_{i}}^{\theta}(\omega)\mathbbm{1}_{[\tau_{i},T]}(u)\bigg{)}\mathrm{d}H_{u}^{1,n,\uparrow}(\omega)$
$\displaystyle=\sum_{i=1}^{k}\Delta
S_{\tau_{i}}^{\theta}(\omega)\big{(}H_{t}^{1,n,\uparrow}(\omega)-H_{\tau_{i}\wedge
t}^{1,n,\uparrow}(\omega)\big{)},$ (A.7)

and

$\displaystyle\int_{0}^{t}\bigg{(}\sum_{i=1}^{k}\Delta
S_{\tau_{i}}^{\theta}(\omega)\mathbbm{1}_{[\tau_{i},T]}(u)\bigg{)}\mathrm{d}H_{u}^{1,\uparrow}(\omega)$
$\displaystyle=\sum_{i=1}^{k}\Delta
S_{\tau_{i}}^{\theta}(\omega)\big{(}H_{t}^{1,\uparrow}(\omega)-H_{\tau_{i}\wedge
t}^{1,\uparrow}(\omega)\big{)}.$ (A.8)

Here, we again exploit our definition of the stochastic integral at left jumps
of the integrator given below the self-financing condition (2.1). Since
$H_{t}^{1,n,\uparrow}(\omega)\to H_{t}^{1,\uparrow}(\omega)$ for all
$t\in[0,T]$, we thus have that

$\displaystyle\int_{0}^{t}\bigg{(}\sum_{i=1}^{k}\Delta
S_{\tau_{i}}^{\theta}(\omega)\mathbbm{1}_{[\tau_{i},T]}(u)\bigg{)}\mathrm{d}H_{u}^{1,n,\uparrow}(\omega)\to\int_{0}^{t}\bigg{(}\sum_{i=1}^{k}\Delta
S_{\tau_{i}}^{\theta}(\omega)\mathbbm{1}_{[\tau_{i},T]}(u)\bigg{)}\mathrm{d}H_{u}^{1,\uparrow}(\omega),$

as $n\to\infty$, by (A.7) and (A.8). Together with the above, it thus follows
that

$\displaystyle\int_{0}^{t}S_{u}^{\theta}(\omega)\mathrm{d}H_{u}^{1,n,\uparrow}(\omega)\to\int_{0}^{t}S_{u}^{\theta}(\omega)\mathrm{d}H_{u}^{1,\uparrow}(\omega),\quad
n\to\infty,$

for every $t\in[0,T]$, since $\varepsilon>0$ was arbitrary.

Repeating the same argument as above for $H^{1,n,\downarrow}$,
$H^{1,\downarrow}$ and $(1-\lambda)S^{\theta}$, we also obtain

$\displaystyle\int_{0}^{t}S_{u}^{\theta}(\omega)\mathrm{d}H_{u}^{1,n,\downarrow}(\omega)\to\int_{0}^{t}S_{u}^{\theta}(\omega)\mathrm{d}H_{u}^{1,\downarrow}(\omega),\quad
n\to\infty.$

Since $\omega$ can be chosen arbitrarily from a set with probability $1$, this
proves (4.3).

In order to prove that $H^{1}\in\mathcal{A}(x)$, we first note that all the
$H^{0,\theta,n,\uparrow}$ and $H^{0,\theta,n,\downarrow}$ as defined in (4.2)
are elements of $\mathcal{P}(\mathbb{F})\cap\mathcal{V}^{+}(\mathbb{F},P)$.
This follows from our definition of the stochastic integral and the fact that
$H^{1,n,\uparrow}$ as well as $H^{1,n,\downarrow}$ are elements of
$\mathcal{P}(\mathbb{F})\cap\mathcal{V}^{+}(\mathbb{F},P)$ (see also Remark
1.31 in [17]). The same argument then holds true for $H^{0,\theta,\uparrow}$
and $H^{0,\theta,\downarrow}$. It thus remains to check that
$V^{\mathrm{liq}}(\theta,H^{1})$ satisfies the admissibility condition (2.5)
for all $\theta\in\Theta$. By assumption, the processes
$(H^{1,n})_{n\in\mathbb{N}}$ are $x$-admissible for all $\theta\in\Theta$.
Hence, by (4.1) and (4.3), we get for every $t\in[0,T]$ and for each
$\theta\in\Theta$ that $V_{t}^{\mathrm{liq}}(\theta,H^{1,n})\to
V_{t}^{\mathrm{liq}}(\theta,H^{1})$ almost surely, by the continuity of the
liquidation function (2.5) with respect to $(H^{0,\theta}_{t},H^{1}_{t})$, so
that admissibility condition (2.5) passes to the limit $H^{1}$. We thus have
$H^{1}\in\mathcal{A}(x)$ and this concludes the proof. ∎

To prove Proposition 4.4, we also use the following well-known variant of
Komlós’ theorem (cf. [11], Lemma A1.1).

######  Lemma A.2.

Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of $\mathbb{R}_{+}$-valued,
random variables on a probability space $(\Omega,\mathcal{F},P)$. There is a
sequence $g_{n}\in\mathrm{conv}(f_{n},f_{n+1},\dots)$ of convex combinations
that converges almost surely to a $[0,\infty]$-valued function $g$. If
$\mathrm{conv}(f_{n},f_{n+1},\dots)$ is bounded in
$L^{0}(\Omega,\mathcal{F},P)$, then $g$ is finite almost surely.

We further need the next simple fact about the measurability of limits of
measurable functions (see, e.g., Lemma 3.5 in [31]).

######  Lemma A.3.

Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of measurable functions on a
measure space $(\Omega,\mathcal{F})$. Then, $\liminf_{n\to\infty}f_{n}$ and
$\limsup_{n\to\infty}f_{n}$ are $[-\infty,\infty]$-valued measurable functions
and the set $F:=\left\\{\omega\in\Omega:\text{$f_{n}(\omega)$ converges to a
limit in $\mathbb{R}$}\right\\}$ is $\mathcal{F}$-measurable and given by

$\displaystyle F={}$
$\displaystyle\left\\{\omega\in\Omega:\liminf_{n\to\infty}f_{n}(\omega)=\limsup_{n\to\infty}f_{n}(\omega)\in\mathbb{R}\right\\}.$

Moreover, if $\liminf_{n\to\infty}f_{n}$ and $\limsup_{n\to\infty}f_{n}$ are
measurable with respect to a sub-$\sigma$-algebra
$\mathcal{G}\subseteq\mathcal{F}$, then $F\in\mathcal{G}$.

We are now ready to prove Proposition 4.4. Here, the main difference to the
initial paper by Chau and Rásonyi [7] (cf. Lemma A.1) is the treatment of jump
times. For this, we revisit the proof of [6], Proposition 3.4666Note that,
since we are already considering the liquidation value in (2.7), we do not
need to assume that $H^{1}_{T}=0$. Therefore, we also do not need to assume
that $\mathcal{F}_{T}=\mathcal{F}_{T-}$ and $S_{T}^{\theta}=S_{T-}^{\theta}$
as in Remark 4.2 in [6].. Our key insight here is that we can show that the
set, where the convergence can fail, is the same for all models. It can again
be exhausted by countably many stopping times. However, since the filtration
$\mathbb{F}$ is not assumed to be complete, the treatment of jump times needs
special care. In particular, we need to pass from $\mathbb{F}$ to
$\mathbb{F}^{P}$, in order to apply Theorem IV.117 [12]. As a consequence, the
limit process is also only $P$-indistinguishable from an
$\mathbb{F}$-predictable process and not predictable itself.

###### Proof of Proposition 4.4.

Fix $x>0$ and let $(H^{1,n})_{n\in\mathbb{N}}\subseteq\mathcal{A}(x)$ be a
sequence of admissible, self-financing strategies. In particular,
$H^{1,n}\in\mathcal{P}(\mathbb{F})\cap\mathcal{V}(\mathbb{F},P)$ is a finite
variation process that is $P$-indistinguishable from an
$\mathbb{F}$-predictable process, and $V^{\mathrm{liq}}(\theta,H^{1,n})$
satisfies (2.5) for all $\theta\in\Theta$. As above, we decompose these
processes canonically as
$H_{t}^{1,n}=H_{t}^{1,n,\uparrow}-H_{t}^{1,n,\downarrow}$, with
$H^{1,n,\uparrow}$ and $H^{1,n,\downarrow}$ both being elements of
$\mathcal{P}(\mathbb{F})\cap\mathcal{V}^{+}(\mathbb{F},P)$. By Proposition A.1
we know that $(H_{T}^{1,n,\uparrow})_{n\in\mathbb{N}}$ and
$(H_{T}^{1,n,\downarrow})_{n\in\mathbb{N}}$ as well as their convex
combinations are bounded in $L^{0}(\Omega,\mathcal{F},P)$. Hence, let
$D\coloneqq\big{(}[0,T]\cap\mathbb{Q}\big{)}\cup\\{T\\}$ and use Lemma A.2
together with a diagonalisation procedure to obtain sequences of convex
weights $\alpha_{n}^{j}$ such that for

$\displaystyle\widetilde{H}_{t}^{1,n,\uparrow}=\sum_{j\geq
1}\alpha_{n}^{j}H_{t}^{1,n+j-1,\uparrow},\quad\widetilde{H}_{t}^{1,n,\downarrow}=\sum_{j\geq
1}\alpha_{n}^{j}H_{t}^{1,n+j-1,\downarrow},\quad t\in D,$

there exist $\mathcal{F}_{t}$-measurable random variables
$\widetilde{H}_{t}^{1,\uparrow}$ and $\widetilde{H}_{t}^{1,\downarrow}$, such
that

$\displaystyle\widetilde{H}_{t}^{1,n,\uparrow}\to\widetilde{H}_{t}^{1,\uparrow},\quad\widetilde{H}_{t}^{1,n,\downarrow}\to\widetilde{H}_{t}^{1,\downarrow},\quad\forall
t\in D,$ (A.9)

almost surely. We denote by $\widetilde{\Omega}_{0}$ the event where (A.9)
holds true so that $P[\widetilde{\Omega}_{0}]=1$. Observe now that
$q\mapsto\widetilde{H}_{q}^{1,\uparrow}(\omega)$ is non-negative and non-
decreasing over $D$ for all $\omega\in\widetilde{\Omega}_{0}$. Now, the
$\mathbb{F}$-stopping time $\sigma$, defined as

$\displaystyle\sigma\coloneqq\inf\bigg{\\{}r\in\mathbb{Q}\colon\sup_{q\leq
r,\>q\in D}\widetilde{H}_{q}^{1,\uparrow}=\infty\bigg{\\}},$

satisfies $P[\sigma=\infty]=1$. Hence, by letting
$\bar{\Omega}_{0}=\widetilde{\Omega}_{0}\cap\\{\sigma=\infty\\}$ such that
$P[\bar{\Omega}_{0}]=1$, we may define

$\displaystyle\bar{H}_{t}^{1,\uparrow}(\omega)=\lim_{\begin{subarray}{c}q\downarrow\downarrow
t,\,q\in\mathbb{Q}\end{subarray}}\widetilde{H}_{q}^{1,\uparrow}(\omega),\quad
t\in[0,T),\quad\omega\in\bar{\Omega}_{0},$ (A.10)

and $\bar{H}_{T}^{1,\uparrow}(\omega)=\widetilde{H}_{T}^{1,\uparrow}(\omega)$,
$\omega\in\bar{\Omega}_{0}$. On the other hand, if
$\omega\notin\bar{\Omega}_{0}$, we set $\bar{H}_{t}^{1,\uparrow}(\omega)=0$,
for all $t\in[0,T]$. Note that by Lemma A.3 and the right continuity of the
filtration, the process $\bar{H}^{1,\uparrow}$ obtained in this way is right
continuous and $\mathbb{F}^{P}$-adapted and hence $\mathbb{F}^{P}$-optional.
Indeed, for fixed $t\in[0,T)$, we have that
$\liminf_{\begin{subarray}{c}q\downarrow\downarrow
t,\,q\in\mathbb{Q}\end{subarray}}\widetilde{H}_{q}^{1,\uparrow}$ and
$\limsup_{\begin{subarray}{c}q\downarrow\downarrow
t,\,q\in\mathbb{Q}\end{subarray}}\widetilde{H}_{q}^{1,\uparrow}$ are
$\mathcal{F}_{t}$-measurable by the right-continuity of the filtration. By
Lemma A.3, we have

$\displaystyle F:={}$
$\displaystyle\left\\{\omega\in\Omega:\bar{H}_{t}^{1,\uparrow}(\omega)=\lim_{\begin{subarray}{c}q\downarrow\downarrow
t,\,q\in\mathbb{Q}\end{subarray}}\widetilde{H}_{q}^{1,\uparrow}(\omega)\right\\}$
$\displaystyle={}$
$\displaystyle\left\\{\omega\in\Omega:\liminf_{\begin{subarray}{c}q\downarrow\downarrow
t,\,q\in\mathbb{Q}\end{subarray}}\widetilde{H}_{q}^{1,\uparrow}(\omega)=\limsup_{\begin{subarray}{c}q\downarrow\downarrow
t,\,q\in\mathbb{Q}\end{subarray}}\widetilde{H}_{q}^{1,\uparrow}(\omega)\in\mathbb{R}\right\\}\in\mathcal{F}_{t},$

and hence $P[F]=1$, since $\bar{\Omega}_{0}\subseteq F$ and
$P[\bar{\Omega}_{0}]=1$. However, by letting $\bar{H}_{t}^{1,\uparrow}=0$
outside of $\bar{\Omega}_{0}$, we need to adjoin all the $P$-null sets to
$\mathbb{F}$, and thus we only have that $\bar{H}^{1,\uparrow}$ is
$\mathbb{F}^{P}$-adapted.

We now claim that if $(\omega,t)\in\bar{\Omega}_{0}\times(0,T)$ is such that
$t$ is a continuity point of the function
$s\mapsto\bar{H}_{s}^{1,\uparrow}(\omega)$, then
$\widetilde{H}_{t}^{1,n,\uparrow}(\omega)\to\bar{H}_{t}^{1,\uparrow}(\omega)$.
Indeed, for $\varepsilon>0$ let $q_{1}<t<q_{2}$ be rational numbers such that
$\bar{H}_{q_{2}}^{1,\uparrow}(\omega)-\bar{H}_{q_{1}}^{1,\uparrow}(\omega)<\varepsilon$.
From (A.9), there exists $N=N(\omega)\in\mathbb{N}$ such that

$\displaystyle\big{|}\widetilde{H}_{q_{1}}^{1,n,\uparrow}(\omega)-\bar{H}_{q_{1}}^{1,\uparrow}(\omega)\big{|}<\varepsilon,\quad\big{|}\widetilde{H}_{q_{2}}^{1,n,\uparrow}(\omega)-\bar{H}_{q_{2}}^{1,\uparrow}(\omega)\big{|}<\varepsilon,\quad\forall
n\geq N.$

We then estimate, for all $n\geq N$,

$\displaystyle\big{|}\widetilde{H}_{q_{2}}^{1,n,\uparrow}(\omega)-\widetilde{H}_{q_{1}}^{1,n,\uparrow}(\omega)\big{|}$
$\displaystyle\leq\big{|}\widetilde{H}_{q_{2}}^{1,n,\uparrow}(\omega)-\bar{H}_{q_{2}}^{1,\uparrow}(\omega)\big{|}+\big{|}\bar{H}_{q_{2}}^{1,\uparrow}(\omega)-\bar{H}_{q_{1}}^{1,\uparrow}(\omega)\big{|}$
$\displaystyle\quad+\big{|}\bar{H}_{q_{1}}^{1,\uparrow}(\omega)-\widetilde{H}_{q_{1}}^{1,n,\uparrow}(\omega)\big{|}$
$\displaystyle<3\varepsilon.$

Therefore, using monotonicity of $\widetilde{H}^{1,n,\uparrow}$, we obtain for
all $n\geq N(\omega)$ that

$\displaystyle\big{|}\widetilde{H}_{t}^{1,n,\uparrow}(\omega)-\bar{H}_{t}^{1,\uparrow}(\omega)\big{|}$
$\displaystyle\leq\big{|}\widetilde{H}_{t}^{1,n,\uparrow}(\omega)-\widetilde{H}_{q_{2}}^{1,n,\uparrow}(\omega)\big{|}+\big{|}\widetilde{H}_{q_{2}}^{1,n,\uparrow}(\omega)-\bar{H}_{q_{2}}^{1,\uparrow}(\omega)\big{|}$
$\displaystyle\quad+\big{|}\bar{H}_{q_{2}}^{1,\uparrow}(\omega)-\bar{H}_{t}^{1,\uparrow}(\omega)\big{|}$
$\displaystyle\leq\big{|}\widetilde{H}_{q_{1}}^{1,n,\uparrow}(\omega)-\widetilde{H}_{q_{2}}^{1,n,\uparrow}(\omega)\big{|}+\big{|}\widetilde{H}_{q_{2}}^{1,n,\uparrow}(\omega)-\bar{H}_{q_{2}}^{1,\uparrow}(\omega)\big{|}$
$\displaystyle\quad+\big{|}\bar{H}_{q_{2}}^{1,\uparrow}(\omega)-\bar{H}_{q_{1}}^{1,\uparrow}(\omega)\big{|}$
$\displaystyle<5\varepsilon.$

For $t=T$, the convergence of
$\widetilde{H}_{T}^{1,n,\uparrow}(\omega)\to\bar{H}_{T}^{1,\uparrow}(\omega)$
follows from (A.9) and the identity
$\bar{H}_{T}^{1,\uparrow}=\widetilde{H}_{T}^{1,\uparrow}$ on
$\bar{\Omega}_{0}$.

The process $\bar{H}^{1,\uparrow}$ is not yet the desired limit because we
still have to ensure the convergence at the jumps times of
$\bar{H}^{1,\uparrow}$. Since $\bar{H}^{1,\uparrow}$ is right continuous and
$\mathbb{F}^{P}$-adapted, there exists a sequence
$(\hat{\tau}_{k})_{k\in\mathbb{N}}$ of $[0,T]\cup\\{\infty\\}$-valued
$\mathbb{F}^{P}$-stopping times exhausting the jumps of the process
$\bar{H}^{1,\uparrow}$. This uses Theorem IV.117 [12] and the fact that
$\mathbb{F}^{P}$ is complete. By Theorem IV.59 [12], there also exists a
sequence $(\tau_{k})_{k\in\mathbb{N}}$ of $\mathbb{F}$-stopping times,
satisfying $\tau_{k}=\hat{\tau}_{k}$, $P$-a.s., for all $k\in\mathbb{N}$.
Hence, by passing once more to convex combinations, we may also assume that
$(\widetilde{H}_{\tau_{k}}^{1,n,\uparrow})$ converges almost surely on
$\\{\tau_{k}\leq T\\}$ for every $k\in\mathbb{N}$. We can therefore set

$\displaystyle\widehat{\Omega}_{0}\coloneqq\Big{\\{}\omega\in\bar{\Omega}_{0}\colon\widetilde{H}_{\tau_{k}(\omega)}^{1,n,\uparrow}(\omega)\text{
converges to a limit in $\mathbb{R}$ for all }k\Big{\\}},$ (A.11)

and still have $P[\widehat{\Omega}_{0}]=1$. Note that, for
$\omega\in\widehat{\Omega}_{0}$, the convergence in (A.11) together with
(A.10) implies that $\widetilde{H}_{t}^{1,n,\uparrow}(\omega)$ converges to a
limit in $\mathbb{R}$ for all $t\in[0,T].$ Finally, we define $H^{1,\uparrow}$
by setting
$H^{1,\uparrow}_{t}(\omega)=\lim_{n\to\infty}\widetilde{H}_{t}^{1,n,\uparrow}(\omega)$
on the set

$G:=\left\\{(\omega,t)\in\Omega\times[0,T]:\text{
$\widetilde{H}^{1,n,\uparrow}_{t}(\omega)$ converges to a limit in
$\mathbb{R}$}\right\\},$

and $H^{1,\uparrow}_{t}(\omega)=0$ on $G^{c}$. This yields an
$\mathbb{F}^{P}$-predictable process $H^{1,\uparrow}$ by Lemma A.3, since the
processes $\widetilde{H}^{1,n,\uparrow}$ are $P$-indistinguishable from an
$\mathbb{F}$\- and therefore $\mathbb{F}^{P}$-predictable process, and because
$\mathbb{F}^{P}$ is complete. Moreover, since for
$\omega\in\widehat{\Omega}_{0}$ we have that
$\widetilde{H}^{1,n,\uparrow}_{t}(\omega)$ converges for all $t\in[0,T]$, and
the mapping $t\mapsto\widetilde{H}^{1,n,\uparrow}_{t}(\omega)$ is non-
decreasing for all $n$, we have that $t\mapsto H^{1,\uparrow}_{t}(\omega)$ is
non-decreasing for all $\omega\in\Omega_{0}\subseteq\widehat{\Omega}_{0}$ with
$P[\Omega_{0}]=1$. In particular, we have that
$H^{1,\uparrow}\in\mathcal{P}(\mathbb{F}^{P})\cap\mathcal{V}^{+}(\mathbb{F}^{P},P)$.
Hence, by Proposition I.1.1 [17] (see also Theorem IV.78 in [12]) and the fact
that $\mathcal{V}(\mathbb{F}^{P},P)=\mathcal{V}(\mathbb{F},P)$ (cf., Remark
0.37 in [17]), we finally obtain that
$H^{1,\uparrow}\in\mathcal{P}(\mathbb{F})\cap\mathcal{V}^{+}(\mathbb{F},P)$.

The case $H^{1,\downarrow}$ is treated analogously. In particular, we obtain
two processes, $H^{1,\uparrow}$ and $H^{1,\downarrow}$, both
$P$-indistinguishable from an increasing, $\mathbb{F}$-predictable process
bounded in $L^{0}$, such that

$\displaystyle P\big{[}\widetilde{H}_{t}^{1,n,\uparrow}\to
H_{t}^{1,\uparrow},\ \forall t\in[0,T]\big{]}=1,\quad\text{and}\quad
P\big{[}\widetilde{H}_{t}^{1,n,\downarrow}\to H_{t}^{1,\downarrow},\ \forall
t\in[0,T]\big{]}=1.$

To conclude the proof, define the process $H^{1}$ as $(H_{t}^{1})_{0\leq t\leq
T}=(H_{t}^{1,\uparrow}-H_{t}^{1,\downarrow})_{0\leq t\leq T}.$ Since
$H^{1,\uparrow}$ and $H^{1,\downarrow}$ both are elements of
$\mathcal{P}(\mathbb{F})\cap\mathcal{V}^{+}(\mathbb{F},P)$, the process
$H^{1}$ is in $\mathcal{P}(\mathbb{F})\cap\mathcal{V}(\mathbb{F},P)$. It
remains to check that $H^{1}\in\mathcal{A}(x)$. By construction, the processes
$(\widetilde{H}^{1,n})_{n\in\mathbb{N}}$, defined via the decomposition
$\widetilde{H}^{1,n}=\widetilde{H}^{1,n,\uparrow}-\widetilde{H}^{1,n,\downarrow}$,
are $x$-admissible for all $\theta\in\Theta$. In particular, the process
$H^{1}\in\mathcal{P}(\mathbb{F})\cap\mathcal{V}(\mathbb{F},P)$ and the
sequence $(\widetilde{H}^{1,n})_{n\in\mathbb{N}}\subseteq\mathcal{A}(x)$
satisfy (4.1), so Lemma 4.5 implies that $H^{1}\in\mathcal{A}(x)$. This
finishes the proof. ∎

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