# Signature Methods in Stochastic Portfolio Theory

Christa Cuchiero  Vienna University, Department of Statistics and Operations
Research, Data Science Uni Vienna, Kolingasse 14-16 1, A-1090 Wien, Austria,
<EMAIL_ADDRESS>Janka Möller Vienna University, Department of
Statistics and Operations Research, Kolingasse 14-16, A-1090 Wien, Austria,
<EMAIL_ADDRESS>
The authors gratefully acknowledge financial support through grant Y 1235 of
the Austrian Science Fund.

###### Abstract

In the context of stochastic portfolio theory we introduce a novel class of
portfolios which we call linear path-functional portfolios. These are
portfolios which are determined by certain transformations of linear functions
of a collections of feature maps that are non-anticipative path functionals of
an underlying semimartingale. As main example for such feature maps we
consider the signature of the (ranked) market weights. We prove that these
portfolios are universal in the sense that every continuous, possibly path-
dependent, portfolio function of the market weights can be uniformly
approximated by signature portfolios. We also show that signature portfolios
can approximate the growth-optimal portfolio in several classes of non-
Markovian market models arbitrarily well and illustrate numerically that the
trained signature portfolios are remarkably close to the theoretical growth-
optimal portfolios. Besides these universality features, the main numerical
advantage lies in the fact that several optimization tasks like maximizing
(expected) logarithmic wealth or mean-variance optimization within the class
of linear path-functional portfolios reduce to a convex quadratic optimization
problem, thus making it computationally highly tractable. We apply our method
also to real market data based on several indices. Our results point towards
out-performance on the considered out-of-sample data, also in the presence of
transaction costs.

Keywords: signature methods, linear feature maps in machine learning,
stochastic portfolio theory, growth-optimal portfolio, portfolio selection,
convex quadratic optimization
MSC (2020) Classification: 91G10, 60L10, 90C20, 62P05

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 The Signature of a Continuous Semimartingale
    2. 2.2 Universal Approximation Theorem for Linear Functions on the Signature
    3. 2.3 Global Universal Approximation Theorem for Linear Functions on the Signature
    4. 2.4 The Johnson-Lindenstrauss Lemma
    5. 2.5 The Market and Important Portfolio Specifications
  3. 3 (Linear) Path-Functional Portfolios
  4. 4 Approximation Properties of Signature Portfolios
    1. 4.1 Universal Approximation of the Growth-Optimal Portfolio
    2. 4.2 A Class of Non-Markovian Markets where the Growth-Optimal Portfolio is a Signature Portfolio
  5. 5 Optimization Tasks for Linear Path-Functional Portfolios
    1. 5.1 Mean-Variance Optimization
    2. 5.2 Optimizing the (Expected) Log-(Relative)-Wealth
    3. 5.3 Constraints and Regularization
  6. 6 Transaction Costs for Portfolios with Short-Selling and Without Bank Account
  7. 7 Numerical Results
    1. 7.1 Simulated Data
    2. 7.2 Real Market Data
      1. 7.2.1 Details on the Optimization and Investment Procedure
      2. 7.2.2 Rank-based approach: NASDAQ
      3. 7.2.3 Name-based approach: SMI and S&P500
  8. A Proofs of Section 2
  9. B Proofs of Section 4
  10. C Proofs of Section 5
  11. D Proofs of Section 6

##  1 Introduction

Portfolio theory and portfolio optimization are a core topic in mathematical
finance and have been the subject of vivid research for decades. The most
prominent example is of course the mean-variance optimization problem as
formulated by Harry Markowitz, the founder of _Modern Portfolio Theory_ (see
[54]). His framework takes into account risk-preferences of investors while
offering a tractable form of optimization, namely being convex and quadratic.
Although the choice of constant portfolio weights is arguably oversimplified,
the elegance of the Markowitz model can not be denied. Ever since, researchers
have been striving for more and more realistic models with fewer limiting
assumptions while hoping to preserve tractability.

In the spirit of coming up with more realistic assumptions, for instance
without the need of claiming a specific form of the hardly measurable drift,
Robert Fernholz developed _Stochastic Portfolio Theory_ (SPT) (see [27] and
also [29]). A key feature of SPT, again in the realm of more realistic
modeling assumptions, is the rather relaxed no-arbitrage condition. Indeed,
the price process is only assumed to be a (continuous) semimartingale
satisfying _No Unbounded Profit with Bounded Risk_ and not necessarily the
stronger condition of _No free lunch with vanishing risk_ (see [24]). It is
also central that the portfolios’ performance is measured with respect to a
benchmark portfolio, which is usually chosen to be the market portfolio, i.e.
large indices like the S&P500. At least since the introduction of very liquid
ETFs replicating market indices the out-performance of the market is indeed a
major challenge for investors. In this context Fernholz introduced
_functionally generated portfolios_ and derived the so-called _master formula_
describing the relative wealth process of such portfolios and allowing for the
detection for relative arbitrages with respect to the market portfolio. These
portfolios are constructed via the log-gradient of a function of the market
weights. While appreciating the beauty of the framework and also its
robustness in view of certain optimal worst case long-run growth rates (see
[46, 39]), one may point out that the log-gradient form as well as the neglect
of past information is somewhat restrictive. To reduce these limitations
various generalizations of the original framework have been developed, e.g.,
towards replacing the market weights by market-to-book ratios [47],
considering trading strategies generated by Lyapunov functions [45], adding
extra information in the form of a finite variation process [59] and
incorporating past information in a path-dependent setting [60, 44]. In this
context it is worth mentioning that the latter two papers do not work in a
stochastic setting but rather in a path-wise one, using functional Itô-
calculus as in [14, 25] based on Föllmer integration. An alternative path-wise
formulation allowing for more general strategies (beyond gradient type) and
building on the theory of rough path has been studied in [2].

In this paper, we consider for simplicity the stochastic setting of continuous
semimartingales (even though a path-wise formulation in the spirit of [2]
would also be possible), take inspiration from the functionally generated
portfolios and generalize them to what we call _path-functional portfolios_.
These are portfolios constructed via an auxiliary portfolio $\tau$ and non-
anticipative path-functionals with a general semimartingale as input. In doing
so, we get rid of any gradient-type form of the functionals and allow for a
general benchmark portfolio $\tau$ and past information of the market or other
exogenous relevant factors in form of a general continuous semimartingale. The
choice of continuous semimartingales is here only for the ease of exposition,
but could be replaced by any other (predictable stochastic) process.

One of our main interest concerns _tractable portfolio optimization_ in the
framework of SPT using such path-functional portfolios. To do so, we focus on
_linear path-functional portfolios_ , which are constructed via a linear
combination of possibly path-dependent feature maps and constant optimization
parameters.

This allows us to incorporate modern machine learning techniques for time-
series data, in particular _signature methods_ (see e.g. [49, 48, 42, 52, 6,
9, 11, 57, 4, 17, 20, 16, 33, 13, 1, 18, 19] and the references therein), but
also other tools, like random neural networks [37]. In the current paper, our
focus lies on signature methods, which play an important role in rough path
theory (see e.g. [53, 31]) and whose appeal stems from the _(global) universal
approximation theorem._ It states that linear functions on the signature can
approximate continuous (with respect to certain variation distances) path-
functionals arbitrarily well on compact sets of paths or even globally when
using the setup of weighted spaces (see [22]). This result then motivates the
notion of _signature portfolios,_ which are linear path-functional portfolios
with feature maps being elements of the signature. Indeed, with this choice we
can approximate generic path-functional portfolios including functionally
generated portfolios and the growth-optimal portfolio in a large class of non-
Markovian markets, arbitrary well.

Despite this versatility, signature portfolios and linear path-functional
portfolios in general lead to remarkably tractable optimization tasks. More
concretely, we show that maximizing the (expected) logarithmic wealth as well
as the mean-variance optimization of portfolio returns lead to _convex
quadratic optimization problems._ We would like to highlight that the
Markowitz portfolio itself is included in the class of signature portfolios
and moreover, that any chosen benchmark portfolio $\tau$ can always be
_attained_ by signature portfolios constructed via $\tau$.

In addition to our theoretical results, we optimize signature portfolios in
numerical experiments using simulated and real market data. By means of
simulated data generated from the Black-Scholes model, volatility stabilized
models [28] and so-called signature market models, we illustrate that the
trained signature portfolios (of small degree) are remarkably close to the
theoretical growth-optimal portfolios.

In the real market situation we additionally incorporate transaction costs in
our optimization by proposing a regularization under which the optimization
problems remain convex and quadratic. To deal with high-dimensional market
indices such as the NASDAQ and S&P500, we use dimension reduction techniques
leading to what we call randomized signature portfolios and JL-signature
portfolios, where the former rely on randomized signatures (see [1, 13, 18,
19] and also [12] for a connection to neural signature kernels), while the
latter are based on an explicit Johnson-Lindenstrauss projection. Note that we
do not reduce the dimension of the underlying process, but only of its
signature. To make the computation of the JL-signature portfolios feasible, we
present a memory efficient algorithm to do so. We train the portfolios in both
a log-relative wealth and mean-variance optimization in the NASDAQ, the SMI
and S&P500 markets. We use randomized- and JL-signature portfolios in the
high-dimensional markets (NASDAQ and S&P500) and signature portfolios for the
SMI. In the NASDAQ market the portfolios are computed from the ranked market
weights, while in the other two cases the name-based market weights are used.

We incorporate proportional transaction costs of 1% and 5% respectively in the
name-based markets. In almost all configurations, our trained portfolios
outperform the market portfolios during the out-of-sample period, even under
5% of proportional transaction costs. It is worth noting that in the setting
with transaction costs we still re-balance our portfolios daily whereas the
market portfolio _does not_ pay any transaction costs.

To provide more context to our findings, we highlight two papers which are
particularly related to our research, namely [10] and [33]. In [10], the
authors study portfolio optimization in SPT over a family of rank-based
functionally generated portfolios parameterized by an exponentially concave
function. Their objective is to maximize the relative logarithmic growth rate,
which in their parameterization leads to a convex optimization problem. In
their empirical study they use (ranked) market data of the 100 largest US
stocks, which they manage to out-perform during the out-of-sample period with
their trained portfolios. As they do not include transaction costs in the
learning procedure (which however could be done by adopting e.g. similar
transaction cost treatments as ours), the out-performance does no longer work
with 0.45% of proportional transaction costs. This could also be related to
the fact that the considered portfolios are long only functionally generated
and thus a rather small class. Indeed, the class of portfolios we consider is
more general in several respects: we allow for short-selling, the inclusion of
information from the past and from exogenous signals as well as for a general
benchmark portfolio $\tau$ in the construction of the portfolios. At the same
time the portfolios of [10] can be approximated arbitrarily well using
signature portfolios, which is a result of the universal approximation
theorem.

Very recently, the work of [33] appeared, where the authors study mean-
variance optimization of the wealth process using signature methods. In
contrast to our multiplicative setting, they consider an additive approach
where the trading strategies correspond to numbers of shares, which are then
directly parameterized as linear functions on the signature. This means in
particular that the strategies are only self-financing if a bank-account is
included, while in our case the strategies are always self-financing _without_
a bank-account, since we use portfolio weights. Even though the formulation of
[33], where the _wealth process_ itself is the quantity of interest, is the
standard one in the literature of the continuous-time mean-variance portfolio
selection problem (see e.g. [62]), it differs from the original approach of
Markowitz [54], where mean-variance optimization of the _returns_ was actually
the objective. The latter is advantageous when optimizing (in a model-free
setup) along the observed trajectory since the returns are more likely to be
stationary than the assets themselves. For these reasons we use returns and a
multiplicative approach, which allows in particular to include the _classical
Markowitz portfolio_ (as a very special case) in our setup (which is not
possible in [33]). Another difference is that we consider general _linear_
path-functionals and not only signature portfolios, while still obtaining a
convex quadratic optimization problem. Additionally we include transaction
costs and dimension reduction techniques to be able to deal with
500-dimensional price processes, whereas in the numerical implementations of
[33] only two-dimensional price processes are considered. Let us also point
out that in [33] the signal is augmented with the two-dimensional lead-lag
process to express Itô-integrals as linear functions on the corresponding
signature. We usually only augment with time, which – under certain conditions
on the quadratic variation – still allows to get linear expressions for the
Itô-integrals (see Remark 5.12). As the computation of the signature becomes
expensive for higher dimensions, this might be of relevance.

The remainder of the article is structured as follows. In Section 2 we recall
the notion of the signature of continuous semimartingales, provide two
versions of the universal approximation theorem, state the Johnson-
Lindenstrauss Lemma and specify the financial market setting as well as
classes of important portfolios. Section 3 is dedicated to the introduction of
(linear) path-functional portfolios with signature portfolios as special case,
while Section 4 addresses their approximation properties, in particular in
view of approximating the growth optimal portfolio. In Section 5 the convex
quadratic optimization tasks for the mean-variance and (expected)
log-(relative) wealth are discussed and Section 6 explains the treatment of
proportional transaction costs. Section 7 concludes the paper with a
presentation of the numerical results. Some proofs are gathered in the
Appendix.

##  2 Preliminaries

###  2.1 The Signature of a Continuous Semimartingale

In this section, we first introduce the signature of an
$\mathbb{R}^{n}$-valued continuous semimartingale $X$ and then state the so-
called _universal approximation theorem_ (see Theorem 2.11), which motivates
the use of signature methods. We start by introducing several preliminary
notions needed to define the signature. Similar introductions can for instance
be found in [51, 31].

######  Definition 2.1 ((Truncated) Extended Tensor Algebra and Group).

The signature of $X$, denoted by $\mathbb{X}$, is an element of the extended
tensor algebra $T((\mathbb{R}^{n}))$ defined as

$T((\mathbb{R}^{n})):=\left\\{(a_{0},a_{1},...)|a_{k}\in(\mathbb{R}^{n})^{\otimes
k},k\geq 0\right\\}$

with $(\mathbb{R}^{n})^{\otimes 0}:=\mathbb{R}$. For $\mathbf{a},\mathbf{b}\in
T((\mathbb{R}^{n}))$ we define the following operations:

$\mathbf{a}+\mathbf{b}=(a_{0}+b_{0},a_{1}+b_{1},\dots),$
$\mathbf{a}\otimes\mathbf{b}=(c_{0},c_{1},\dots)\textrm{ for
}c_{k}=\sum_{j=0}^{k}a_{j}\otimes b_{k-j},$ $\lambda\mathbf{a}=(\lambda
a_{0},\lambda a_{1},\dots)\textrm{ for }\lambda\in\mathbb{R}.$

Note that $T((\mathbb{R}^{n}))$ forms a real non-commutative unitial algebra
under these operations with unit element $\mathbf{1}=(1,0,0,\dots)$. Moreover,
$T_{1}((\mathbb{R}^{n}))\subset T((\mathbb{R}^{n}))$, defined as

$T_{1}((\mathbb{R}^{n})):=\left\\{\mathbf{a}\mid\mathbf{a}\in
T((\mathbb{R}^{n}))\textrm{ with }a_{0}=1\right\\}$

is a group under the operation $\otimes$ with unit element
$\mathbf{1}=(1,0,0,\dots)$. For $\mathbf{a}\in T_{1}((\mathbb{R}^{n}))$ its
inverse is defined as

$\mathbf{a}^{-1}=\sum_{k\geq 0}\left(\mathbf{1}-\mathbf{a}\right)^{k}.$

We define the _truncated_ tensor algebra at level $N$ by

$T^{N}(\mathbb{R}^{n}):=\left\\{(a_{0},a_{1},...,a_{N})|a_{k}\in(\mathbb{R}^{n})^{\otimes
k},0\leq k\leq N\right\\}$

and denote by $T_{1}^{N}(\mathbb{R}^{n})$ the corresponding _truncated_ tensor
group. We denote by $(e_{1},...,e_{n})$ the canonical basis of
$\mathbb{R}^{n}$. Let $I=(i_{1},...,i_{m})$ be a multi-index and denote its
length by $|I|=m$, then $e_{I}:=e_{i_{1}}\otimes...\otimes e_{i_{m}}$ a basis
element of $(\mathbb{R}^{n})^{\otimes m}$. We use the convention that $|I|=0$
means that $I=\emptyset$, the empty word. In this formulation, we can write
any $\mathbf{a}\in T^{N}(\mathbb{R}^{n})$ as

$\mathbf{a}=\sum_{0\leq|I|\leq N}\langle e_{I},\mathbf{a}\rangle e_{I},$

where $\langle\cdot,\cdot\rangle$ is defined as the inner product of
$(\mathbb{R}^{n})^{\otimes k}$ for each $k\geq 0$. Finally we introduce the
two canonical projections

$\displaystyle\Pi_{k}:T((\mathbb{R}^{n}))$
$\displaystyle\rightarrow(\mathbb{R}^{n})^{\otimes k}$
$\displaystyle\mathbf{a}$ $\displaystyle\mapsto a_{k},$
$\displaystyle\Pi_{\leq N}:T((\mathbb{R}^{n}))$ $\displaystyle\rightarrow
T^{N}(\mathbb{R}^{n})$ $\displaystyle\mathbf{a}$
$\displaystyle\mapsto(a_{0},...,a_{N}).$

Of course, these projections can also be applied on the truncated tensor
algebra, i.e.
$\Pi_{k}:T^{N}(\mathbb{R}^{n})\rightarrow(\mathbb{R}^{n})^{\otimes k}$ for
$k\leq N$ and $\Pi_{\leq N}:T^{M}(\mathbb{R}^{n})\rightarrow
T^{N}(\mathbb{R}^{n})$ for $M\geq N$.

We define accordingly $T_{0}^{N}(\mathbb{R}^{N}):=\\{\mathbf{a}\in
T^{N}(\mathbb{R}^{d})\lvert a_{0}=0\\}$. In fact, it holds that
$T_{1}^{N}(\mathbb{R}^{d})$ is a Lie group under the tensor product $\otimes$
and $T_{0}^{N}(\mathbb{R}^{d})$ is a Lie algebra with Lie bracket
$T_{0}^{N}(\mathbb{R}^{d})\times
T_{0}^{N}(\mathbb{R}^{d})\ni(\mathbf{a},\mathbf{b})\mapsto[\mathbf{a},\mathbf{b}]:=\Pi_{\leq
N}(\mathbf{a}\otimes\mathbf{b}-\mathbf{b}\otimes\mathbf{a})\in
T_{0}^{N}(\mathbb{R}^{d})$. The two are related by the exponential map

$\displaystyle\exp:T_{0}^{N}(\mathbb{R}^{d})$ $\displaystyle\rightarrow
T_{1}^{N}(\mathbb{R}^{d})$ $\displaystyle\mathbf{a}$
$\displaystyle\mapsto\Pi_{\leq
N}\left(1+\sum_{k=1}^{N}\frac{\mathbf{a}^{\otimes k}}{k!}\right).$

We call
$\mathfrak{g}^{N}(\mathbb{R}^{d})=\mathbb{R}^{d}\oplus[\mathbb{R}^{d},\mathbb{R}^{d}]\oplus\dots\oplus[\mathbb{R}^{d},[\dots,[\mathbb{R}^{d},\mathbb{R}^{d}]]]$
the _free step-N nilpotent Lie algebra_ and
$G^{N}(\mathbb{R}^{d})=\exp(\mathfrak{g}^{N}(\mathbb{R}^{d}))$ the _free
nilpotent group of step N_.

We are now ready to define the signature of a continuous semimartingale. For
an introduction of the signature of a continuous semimartingale in a similar
spirit, see [17]. To this end, consider a finite time-horizon $T>0$ and some
filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P})$.

######  Definition 2.2 (Signature of a Continuous Semimartingale).

For an $\mathbb{R}^{n}$-valued continuous semimartingale $X$, we denote its
signature by $\mathbb{X}$ and define $\mathbb{X}$ as the process in
$T_{1}((\mathbb{R}^{n}))$ whose value at time $t$ is given via

$\displaystyle\Pi_{k}(\mathbb{X}_{t})=\int_{0\leq u_{1}\leq\dots\leq u_{k}\leq
t}\otimes\circ dX_{u_{1}}\dots\otimes\circ dX_{u_{k}},$

where $\circ$ denotes Stratonovich integration. Accordingly, we define the
truncated signature at level $N$, which we denote by $\mathbb{X}^{N}$ as

$\mathbb{X}^{N}_{t}=\Pi_{\leq N}(\mathbb{X}_{t}).$

Note that $\mathbb{X}^{N}_{t}$ takes values in $G^{N}(\mathbb{R}^{n})$ for all
$t\in[0,T]$. We define the increments of the signature $\mathbb{X}_{s,t}$ by
$\mathbb{X}_{s,t}:=\mathbb{X}_{s}^{-1}\otimes\mathbb{X}_{t}$ and note that

$\displaystyle\Pi_{k}(\mathbb{X}_{s,t})=\int_{s\leq u_{1}\leq\dots\leq
u_{k}\leq t}\otimes\circ dX_{u_{1}}\dots\otimes\circ dX_{u_{k}},$

in particular, $\Pi_{1}(\mathbb{X}_{s,t})=X_{t}-X_{s}$.

Let us now state a few important and useful properties about the signature.

######  Lemma 2.3 (Shuffle Product Property).

For two multiindices $I$, $J$ we define the shuffle product $\shuffle$ as

$e_{I}\shuffle e_{\emptyset}=e_{\emptyset}\shuffle e_{I}=e_{I}$ $(e_{I}\otimes
e_{i})\shuffle(e_{J}\otimes e_{j})=(e_{I}\shuffle(e_{J}\otimes e_{j}))\otimes
e_{i}+((e_{I}\otimes e_{i})\shuffle e_{J})\otimes e_{j}.$

Then, for any two multiindices $I$, $J$ and any continuous semimartingale $X$,
it holds that

$\langle e_{I},\mathbb{X}\rangle\langle e_{J},\mathbb{X}\rangle=\langle
e_{I}\shuffle e_{J},\mathbb{X}\rangle.$

###### Proof.

This follows directly from the properties of the Stratonovich integral. ∎

######  Lemma 2.4 (Uniqueness of the Signature).

Let $X$ be a continuous semimartingale with at least one strictly monotone
component. Then the signature uniquely determines $X$ up to vertical
translations of the trajectories.

###### Proof.

This follows from the fact, that the signature determines the underlying
semimartingale uniquely up to tree-like equivalences, see [8]. A more direct
proof, in the case where that component is the time, can be found e.g. in
[20]. ∎

We now define a class of functions which plays a major role in this paper,
namely _linear functions on the signature_ :

######  Definition 2.5 (Linear Functions on the Signature).

Let $X$ be an $\mathbb{R}^{n}$-valued continuous semimartingale and
$\mathbb{X}$ its signature. We call a function
$L:T((\mathbb{R}^{n}))\rightarrow\mathbb{R}$ a _linear function on the
signature_ if $L$ is given by

$L(\mathbb{X}_{t})=\sum_{0\leq|I|\leq N}l_{I}\langle
e_{I},\mathbb{X}_{t}\rangle$

for some coefficients $l_{I}\in\mathbb{R}$ and any $N\in\mathbb{N}$. It is
important to note that all $l_{I}$ are _constant in time_. We denote the set
of all linear functions on the signature by $\mathfrak{L}$.

###  2.2 Universal Approximation Theorem for Linear Functions on the Signature

Linear functions on the signature are dense in a set of certain continuous
path-functionals defined on compact sets of paths. We call this the _universal
approximation property_ and state it rigorously in Theorem 2.11. To this end
we need the following preparations which have been formulated in a similar
manner in [6, 42, 50].

Let us first define the _2-variation metric_ of paths $\mathbf{x},\mathbf{y}$
with values in the group $T_{1}^{N}(\mathbb{R}^{n})$ by

$d_{2-var;[0,t]}(\mathbf{x},\mathbf{y})=\max_{k\in{1,...,N}}\sup_{D\in\mathcal{D}}\left(\sum_{t_{i},t_{i-1}\in
D}|\Pi_{k}(\mathbf{x}_{t_{i-1},t_{i}}-\mathbf{y}_{t_{i-1},t_{i}})|^{\frac{2}{k}}\right)^{\frac{k}{2}},$
(2.1)

where $\mathcal{D}$ is the set of all partitions of $[0,t]$ and $|\cdot|$ is
the Euclidean distance. The definition of increments of paths with values in
the group $T_{1}^{N}(\mathbb{R}^{n})$ is as for the signature in Definition
2.2, namely $\mathbf{x}_{s,t}:=\mathbf{x}_{s}^{-1}\otimes\mathbf{x}_{t}$.

######  Definition 2.6 (Lifted Paths).

Let $\varphi:[0,T]\rightarrow\mathbb{R}$ be a strictly monotone increasing
function. For $N\geq 2$ and $t\in[0,T]$ we define by

$\displaystyle\mathcal{G}^{N}_{t}(\varphi,x)=\\{$
$\displaystyle\hat{\mathbb{X}}^{N}_{[0,t]}(\omega)\,|\,\hat{X}_{s}=(\varphi(s),X_{s}),\omega\in\bar{\Omega},$
$\displaystyle X\textrm{ is an $\mathbb{R}^{n}$-valued
cont.~{}semimart.~{}with }X_{0}=x\\}$

the space of _lifted paths_ $\hat{\mathbb{X}}^{N}_{[0,t]}(\omega)$ of all
$\mathbb{R}^{n}$-valued continuous semimartingales with initial value $x$ and
$\omega\in\bar{\Omega}$. Here, $\bar{\Omega}$ is a probability-one set on
which the metric given in (2.1) is well-defined for all $t\in[0,T]$ and on
which the Lyons’ extension theorem (see Corollary 2.10) holds for all $N\geq
2$. Note that we denote by $\hat{\mathbb{X}}$ the signature of $\hat{X}$ and
we use the subscript $[0,t]$ to emphasize that we take the _entire path_
between $[0,t]$. We equip $\mathcal{G}_{t}^{N}(\varphi,x)$ with the metric
$d_{2-var;[0,t]}$ (now for paths with values in
$T_{1}^{N}(\mathbb{R}^{n+1})$).

######  Definition 2.7 (Space of Lifted (Stopped) Paths).

We define the space of _lifted stopped paths_ via

$\displaystyle{\Lambda}^{N}_{T}(\varphi,x)=\bigcup_{t\in[0,T]}\mathcal{G}_{t}^{N}(\varphi,x)$

We then equip ${\Lambda}_{T}^{N}(\varphi,x)$ with the metric $d_{\Lambda}$
with its form being based on [14, 25]

$\displaystyle
d_{\Lambda}\left(\hat{\mathbb{X}}^{N}_{[0,t]}(\omega_{1}),\hat{\mathbb{Y}}^{N}_{[0,s]}(\omega_{2})\right)$
$\displaystyle\quad=|t-s|+d_{2-var;[0,t\vee
s]}\left(\widehat{\mathbb{X}^{t}}^{N}_{[0,t\vee
s]}(\omega_{1}),\widehat{\mathbb{Y}^{s}}^{N}_{[0,t\vee
s]}(\omega_{2})\right),$

where we denote by $\widehat{X^{t}}$ the process
$(\widehat{X^{t}})_{u}=(\varphi(u),X^{t}_{u})$, i.e. the process where we stop
the semimartingale $X$ at time $t$ but not the time-augmentation $\varphi$ and
write $\widehat{\,\mathbb{X}^{t}}$ for the signature of $\widehat{X^{t}}$. See
[6] for a similar definition in the setting of general rough paths.

######  Remark 2.8.

Note that if
$\hat{\mathbb{X}}^{N}_{[0,t]}(\omega)\in\Lambda_{T}^{N}(\varphi,x)$, then
$\widehat{\,\mathbb{X}^{t}}^{N}_{[0,T]}(\omega)\in\mathcal{G}_{T}^{N}(\varphi,x)$
and hence
$\widehat{\,\mathbb{X}^{t}}^{N}_{[0,T]}(\omega)\in\Lambda_{T}^{N}(\varphi,x)$.
This is the reason, we do not stop the time-augmentation $\varphi$. Indeed,
$(\hat{\mathbb{X}}^{N})^{t}_{[0,T]}(\omega)\notin\mathcal{G}_{T}^{N}(\varphi,x)$,
since $\varphi$ is required to be strictly monotone.

Based on [14] we formulate the notion of non-anticipative path-functionals in
our setting:

######  Definition 2.9 (Non-Anticipative Path-Functionals).

We call functionals $f:{\Lambda}^{2}_{T}(\varphi,x)\rightarrow\mathbb{R}$
_non-anticipative path-functionals_.

The following continuity result for the signature is crucial for the universal
approximation property.

######  Corollary 2.10 (Lyon’s Lift).

For all $t\in[0,T]$ and $N\geq 2$ there exists a unique bijection

$\displaystyle
S^{N}:\mathcal{G}_{t}^{2}(\varphi,x)\mapsto\mathcal{G}_{t}^{N}(\varphi,x)$

with inverse $\Pi_{\leq 2}$ and $S^{N}$ is continuous on bounded sets. We call
$S^{N}$ _Lyons’ lift_. Moreover, for any continuous semimartingale $X$ we have
for almost all $\omega\in\Omega$ that

$S^{N}\left(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\right)=\hat{\mathbb{X}}^{N}_{[0,t]}(\omega).$

###### Proof.

The first part of the theorem follows from [31, Theorem 8.10, Theorem 9.5 and
Corollary 9.11] and the second statement follows from [31, Proposition 17.1
and Exercise 17.2]. ∎

Relying on two lemmas proved in Appendix A, we can now show the universal
approximation theorem for linear functions on the signature. We here state a
version that holds uniformly in time. For this we also provide a rigorous
proof in Appendix A which is – up to our knowledge – not available in the
literature. For similar assertions we however refer to [6, 42]. In the setting
of càdlàg rough paths an analogous result has been proved in [20], however
with different topologies.

######  Theorem 2.11.

Let $K\subset\mathcal{G}_{T}^{2}(\varphi,x)$ be a compact subset. For any
continuous non-anticipative path-functional
$f:\Lambda_{T}^{2}(\varphi,x)\rightarrow\mathbb{R}$ and for every $\epsilon>0$
there exists a linear function on the signature $L$ such that for almost all
$\omega\in\Omega$

$\sup_{(t,\hat{\mathbb{X}}^{2}_{[0,T]}(\omega))\in[0,T]\times
K}|f(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega))-L(\hat{\mathbb{X}}_{t})(\omega)|<\epsilon.$

To show the applicability of this result let us give some examples of non-
anticipative path-functionals.

######  Example 2.12 (Examples of Continuous Non-Anticipative Path-
Functionals).

  1.   2. 1.

Solutions of SDEs driven by $X$: For $\gamma>2$, an $\mathbb{R}^{n}$-valued
continuous semimartingale $X$ and an $\mathbb{R}^{m}$-valued continuous
process of bounded variation $H$, consider the SDE

$dY_{t}=\sum_{i=1}^{n}V_{i}(Y_{t})\circ
dX^{i}_{t}+\sum_{j=1}^{m}W_{j}(Y_{t})dH^{j}_{t},\hskip
14.22636ptY_{0}=y_{0}\in\mathbb{R}^{d},$

where $\\{V_{i}\\}_{1\leq i\leq n}$ is a collection of vector fields in
$\mathrm{Lip}^{\gamma}(\mathbb{R}^{d})$ and $\\{W_{i}\\}_{1\leq i\leq m}$ is a
collection of $\mathrm{Lip}^{1}(\mathbb{R}^{d})$ vector fields.111Following
the definition in [31, Definition 10.2] a vector field $V$ is in
$\mathrm{Lip}^{\gamma}$ if $V$ is $\lfloor\gamma\rfloor$ times continuously
differentiable and if the supremum norm of $V$ and the supremum norm of its
first $\lfloor\gamma\rfloor$ derivatives admits a common bound. Set $Z=(X,H)$
and $z=(X_{0},H_{0})$ and define $D_{Z}$ to be the set

$D_{Z}=\bigcup_{t\in[0,T]}\left\\{{\hat{\mathbb{Z}}}_{[0,t]}^{2}(\omega)|\omega\in\bar{\Omega}\right\\}\subset\Lambda_{T}^{2}(\varphi,z).$

Then the components of the solution to this SDE $(Y^{i}_{t})_{t\in[0,T]}$, can
be seen as a map

$\displaystyle Y^{i}_{t}(\omega):\,$ $\displaystyle
D_{Z}\rightarrow\mathbb{R}$
$\displaystyle{\hat{\mathbb{Z}}}_{[0,t]}^{2}(\omega)\mapsto Y_{t}^{i}(\omega)$

and it holds almost surely (i.e. on $\bar{\Omega}$) that $Y_{t}^{i}\in
C(D_{Z};\mathbb{R})$ for every $t\in[0,T]$. The continuity follows from the
fact that the solution to the corresponding rough differential equation solves
the SDE almost surely by [31, Theorem 17.3] and the RDE solution is continuous
[31, Corollary 10.28 & Theorem 12.10] in the lifted input path. Hence, we can
apply Theorem 2.11 and obtain that for any compact set
$K\subset\\{\hat{\mathbb{Z}}^{2}_{[0,T]}(\omega)|\omega\in\bar{\Omega}\\}$ and
all $\epsilon>0$ it holds that for all $i\in\\{1,\dots,d\\}$ there exists a
linear function $L^{i}$ on the signature such that

$\sup_{(t\in[0,T],\hat{\mathbb{Z}}^{2}_{[0,T]}(\omega))\in
K}\left|Y_{t}^{i}(\omega)-L^{i}(\hat{\mathbb{Z}}_{t}(\omega))\right|<\epsilon.$

  3. 2.

Continuous Functions Without Path-Dependence: Consider an arbitrary but fixed
$\mathbb{R}^{n}$-valued continuous semimartingale $X$ with initial value
$X_{0}=x$. For any $t\in[0,T]$ let

$\displaystyle h:\mathbb{R}^{d+1}$ $\displaystyle\rightarrow\mathbb{R}$
$\displaystyle\hat{X}_{t}(\omega)$ $\displaystyle\mapsto
h\left(\hat{X}_{t}(\omega)\right)$

be a continuous function. Such a function can for almost all $\omega\in\Omega$
be extended to a non-anticipative path-functional $\tilde{h}$ on
$\Lambda_{T}^{2}(\varphi,x)$. To be precise, simply define $\tilde{h}$ as

$\displaystyle\tilde{h}:\,\Lambda_{T}^{2}(\varphi,x)$
$\displaystyle\rightarrow\,\mathbb{R}$
$\displaystyle\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)$
$\displaystyle\mapsto\,\tilde{h}\left(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\right):=h\left((\hat{X}_{[0,t]})_{\odot}(\omega)\right)=h\left(\hat{X}_{t}(\omega)\right).$

In particular, $\tilde{h}\in
C\left(\Lambda_{T}^{2}(\varphi,x);\mathbb{R}\right)$. To see this, note that
for any
$\hat{\mathbf{x}}_{[0,t]},\hat{\mathbf{y}}_{[0,s]}\in\Lambda_{T}^{2}(\varphi,x)$
it holds that

$\displaystyle
d_{\Lambda}(\hat{\mathbf{x}},\hat{\mathbf{y}})\geq|\hat{x}_{0,t}-\hat{y}_{0,s}|=|\hat{x}_{t}-\hat{y}_{s}|$

with $\hat{x}=\Pi_{1}(\hat{\mathbf{x}})$, $\hat{y}=\Pi_{1}(\hat{\mathbf{y}})$,
where we replaced the $\max_{k\in\\{1,\dots,N\\}}$ by $k=1$ and
$\sup_{D\in\mathcal{D}}$ by $D=[0,t\vee s]$ in the definition of
$d_{2-var;[0,t\vee s]}$ to obtain the inequality. The last equality follows
from the fact that all paths in $\Lambda_{T}^{2}(\varphi,x)$ have the same
initial value. Due to the continuity of $h$, for every $\epsilon>0$ there
exists a $\delta>0$ such that for all $t_{1},t_{2}\in[0,T]$ and all paths
$X(\omega_{1}),Y(\omega_{2})$ satisfying
$\left|\hat{X}_{t_{1}}(\omega_{1})-\hat{Y}_{t_{2}}(\omega_{2})\right|<\delta$
we have

$\left|h\left(\hat{X}_{t_{1}}(\omega_{1})\right)-h\left(\hat{X}_{t_{2}}(\omega_{2})\right)\right|<\epsilon.$

By the above estimate,
$d_{\Lambda}\left(\hat{\mathbb{X}}^{2}_{[0,t_{1}]}(\omega_{1}),\hat{\mathbb{X}}^{2}_{[0,t_{2}]}(\omega_{2})\right)<\delta$
therefore implies

$\displaystyle\left|\tilde{h}\left(\hat{\mathbb{X}}^{2}_{[0,t_{1}]}(\omega_{1})\right)-\tilde{h}\left(\hat{\mathbb{X}}^{2}_{[0,t_{2}]}(\omega_{2})\right)\right|<\epsilon$

and thus proves the claim.

###  2.3 Global Universal Approximation Theorem for Linear Functions on the
Signature

In this subsection, we state an alternative universal approximation theorem
which is due to [22]. We will refer to Theorem 2.18 also as _global_ universal
approximation theorem because we are no longer limited to compact sets,
however this comes at the price of approximating with respect to weighted
norms instead of the uniform norm. The following is based on the statements
and the results of [22] adapted to the setting considered in this paper.
However, we do formulate everything in the path-functional setting here,
leading to a approximation _uniformly in time_ which has not been done in
[22].

Let us start by introducing the following metrics:

######  Definition 2.13 (The Carnot-Carathéodory Norm and Homogeneous
$\alpha$-Hölder Metric).

  1. 1.

Let $\mathbf{a}\in G^{N}(\mathbb{R}^{n})$, we define the _Carnot-Carathéodory_
norm $\|\cdot\|_{cc}$ by

$\displaystyle\|\mathbf{a}\|_{cc}:=\inf\Big{\\{}\int_{0}^{T}\|dX_{t}\|\,\Big{\lvert}\,$
$\displaystyle X:[0,T]\rightarrow\mathbb{R}^{d}\textrm{ continuous and of
finite variation,}$ $\displaystyle\textrm{such that
}\mathbb{X}^{N}_{T}=\mathbf{a}\Big{\\}}.$

Moreover, for $\mathbf{a},\mathbf{b}\in G^{N}(\mathbb{R}^{n})$ the
corresponding metric $d_{cc}(\cdot,\cdot)$ is induced by

$d_{cc}(\mathbf{a},\mathbf{b}):=\|\mathbf{a}^{-1}\otimes\mathbf{b}\|_{cc}\,.$

  2. 2.

Recall the space of lifted path $\mathcal{G}_{t}^{N}(\varphi,x)$ for $N\geq 2$
and $t\in[0,T]$. For two lifted paths
$\hat{\mathbb{X}}^{N}_{[0,t]}(\omega_{x}),\hat{\mathbb{Y}}^{N}_{[0,t]}(\omega_{y})\in\mathcal{G}_{t}^{N}(\varphi,x)$
we define the _homogeneous_ $\alpha$_-Hölder metric_
$d_{\alpha,[0,t]}(\cdot,\cdot)$ for $\alpha<\frac{1}{2}$ by

$d_{\alpha,[0,t]}\left(\hat{\mathbb{X}}^{N}_{[0,t]}(\omega_{x}),\hat{\mathbb{Y}}^{N}_{[0,t]}(\omega_{y})\right)=\sup_{\begin{subarray}{c}r,s\in[0,t]\\\
r<s\end{subarray}}\frac{d_{cc}\left(\hat{\mathbb{X}}^{N}_{r,s}(\omega_{x}),\hat{\mathbb{Y}}^{N}_{r,s}(\omega_{y})\right)}{|r-s|^{\alpha}}.$

######  Notation 2.14.

In the following we shall denote by
$\mathcal{G}_{t}^{N,\alpha}(\varphi,x):=(\mathcal{G}_{t}^{N}(\varphi,x),d_{\alpha,[0,t]})$
the space of lifted paths equipped with the homogeneous $\alpha$-Hölder metric
for some $\alpha<\frac{1}{2}$. Moreover, we write
$\Lambda^{N,\alpha}_{T}(\varphi,x):=(\Lambda^{N}_{T}(\varphi,x),d_{\Lambda^{\alpha}})$
for the corresponding space of lifted stopped paths equipped with

$d_{\Lambda^{\alpha}}\left(\hat{\mathbb{X}}^{N}_{[0,t]}(\omega_{x}),\hat{\mathbb{Y}}^{N}_{[0,s]}(\omega_{y})\right):=|t-s|+d_{\alpha,[0,t\vee
s]}\left(\widehat{\,\mathbb{X}^{t}}^{N}_{[0,t\vee
s]}(\omega_{x}),\widehat{\mathbb{Y}^{s}}^{N}_{[0,t\vee
s]}(\omega_{y})\right).$

We shall assume without loss of generality that the set $\bar{\Omega}$ in the
definition of $\mathcal{G}_{t}^{N}(\varphi,x)$ is such that the homogeneous
$\alpha$-Hölder norm is finite for all lifted paths and all
$\alpha<\frac{1}{2}$.

######  Remark 2.15.

Due to the fact that we are now considering different topologies on the space
of lifted (stopped) paths, the continuity statements of Lemma A.2, Corollary
2.10 and Lemma A.3 need to be proved again. In fact, they do hold analogously:

  * •

The proof that the map $\lambda$ defined in Lemma A.2 is again continuous in
the $\alpha$-Hölder topology works very similar to the one of Lemma A.2, where
one ultimately chooses $\delta>0$ such that

$\delta+d_{\alpha,[0,t+\delta]}\left(\widehat{\mathbb{Y}^{s}}^{N}_{[0,t+\delta]}(\omega_{y}),\widehat{\,\mathbb{Y}^{t}}^{N}_{[0,t+\delta]}(\omega_{y})\right)<\epsilon.$

  * •

The continuity of Lyons’ lift holds by [31, Theorem 9.5, Corollary 9.11].

  * •

The proof of Lemma A.3 holds analogously in the $\alpha$-Hölder topology.

In order to formulate the global universal approximation theorem we need the
notion of a weighted space and certain weighted function spaces being
generalizations of continuous functions.

######  Definition 2.16 (Weighted Space and Weighted Function Space).

Let $(X,\tau_{X})$ be a completely regular Hausdorff space and consider a
function $\psi:X\rightarrow(0,\infty)$. We call the pair $(X,\psi)$ a
_weighted space_ if it holds that $K_{R}:=\psi^{-1}((0,R])=\\{x\in
X\lvert\psi(x)\leq R\\}$ is $\tau_{X}$-compact for all $R>0$. Moreover, for a
weighted space $(X,\psi)$, we define

$B_{\psi}(X)=\left\\{f:X\rightarrow\mathbb{R}\,\lvert\,\sup_{x\in
X}\frac{|f(x)|}{\psi(x)}<\infty\right\\}$

and equip it with $\|f\|_{\mathcal{B}_{\psi}(X)}:=\sup_{x\in
X}\frac{|f(x)|}{\psi(x)}$. We denote by $\mathcal{B}_{\psi}(X)$ the closure of
the space of bounded continuous functions $C_{b}(X)$ under
$\|f\|_{\mathcal{B}_{\psi}(X)}$. We refer to $\mathcal{B}_{\psi}(X)$ as
weighted function space.

The global universal approximation theorem via linear functions on the
signature hinges on an application of the weighted Stone-Weierstrass theorem
[22, Theorem 3.6] applied to $\mathcal{B}_{\psi}(\Lambda_{T}^{2}(\varphi,x))$
for the specific weight function222Note that in contrast to [22, Theorem 5.4]
we can choose $\xi>1$ (and do not need to require $\xi>2$), as we are working
in a semimartingale setting. We refer to the proof of Theorem 2.18 for a more
detailed explanation.

$\displaystyle\psi(\hat{\mathbb{X}}^{N}_{[0,t]}(\omega)):=\exp\left(\zeta\|\widehat{\,\mathbb{X}^{t}}^{N}_{[0,T]}(\omega)\|^{\xi}_{\alpha,[0,T]}\right)$
(2.2)

for $\hat{\mathbb{X}}^{N}_{[0,t]}(\omega)\in\Lambda_{T}^{2}(\varphi,x)$,
$\zeta>0$, $\xi>1$, $\alpha<\frac{1}{2}$ and $\|\cdot\|_{\alpha,[0,T]}$ being
the norm induced by $d_{\alpha,[0,T]}$. To this end we have to prove that
$\Lambda_{T}^{2}(\varphi,x)$ is indeed a weighted space. This can only be
achieved if we use a weaker topology than the $\alpha$-Hölder metric. In
analogy to the Hölder spaces considered in [22] we thus equip
$\Lambda_{T}^{2}(\varphi,x)$ with $d_{\Lambda^{\alpha^{\prime}}}$ for some
$0\leq\alpha^{\prime}<\alpha<\frac{1}{2}$ and obtain the following lemma
proved in Appendix A.

######  Lemma 2.17.

Let $\alpha<\frac{1}{2}$ and consider the weight function $\psi$ given in
(2.2).

Then $\Lambda_{T}^{2,\alpha^{\prime}}(\varphi,x)$, i.e.
$\Lambda_{T}^{2}(\varphi,x)$ equipped with $d_{\Lambda^{\alpha^{\prime}}}$ for
$0\leq\alpha^{\prime}<\alpha$, is a weighted space.

We are finally ready to formulate the global universal approximation theorem
for non-anticipative path-functionals. Its proof is also be given in Appendix
A.

######  Theorem 2.18 (Global Universal Approximation Theorem).

Let $0\leq\alpha^{\prime}<\alpha<\frac{1}{2}$ and consider the weight function
$\psi$ given in (2.2). Then, for every non-anticipative path-functional
$f\in\mathcal{B}_{\psi}\left(\Lambda^{2,\alpha^{\prime}}_{T}(\varphi,x)\right)$
(i.e. $\Lambda^{2}_{T}(\varphi,x)$ is equipped with
$d_{\Lambda^{\alpha^{\prime}}}$) and for every $\epsilon>0$ there exists a
linear function on the signature $L$ such that for almost all
$\omega\in\Omega$

$\sup_{\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\in\,\Lambda_{T}^{2}(\varphi,x)}\frac{\left|f(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega))-L(\hat{\mathbb{X}}_{t})(\omega)\right|}{\psi(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega))}<\epsilon.$

###  2.4 The Johnson-Lindenstrauss Lemma

The Johnson-Lindenstrauss Lemma [41] is an important mathematical result,
which is interesting for machine learning and data science, as it can be seen
as a dimension reduction technique. There are many variants of the Johnson-
Lindenstrauss lemma, we will here use a version stated in [55, Theorem 3.1]:

######  Theorem 2.19.

Let $n\in\mathbb{N}$, $\epsilon\in(0,\frac{1}{2})$, $\delta\in(0,1)$ and set
$k=C\epsilon^{-2}\log(\frac{\delta}{2})$ for a suitable constant $C$. Then
consider a random matrix $A$ of dimensions $k\times n$, whose entries $A_{ij}$
are i.i.d. random variables with $\mathbb{E}[A_{ij}]=0$,
$\mathrm{Var}[A_{ij}]=\frac{1}{k}$ and sub-Gaussian tails. Then it holds for
all $x\in\mathbb{R}^{n}$ that

$\mathbb{P}\left[(1-\epsilon)\|x\|\leq\|Ax\|\leq(1+\epsilon)\|x\|\right]\geq
1-\delta.$

######  Remark 2.20.

Note that since the above holds for all $x\in\mathbb{R}^{n}$, it immediately
follows that it holds in particular for all $x-y$, where
$x,y\in\mathbb{R}^{n}$. Hence, Theorem 2.19 can be understood to preserve
distances up to $(1\pm\epsilon)$.

######  Remark 2.21.

In this paper, we will use Theorem 2.19 to obtain a random projection of the
truncated signature. We will choose the components $A_{ij}$ to be i.i.d. with
$A_{ij}\sim\mathcal{N}(0,\frac{1}{k})$.

###  2.5 The Market and Important Portfolio Specifications

Consider a finite time-horizon $T>0$ and some filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P})$. We assume
that the market consists of $d$ companies, with market capitalizations given
by a vector $(S_{t})_{t\in[0,T]}=(S^{1}_{t},\ldots,S^{d}_{t})_{t\in[0,T]}$ of
$d$ positive continuous semimartingales with fixed initial value
$S_{0}=s_{0}$. We assume this market fulfils the _No Unbounded Profit with
Bounded Risk (NUPBR)_ [38] condition. Furthermore, we assume that each company
has issued a single infinitely divisible share, hence
$S^{1}_{t},\ldots,S^{d}_{t}$ correspond to the prices of these shares. We
denote the relative capitalization of the companies, also called _market
weights_ by $\mu_{t}=(\mu_{t}^{1},...,\mu_{t}^{d})$ with

$\mu_{t}^{i}=\frac{S_{t}^{i}}{S_{t}^{1}+\cdots+S_{t}^{d}}.$

######  Definition 2.22.

We call a vector $\pi_{t}=(\pi^{1}_{t},...,\pi^{d}_{t})$ of predictable
processes fulfilling $\sum_{i=1}^{d}\pi_{t}^{i}\equiv 1$ and being integrable
with respect to $R$, with
$R_{t}^{i}:=\int_{0}^{t}\frac{dS_{u}^{i}}{S_{u}^{i}}$, an _( $R$-integrable)
portfolio_.

The component $\pi^{i}_{t}$ denotes the proportion of wealth invested in stock
$i$ at time $t$, hence every portfolio is self-financing. Consider investing
with a portfolio $\pi$ over a time-horizon $[t_{0},t_{1}]$ for $0\leq
t_{0}<t_{1}\leq T$. We denote its wealth process portfolio $\pi$ with initial
wealth $w$ by $(W^{\pi,w}_{t})_{t\in[t_{0},t_{1}]}$. Without loss of
generality we will assume from now on that the initial investment is always
one unit of currency, i.e. that $w=1$ and therefore denote
$W^{\pi}_{t}:=W^{\pi,1}_{t}$. The wealth process of a portfolio fulfils the
stochastic differential equation

$\frac{dW_{t}^{\pi}}{W_{t}^{\pi}}=\sum_{i=1}^{d}\pi^{i}_{t}\frac{\textrm{d}S_{t}^{i}}{S_{t}^{i}}$
(2.3)

which is understood in Itô-sense.

A special portfolio is the _market portfolio_ whose portfolio weights are
given by the relative capitalizations $\mu_{t}$, whence we also refer to
$\mu_{t}$ as the market weights. The wealth of the market portfolio is given
by

$W^{\mu}_{t}=\frac{S_{t}^{1}+\ldots+S_{t}^{d}}{S_{t_{0}}^{1}+\ldots+S_{t_{0}}^{d}}$

since we require $W_{t_{0}}^{\mu}=1$.

We define the relative wealth process of a portfolio $\pi$ to be

$V_{t}^{\pi}:=\frac{W_{t}^{\pi}}{W_{t}^{\mu}}.$

By using Itô’s-formula, it follows directly from (2.3) that

$\frac{dV_{t}^{\pi}}{V_{t}^{\pi}}=\sum_{i=1}^{d}\pi^{i}_{t}\frac{d\mu_{t}^{i}}{\mu_{t}^{i}}.$
(2.4)

Furthermore, by applying Itô’s formula again, we obtain

$d\ln(V_{t}^{\pi})=\sum_{i=1}^{d}\frac{\pi^{i}_{t}}{\mu_{t}^{i}}d\mu_{t}^{i}-\frac{1}{2}\sum_{i,j=1}^{d}\frac{\pi^{i}_{t}}{\mu_{t}^{i}}\frac{\pi^{j}_{t}}{\mu_{t}^{j}}d[\mu^{i},\mu^{j}]_{t},$
(2.5)

where $[X,Y]_{t}$ denotes the co-variation process of continuous
semimartingales $X,Y$.

######  Notation 2.23.

We denote by $\Delta^{n}$ the unit simplex of $\mathbb{R}^{n}$ and by
$\Delta_{+}^{n}$ its interior. We call a portfolio $\pi$ long-only, if
$\pi_{t}\in\Delta^{d}$ for all $t\in[0,T]$ almost surely.

We now turn to some portfolios which are of special interest, those are the
_numeraire_ , _log-optimal_ and _growth-optimal_ portfolio.

######  Definition 2.24 (Numeraire Portfolio).

We call a portfolio $\rho$ the _numeraire portfolio_ if
$\frac{W^{\pi}}{W^{\rho}}$ is a supermartingale for any other portfolio $\pi$.

######  Theorem 2.25.

In a semimartingale market the NUPBR condition (see e.g. [38]) holds if and
only if the numeraire portfolio $\rho$ exists and $W^{\rho}_{T}<\infty$ almost
surely.

###### Proof.

See [43, Theorem 4.12], where in our case the predictable closed convex
constraints $\mathfrak{C}$ are

$\mathfrak{C}=\left\\{\pi\in\mathbb{R}^{d}\,\middle|\,\sum_{i=1}^{d}\pi^{i}=1\right\\}.$

Note that [43, Theorem 4.12] considers the setting with bank-account, however
our constraint set $\mathfrak{C}$ prohibits us from investing into the bank
account, i.e. $\pi^{0}\equiv 0$ (in the notation of [43]). ∎

To connect the numeraire portfolio with the log-optimal one introduced in
Definition 2.27 below, we recall Proposition 4.19 from [43].

######  Proposition 2.26.

Let $U:(0,\infty)\rightarrow\mathbb{R}$ be concave and strictly increasing.
The utility maximization problem

$\sup_{\pi}\mathbb{E}\left[U(W_{T}^{\pi})\right]$

has no solution or has infinitely many solutions, if the NUPBR condition does
not hold.

######  Definition 2.27.

Consider the log-utility optimization problem

$\sup_{\pi}\mathbb{E}\left[\log(W_{T}^{\pi})\right].$

If it is well-posed, we call the corresponding optimal portfolio _log-optimal_
portfolio.

The following lemma can also be found in [43].

######  Lemma 2.28.

Under the NUPBR condition, the log-utility optimization problem admits a
solution, if it is finite, in which case the numeraire portfolio is the log-
optimal portfolio.

Finally, let us introduce the notion of the growth-optimal portfolio.

######  Definition 2.29.

Consider a market model

$dS_{t}=\operatorname{diag}(S_{t})(a_{t}dt+\Sigma_{t}dB_{t}),$

where $a_{t}$ is a $d$-dimensional vector, $\Sigma_{t}$ a $d\times m$-matrix
for $m\geq d$ and $B$ a $m$-dimensional Brownian motion. Assume $a_{t}$,
$B_{t}$ are predictable processes and satisfy

$\sum_{i=1}^{d}\int_{0}^{T}|a_{t}^{i}|dt+\sum_{i=1}^{d}\sum_{j=1}^{m}\int_{0}^{T}|\Sigma_{t}^{ij}|^{2}dt<\infty\hskip
8.5359pt\mathbb{P}\textrm{-a.s.}$

and $\Sigma_{t}\Sigma_{t}^{\mathsf{T}}$ is almost surely invertible for all
$t\in[0,T]$. The log-wealth-process of a portfolio $\pi$ in this market model
is then given by

$d\ln(W^{\pi}_{t})=\left(\pi_{t}^{\mathsf{T}}a_{t}-\frac{1}{2}\pi_{t}^{\mathsf{T}}\Sigma_{t}\Sigma_{t}^{\mathsf{T}}\pi_{t}\right)dt+\pi_{t}^{\mathsf{T}}\Sigma_{t}dW_{t}.$

We call
$g^{\pi}_{t}=\pi_{t}^{\mathsf{T}}a_{t}-\frac{1}{2}\pi_{t}^{\mathsf{T}}\Sigma_{t}\Sigma_{t}^{\mathsf{T}}\pi_{t}$
the portfolios _growth-rate_ and the portfolio with maximal growth-rate, if it
exists, the _growth-optimal_ portfolio.

######  Lemma 2.30.

Consider a market model given in Definition 2.29 and let us define the
(instantaneous) market price of risk

$\theta_{t}:=\Sigma_{t}^{\mathsf{T}}(\Sigma_{t}\Sigma_{t}^{\mathsf{T}})^{-1}a_{t}.$

The growth-optimal portfolio exists if and only if
$\int_{0}^{T}\|\theta_{t}\|^{2}dt<\infty$ almost surely. Moreover, the log-
optimal portfolio exists if and only if
$\mathbb{E}\left[\int_{0}^{T}\|\theta_{t}\|^{2}dt\right]<\infty$. Moreover, if
the growth-optimal portfolio $\pi^{(g)}$ exists, it is given by

$\pi^{(g)}_{t}=(\Sigma_{t}\Sigma_{t}^{\mathsf{T}})^{-1}\left(a_{t}-\kappa\mathbf{1}\right)\textrm{
for
}\kappa=\frac{\sum_{i}((\Sigma_{t}\Sigma_{t}^{\mathsf{T}})^{-1}a_{t})_{i}-1}{\sum_{i,j}(\Sigma_{t}\Sigma_{t}^{\mathsf{T}})^{-1}_{i,j}}$

where $\mathbf{1}=(1,...,1)^{\mathsf{T}}$.

###### Proof.

Regarding the existence of the growth-optimal and log-optimal portfolios, see
[38], also for the general case of continuous semimartingale markets. The
specific form of the growth-optimal portfolio is due to [30]. ∎

######  Lemma 2.31.

Consider the setting of a market as in Definition 2.29. Then the growth-
optimal portfolio exists if and only if the numeraire portfolio exists in
which case they are the same333Assuming that their generated wealth is almost
surely finite this holds if and only if NUPBR is satisfied (see Theorem
2.25).. Moreover, if the log-utility maximization problem is finite, then they
also coincide with the log-optimal portfolio.

###### Proof.

See [30] and [38]. The last assertion then follows from Lemma 2.28 ∎

In SPT, there is another class of portfolios of special interest, those are
_functionally generated portfolios_ , see [27].

######  Definition 2.32 (Functionally Generated Portfolios).

Let $U$ be a neighbourhood of $\Delta^{d}$ and consider a $C^{2}$-function
$G:U\rightarrow\mathbb{R}_{+}$ such that $x_{i}D_{i}\log G(x)$ is bounded on
$\Delta^{d}$. Then $G$ defines the functionally generated portfolio via

$\pi^{i}_{t}=\mu_{t}^{i}\left(D_{i}\log{G(\mu_{t})}+1-\sum_{j=1}^{d}\mu_{t}^{j}D_{j}\log{G(\mu_{t})}\right).$

The function $G$ is called the _portfolio generating function_ and if it is
concave, $\pi$ is a long-only portfolio.

Before we conclude this section, we introduce the _ranked market-weights_
which are of particular interest in SPT, due to the remarkable stability of
the capital distribution curves, see for example [27, 45].

######  Definition 2.33 (Ranked Capitalization and Market Weight Processes).

For an $\mathbb{R}^{n}_{+}$-valued continuous semimartingale $X$ we denote its
_ranked process_ by $\mathbf{X}=(X^{(1)},...,X^{(n)})$ which is defined as

$\max_{i\in\\{1,\dots,n\\}}X_{t}^{i}=X^{(1)}_{t}\geq X^{(2)}_{t}\geq\dots\geq
X^{(n-1)}_{t}\geq X^{(n)}_{t}=\min_{i\in\\{1,\dots,n\\}}X_{t}^{i},$

where we break ties by allocating a lower rank to smaller labels. To be
precise, if $X_{t}^{i}=X_{t}^{j}$ with $i\geq j$, we set
$X_{t}^{(r_{i})}=X_{t}^{i}$ and $X_{t}^{(r_{j})}=X_{t}^{j}$ where $r_{i}\geq
r_{j}$.

Accordingly, we denote by $\boldsymbol{\mu}$ the _ranked market weights_ and
by $\mathbf{S}$ the _ranked capitalization process_.

######  Corollary 2.34.

Given an $\mathbb{R}^{n}$-valued continuous semimartingale $X$, its ranked
process is again an $\mathbb{R}^{n}$-valued continuous semimartingale.

###### Proof.

See [5, Theorem 2.2]. ∎

##  3 (Linear) Path-Functional Portfolios

Before we present our application of signature methods in the context of SPT
and portfolio optimization, we generalize the concept of functionally
generated portfolios by introducing so-called _path-functional portfolios_.

######  Definition 3.1 (Path-Functional Portfolios).

We introduce _path-functional portfolios_ which are generated by a family of
non-anticipating path functionals $\\{f^{i}\\}_{1\leq i\leq d}$ depending on
time and the path of $X$ and which are constructed via an auxiliary portfolio
$\tau$, where $\tau$ is required to be uniformly bounded. Moreover, we assume
that the non-anticipating path functionals $\\{f^{i}\\}_{1\leq i\leq d}$
fulfil the necessary integrability conditions such that the following
processes $\pi$ are portfolios in the sense of Definition 2.22. We consider
two types of such portfolios:

  1. (I)

$\pi_{t}^{i}(\tau,{X})=\tau_{t}^{i}\left(f^{i}\left(t,X_{[0,t]}\right)+1-\sum_{j=1}^{d}\tau_{t}^{i}f^{j}\left(t,X_{[0,t]}\right)\right)$

  2. (II)

$\pi_{t}^{i}(\tau,{X})=f^{i}\left(t,X_{[0,t]}\right)+\tau_{t}^{i}\left(1-\sum_{j=1}^{d}f^{j}\left(t,X_{[0,t]}\right)\right)$

where we use the notation of the subscript ${[0,t]}$ to make the dependence of
on the entire path explicit. Moreover, we refer to the functionals
$\\{f^{i}\\}_{1\leq i\leq d}$ as the _portfolio controlling functions_.

######  Remark 3.2 (Relation to (Classical) Stochastic Portfolio Theory).

In the above definition, we consider a general continuous semimartingale and a
general auxiliary portfolio $\tau$ in the construction of the path-functional
portfolios. In the spirit of SPT, one would choose $\tau=\mu$ the market
portfolio and $X=\mu$, the process of market weights. One can then recover the
classical functionally generated portfolios from the path-functional
portfolios of type $I$ by setting $X=\mu$ and choosing
$f^{i}(t,\mu_{[0,t]})=f^{i}(\mu_{t})=D_{i}\log(G(\mu_{t}))$.

######  Remark 3.3 (Path-Functional Portfolios of Type $II$).

A simple example of a path-functional portfolio of type $II$ would be any
portfolio generated by a deep (recurrent) neural network, see for example
[61]. Denote by $\mathcal{NN}^{i}(t,X_{[0,t]})$ the $i$-th output of a deep
(recurrent) neural network. If

  * •

the last layer of the neural network is for example a softmax function and the
portfolio weights are defined by
$\pi^{(\mathcal{NN}),i}_{t}=\mathcal{NN}^{i}(t,X_{[0,t]})$, or,

  * •

the portfolio weights are constructed as
$\pi^{(\mathcal{NN}),i}_{t}=\frac{\mathcal{NN}^{i}(t,X_{[0,t]})}{\sum_{j=1}^{d}\mathcal{NN}^{j}(t,X_{[0,t]})}$,

then, the portfolios $\pi^{(\mathcal{NN})}$ are in both cases path-functional
portfolios of type $II$ but not of type $I$. Note that the choice of $\tau$ is
irrelevant because $\sum_{j=1}\pi^{(\mathcal{NN}),j}_{t}=1$.

######  Remark 3.4 (Conversion between Path-Functional Portfolios of Type
I&II).

Note that for a fixed auxiliary portfolio $\tau$ with weight-processes which
are continuous semimartingales and for a continuous semimartingale
$(X_{t})_{t\in[0,T]}$ any path-functional portfolio of type $I$ with portfolio
controlling functions $f^{(I),i}:X_{[0,t]}\mapsto f^{(I),i}(X_{[0,t]})$ is a
path-functional portfolio of type $II$ with portfolio controlling function
$f^{(II),i}:(\tau,X)_{[0,t]}\mapsto\tau_{t}^{i}f^{(I),i}(X_{[0,t]})$. This
also holds vice versa, if the components of $\tau$ are all non-zero. Indeed,
any path-functional portfolio of type $II$ with portfolio controlling function
$f^{(II),i}:X_{[0,t]}\mapsto f^{(II),i}(X_{[0,t]})$ is a path-functional
portfolio of type $I$ with portfolio controlling function
$f^{(I),i}:(\tau,X)_{[0,t]}\mapsto\frac{f^{(II),i}(X_{[0,t]})}{\tau_{t}^{i}}$.
However, note that this conversion always involves adding the process of the
auxiliary portfolio $\tau$ to the input. Not only does this enlarge the
dimensions of the inputs, but also is $\tau$ then required to be a continuous
semimartingale. Hence, this conversion is not always possible.

We now introduce a special class of path-functional portfolios which are
particularly useful for optimizing path functional portfolios, as we will make
more explicit in Section 5. This class is the one of _linear path-functional
portfolios_.

######  Definition 3.5 (Linear Path-Functional Portfolios).

For a path-functional portfolio of type $I$ or $II$, we call it a _linear_
path-functional portfolio of the corresponding type, if the portfolio
controlling functions are of the form

$f^{i}(t,X_{[0,t]})=\sum_{\nu\in\mathcal{V}}l_{\nu}^{i}\phi^{\nu}(t,X_{[0,t]})$

where $\\{\phi^{\nu}\\}_{\nu\in\mathcal{V}}$ is a collection of feature maps
which are themselves non-anticipating path functionals, $\mathcal{V}$ is a
finite set of features and $l_{\nu}^{i}\in\mathbb{R}$ for each $i,\nu$ are
constant (optimization) parameters.

######  Remark 3.6 (Significance of the Auxiliary Portfolio $\tau$).

Considering linear path-functional portfolios gives us a first idea on how to
do portfolio optimization in that context. Namely, optimizing the parameters
$\\{l_{\nu}^{i}\\}_{{1\leq i\leq d,\,\nu\in\mathcal{V}}}$ for a given
collection of feature maps. Note, that the auxiliary portfolio $\tau$ can
always be attained in this way by

  * •

setting $l_{\nu}^{i}=l_{\nu}^{j}$ for all $i,j\in\\{1,\dots,d\\}$,
$\nu\in\mathcal{V}$ for portfolios of type $I$

  * •

setting $l_{\nu}^{i}=0$ for all $i\in\\{1,\dots,d\\}$, $\nu\in\mathcal{V}$ for
portfolios of type $II$.

Hence, if one wants to learn a linear path-functional portfolio which
outperforms a given benchmark portfolio $\bar{\pi}$, one should use the
benchmark portfolio as the auxiliary portfolio, i.e. set $\tau=\bar{\pi}$,
since then the benchmark portfolio is included in the family of portfolios one
optimizes over. In the context of stochastic portfolio theory, we therefore
often choose $\tau=\mu$, because we aim to outperform the market portfolio.

Among all linear path-functional portfolios, we are in particularly interested
in those, whose feature maps are (randomized) elements of the signature,
resulting in _(JL- or randomized-)signature portfolios_. We make this more
precise in the following definition:

######  Definition 3.7 (Signature Portfolios, JL- and Randomized-Signature
Portfolios).

For a given auxiliary portfolio $\tau$ and any $\mathbb{R}^{n}$-valued
continuous semimartingale $X$, we call linear path-functional portfolios _(JL-
or randomized-)signature portfolios_ if the feature maps are elements of the
(JL- or randomized-)signature of the time-augmented semimartingale $\hat{X}$.
More precisely, let $\hat{X}_{t}=(\varphi(t),X_{t})$ be the time-augmented
process of $X$, where $\varphi$ is a strictly increasing function. Recall that
$\hat{\mathbb{X}}^{N}$ denotes the signature of $\hat{X}$ truncated at level
$N$. We then consider the following portfolios:

  * •

_Signature Portfolios:_ A _signature portfolio of degree $N$_ is a linear
path-functional portfolio with
$\mathcal{V}=\\{I\,\mid\,I=(i_{1},...,i_{m})\in\\{1,\dots,n\\}^{m}\textrm{ for
}0\leq m\leq N\\}$ (the set of multiindices up to length $N$) and with feature
maps

$\phi^{I}(t,X_{[0,t]})=\langle e_{I},\hat{\mathbb{X}}^{N}_{t}\rangle=\langle
e_{I},\hat{\mathbb{X}}_{t}\rangle.$

In other words, signature portfolios are path-functional portfolios, where the
portfolio controlling functions are linear functions on the signature.

  * •

_JL-Signature Portfolios:_ A _JL-signature portfolio of dimension $(P,N)$_ is
a linear path-functional portfolio with $\mathcal{V}=\\{1,\dots,P\\}$, where
$P$ is the dimension of the projection, and with feature maps

$\phi^{p}(t,X_{[0,t]})=\langle A^{p},\hat{\mathbb{X}}^{N}_{t}\rangle,$

where $A^{p}$ is the $p$-th column of the Johnson-Lindenstrauss projection
introduced in Theorem 2.19. Hence, JL-signature portfolios are path-functional
portfolios, where the portfolio controlling functions are linear functions on
the Johnson-Lindenstrauss projected signature.

  * •

_Randomized-Signature Portfolios:_ Let $\mathcal{S}$ denote the solution to

$d\mathcal{S}_{t}=\sum_{i=1}^{n+1}\sigma\left(b^{(i)}+A^{(i)}\mathcal{S}_{t}\right)\circ
d\hat{X}_{t}^{i}\hskip
28.45274pt\mathcal{S}_{0}=(1,0,0,\dots)\in\mathbb{R}^{p},$

where $\sigma$ is an activation function, $b^{(i)}\in\mathbb{R}^{p}$,
$A^{(i)}\in\mathbb{R}^{p\times p}$ for all $i\in\\{1,...,n+1\\}$ and $\hat{X}$
is the time-augmented process of an $\mathbb{R}^{n}$-valued continuous
semimartingale $X$. We call $\mathcal{S}(\hat{X})$ the randomized signature of
dimension $p$ of $\hat{X}$, see [1, 13, 18, 19]. For a connection with neural
signature kernels and controlled ResNets we refer to [12]. A _randomized-
signature portfolio of dimension $P$_ is a linear path-functional portfolio
with $\mathcal{V}=\\{1,\dots,P\\}$, where $P$ is the dimension of the
randomized signature, and with feature maps

$\phi^{p}(t,X_{[0,t]})=\langle e_{p},\mathcal{S}_{t}\rangle.$

Hence, randomized-signature portfolios are path-functional portfolios, where
the portfolio controlling functions are linear combinations of the elements of
the randomized signature.

######  Remark 3.8.

We have highlighted signature, JL- and randomized-signature portfolios as
linear path-functional portfolios of special interest in this paper. However,
we would like to mention some other examples of linear path-functional
portfolios whose portfolio controlling functions are

  * •

increments of the signature. That is, for a fixed time span $\Theta$, we
observe a rolling window of length $\Theta<T$ and forget the information
before that, i.e.

$\phi^{I}(t,X_{[0,t]})=\langle
e_{I},\hat{\mathbb{X}}_{0\vee(t-\Theta),t}\rangle.$

We will refer to such portfolios as signature portfolios _with rolling
windows_. We shall not put particular emphasis on these portfolios in this
paper, since our approximation results in Section 4 are tailored to signature
portfolios _without_ rolling windows and since portfolios with rolling windows
are numerically less tractable, as we outline in Section 7.

  * •

random neural networks, which are neural networks where the parameters of the
hidden layers are not trained but randomly sampled and only the (linear) read-
out layer is trained, see [37]. Training the read-out layer, exactly amounts
to training the parameters $\\{l_{\nu}^{i}\\}_{1\leq i\leq
d,\nu\in\mathcal{V}}$ of path-functional portfolios. Such neural networks may
have an infinite-dimensional input space, corresponding in our context to path
spaces, as e.g. considered in [22] or [7].

  * •

given by reservoir computers, see for instance [36]. A special class of which
are Echo State Networks [40, 35, 34] or Quantum Reservoir Computers [56, 32].
Note, that there is an interesting connection between the Johnson-
Lindenstrauss projection of signatures, randomized signature and reservoir
computing, which is worked out in [18].

  * •

given by a constant, which leads to the case of constant portfolio weights
when using a constant auxiliary portfolio $\tau$. In this case we recover the
Markowitz portfolio optimization as a very special case in Subsection 5.1.
Moreover, since the first element of the signature is constant, this class of
portfolios is also included in the class of signature portfolios.

##  4 Approximation Properties of Signature Portfolios

This section is dedicated to prove universal approximation properties of
signature portfolios. Let us recall that for a compact subset
$K\subset\mathcal{G}_{T}^{2}(\varphi,x)$ the set

$\displaystyle\mathcal{K}_{\Lambda}=\lambda([0,T]\times
K)=\left\\{\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\mid\,t\in[0,T]\textrm{ and
}\hat{\mathbb{X}}^{2}_{[0,T]}(\omega)\in K\right\\}$ (4.1)

is a compact subset of $\Lambda_{T}^{2}(\varphi,x)$ by the continuity of
$\lambda$, see Lemma A.2. Furthermore, recall that in Definition 3.1 we have
introduced portfolio controlling functions $\\{f^{i}\\}_{1\leq i\leq d}$ as
non-anticipative path-functionals taking as input the paths of some
semimartingale $X$, i.e. the $\\{f^{i}\\}_{1\leq i\leq d}$ were of form
$f^{i}(t,X_{[0,t]})$. In the following, we will require a more specific form,
namely portfolio controlling functionals depending on the lifted path, that is
$f^{i}(\hat{\mathbb{X}}^{2}_{[0,t]})$, where the dependence on the lifted path
is important for the continuity statements. However, this is not a
contradiction to the general definition, in particular since
$\hat{\mathbb{X}}^{2}$ is a continuous semimartingale itself. Note that the
functions $\\{f^{i}\\}_{1\leq i\leq d}$ can of course only depend on the first
level $X_{[0,t]}$ as well.

######  Theorem 4.1.

Consider an arbitrary but fixed auxiliary portfolio $\tau$ and a path-
functional portfolio $\pi(\tau,\cdot)$ of type $I$ (type $II$) with portfolio
controlling functions $f^{i}\in C(\mathcal{K}_{\Lambda};\mathbb{R})$ for all
$i\in\\{1,\dots,d\\}$. Then for every $\epsilon>0$ there exists a signature
portfolio $\pi^{*}(\tau,\cdot)$ of type I (type II) such that it holds for all
$i\in\\{1,\dots,d\\}$

$\sup_{(t,\hat{\mathbb{X}}^{2}_{[0,T]}(\omega))\in[0,T]\times
K}|\pi^{i}(\tau,\hat{X}_{[0,t]})(\omega)-\pi^{*,i}(\tau,\hat{X}_{[0,t]})(\omega)|<\epsilon$

for almost all $\omega\in\Omega$.

Moreover, take $0\leq\alpha^{\prime}<\alpha<\frac{1}{2}$, an arbitrary but
fixed auxiliary portfolio $\tau$ and a path-functional portfolio
$\pi(\tau,\cdot)$ of type I (type II) with portfolio controlling functions
such that for all $i\in\\{1,\dots,d\\}$
$f^{i}\in\mathcal{B}_{\psi}(\Lambda^{N,\alpha^{\prime}}_{T}(\varphi,x))$. Then
for any continuous semimartingale $X$ with $X_{0}=x$ fulfilling
$\mathbb{E}\left[\sup_{t\in[0,T]}\exp(\zeta\|\widehat{\,\mathbb{X}^{t}}^{2}_{[0,T]}\|^{\xi}_{\alpha,[0,T]})\right]<\infty$
with $\zeta>0$, $\xi>1$ and for every $\epsilon,\delta>0$ there exists a
signature portfolio $\pi^{*}(\tau,\cdot)$ of type I (type II) such that it
holds for all $i\in\\{1,\dots,d\\}$

$\mathbb{P}\left[\sup_{t\in[0,T]}\Big{|}\pi^{i}(\tau,\hat{X}_{[0,t]})-\pi^{*,i}(\tau,\hat{X}_{[0,t]})\Big{|}>\epsilon\right]<\delta.$

###### Proof.

We present the proof for the case of portfolios of type $I$, however the proof
is analogous for the case of type $II$.

Let us start by showing the first statement of the theorem. By the universal
approximation result (Theorem 2.11), we know that for each $\\{f^{i}\\}_{1\leq
i\leq d}$ there exits a linear function on the signature of $\hat{X}$ that
approximates $f^{i}$ arbitrarily well, on compact sets. Those linear functions
can be chosen as the portfolio controlling functions $f^{i,\\*}$ of a
signature portfolio $\pi^{*}$. Hence, there exists for each
$\epsilon^{\prime}>0$ a signature portfolio $\pi^{*}$ of type $I$ such that
for all $i\in\\{1,...,d\\}$ it holds that

$\sup_{(t,\hat{\mathbb{X}}^{2}_{[0,T]}(\omega))\in[0,T]\times
K}|f^{i}(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega))-f^{*,i}(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega))|<\epsilon^{\prime}$
(4.2)

for almost all $\omega\in\Omega$.

Now, we show that this translates to the portfolio weights. The statement is
trivial for all $\omega\in\Omega$ where $\hat{\mathbb{X}}^{2}_{[0,T]}\notin
K$.

On $\Omega^{(K)}:=\\{\omega\mid\hat{\mathbb{X}}^{2}_{[0,T]}(\omega)\in K\\}$,
it is implied by (4.2) that

$\displaystyle\sup_{t\in[0,T]}\left|\pi^{i}_{t}-\pi^{*,i}_{t}\right|=\sup_{t\in[0,T]}\Bigg{|}\tau^{i}_{t}\left(f^{i}(\hat{\mathbb{X}}^{2}_{[0,t]})+1-\sum_{j=1}^{d}\tau_{t}^{j}f^{j}(\hat{\mathbb{X}}^{2}_{[0,t]})\right)$
$\displaystyle-\tau^{i}_{t}\left(f^{*,i}(\hat{\mathbb{X}}^{2}_{[0,t]})+1-\sum_{j=1}^{d}\tau_{t}^{j}f^{*,j}(\hat{\mathbb{X}}^{2}_{[0,t]})\right)\Bigg{|}$
$\displaystyle=\sup_{t\in[0,T]}\left|\tau_{t}^{i}\left(f^{i}(\hat{\mathbb{X}}^{2}_{[0,t]})-f^{*,i}(\hat{\mathbb{X}}^{2}_{[0,t]})\right)-\sum_{j=1}^{d}\tau_{t}^{j}\left(f^{j}(\hat{\mathbb{X}}^{2}_{[0,t]})-f^{*,j}(\hat{\mathbb{X}}^{2}_{[0,t]})\right)\right|$
$\displaystyle\leq\sup_{t\in[0,T]}\left|\tau_{t}^{i}\left(f^{i}(\hat{\mathbb{X}}^{2}_{[0,t]})-f^{*,i}(\hat{\mathbb{X}}^{2}_{[0,t]})\right)\right|+\sum_{j=1}^{d}\sup_{t\in[0,T]}\left|\tau_{t}^{j}\left(f^{j}(\hat{\mathbb{X}}^{2}_{[0,t]})-f^{*,j}(\hat{\mathbb{X}}^{2}_{[0,t]})\right)\right|$
$\displaystyle\leq\,M\sup_{t\in[0,T]}\left|f^{i}(\hat{\mathbb{X}}^{2}_{[0,t]})-f^{*,i}(\hat{\mathbb{X}}^{2}_{[0,t]})\right|+M\sum_{j=1}^{d}\sup_{t\in[0,T]}\left|f^{j}(\hat{\mathbb{X}}^{2}_{[0,t]})-f^{*,j}(\hat{\mathbb{X}}^{2}_{[0,t]})\right|$
$\displaystyle<M(d+1)\epsilon^{\prime}$

holds almost surely, where $M$ is the bound of $\tau$. The result follows by
choosing $\epsilon^{\prime}=\frac{\epsilon}{M(d+1)}$.

The proof of the second approximation statement follows by similar arguments.
Recall Theorem 2.18 and use the same reasoning as above to translate the
approximation result from the portfolio controlling functions to the portfolio
weights itself, i.e. we obtain that there exists for each
$\epsilon^{\prime}>0$ a signature portfolio $\pi^{*}$ of type $I$ such that
for all $i\in\\{1,...,d\\}$ it holds that

$\sup_{t\in[0,T]}\frac{|\pi^{i}(\tau,\hat{X}_{[0,t]})-\pi^{*,i}(\tau,\hat{X}_{[0,t]})|}{\exp(\zeta\|\widehat{\,\mathbb{X}^{t}}^{2}_{[0,T]}\|^{\xi}_{\alpha,[0,T]})}<\epsilon^{\prime}$
(4.3)

almost surely. Therefore,

$\displaystyle\mathbb{P}\left[\sup_{t\in[0,T]}\Big{|}\pi^{i}(\tau,\hat{X}_{[0,t]})-\pi^{*,i}(\tau,\hat{X}_{[0,t]})\Big{|}>\epsilon\right]$
$\displaystyle\leq\epsilon^{-1}\cdot\mathbb{E}\left[\sup_{t\in[0,T]}\Big{|}\pi^{i}(\tau,\hat{X}_{[0,t]})-\pi^{*,i}(\tau,\hat{X}_{[0,t]})\Big{|}\right]$
$\displaystyle<\frac{\epsilon^{\prime}}{\epsilon}\mathbb{E}\left[\sup_{t\in[0,T]}\exp(\zeta\|\widehat{\,\mathbb{X}^{t}}^{2}_{[0,T]}\|^{\xi}_{\alpha,[0,T]})\right],$

where the first inequality is due to Markov’s inequality and the second one
follows from (4.3). Hence the statement is obtained by setting
$\delta=\frac{\epsilon^{\prime}}{\epsilon}\mathbb{E}\left[\sup_{t\in[0,T]}\exp(\zeta\|\widehat{\,\mathbb{X}^{t}}^{2}_{[0,T]}\|^{\xi}_{\alpha,[0,T]})\right]$
and choosing $\epsilon^{\prime}$ accordingly.

∎

######  Remark 4.2.

Note, that by Corollary 2.12, Theorem 4.1 also ensures that signature
portfolios approximate classical functionally generated portfolios arbitrarily
well, on compact sets. In particular, this includes the functionally generated
portfolios as introduced by Fernholz [27] and those considered for functional
portfolio optimization in [10].

Having obtained a universal approximation theorem of path-functional
portfolios, we will now apply it to obtain universal approximation results of
the growth-optimal portfolio by signature portfolios in several markets.

###  4.1 Universal Approximation of the Growth-Optimal Portfolio

In this subsection, we present a large class of markets where the growth-
optimal portfolio can be regarded as a path-functional portfolio and show that
it can therefore be approximated by a signature portfolio.

The following statements are to be understood in the market setting outlined
in Definition 2.29 and we therefore assume that the corresponding necessary
conditions are satisfied, in particular those ensuring the existence of the
growth-optimal portfolio (Lemma 2.30). Moreover, recall also the definition of
the compact set $\mathcal{K}_{\Lambda}$ given in (4.1).

######  Theorem 4.3.

Consider the following class of markets with $d$ stocks:

$dS_{t}=\operatorname{diag}(S_{t})(a(\hat{\mathbb{X}}^{2}_{[0,t]})dt+\Sigma(\hat{\mathbb{X}}^{2}_{[0,t]})dB_{t}),$
(4.4)

where the components of $a$, $\Sigma$ are non-anticipative path-functionals in
$C(\mathcal{K}_{\Lambda};\mathbb{R})$ and $X$ some $\mathbb{R}^{n}$-valued
semimartingale with $n\in\mathbb{N}$. Then for any auxiliary portfolio $\tau$
there exists a signature portfolio of type $II$ which approximates the weights
of the growth-optimal portfolio almost surely arbitrarily well on
$\mathcal{K}_{\Lambda}$. Moreover, if the components of the auxiliary
portfolio $\tau$ are deterministic, then there also exists a signature
portfolio of type $I$ which approximates the weights of the growth-optimal
portfolio almost surely arbitrarily well on $\mathcal{K}_{\Lambda}$.

Alternatively, let $0\leq\alpha^{\prime}<\alpha$ and let $a,\Sigma$ be such
that the components of $\pi^{(g)}$ are non-anticipative path-functionals in
$\mathcal{B}_{\psi}(\Lambda_{T}^{N,\alpha^{\prime}}(\varphi,x))$.444Note that
a sufficient condition for this to hold is if the components of $a$ are non-
anticipative path-functionals in
$\mathcal{B}_{\psi}(\Lambda_{T}^{N,\alpha^{\prime}}(\varphi,x))$ and $\Sigma$
is constant. Then it holds for any continuous semimartingale $X$ with
$X_{0}=x$, fulfilling
$\mathbb{E}\left[\sup_{t\in[0,T]}\exp(\zeta\|\widehat{\,\mathbb{X}^{t}}^{2}_{[0,T]}\|^{\xi}_{\alpha,[0,T]})\right]<\infty$
with $\zeta>0$, $\xi>1$ and for every $\epsilon,\delta>0$ that there exists a
signature portfolio $\pi^{*}(\tau,\cdot)$ for type $II$ for any auxiliary
portfolio $\tau$ (or of type $I$ for a deterministic portfolio $\tau$) such
that it holds for all $i\in\\{1,\dots,d\\}$

$\mathbb{P}\left[\sup_{t\in[0,T]}\Big{|}\pi^{(g),i}(\tau,\hat{X}_{[0,t]})-\pi^{*,i}(\tau,\hat{X}_{[0,t]})\Big{|}>\epsilon\right]<\delta,$

where $\pi^{(g)}$ is the growth-optimal portfolio of the respective market.

######  Remark 4.4.

Note that the semimartingale $X$ in (4.4) could be (a functions of) $S$, so
that we are actually dealing with path-dependent SDEs. In the particular case
where $a$ and $\Sigma$ are entire functions of the signature as introduced in
[23] we are then in the tractable setup of _signature SDEs._

###### Proof of Theorem 4.3.

Recalling Lemma 2.30, it is straightforward that for any auxiliary portfolio
$\tau$, the growth-optimal portfolio $\pi^{(g)}$ is a path-functional
portfolio of type $II$ with portfolio controlling functions

$f^{(g,II),i}=\pi^{(g),i}.$

Likewise, $\pi^{(g)}$ is a path-functional portfolio of type $I$ for any
deterministic auxiliary portfolio and portfolio controlling functions

$f^{(g,I),i}=\tau^{i}_{\cdot}\cdot\pi^{(g),i}.$

By Lemma B.1 and the form of $\pi^{(g),i}$ Theorem 4.1 is applicable for
$\pi^{(g),i}$ (viewed as a path-functional portfolio of type $I$ resp. type
$II$). Hence, the statements follow. ∎

######  Remark 4.5.

If $X=\mu$ or $X=S$, the growth optimal is also a path-functional portfolio of
type $I$ with auxiliary portfolio $\tau=\mu$.

######  Remark 4.6 (Examples of Markovian Markets).

We here give two prominent examples of markets, to which Theorem 4.3 can be
applied.

  * •

Black-Scholes Market:

$\textrm{d}S_{t}=\operatorname{diag}(S_{t})(adt+\Sigma dB_{t}),$

where $a$ and $\Sigma$ are constant.

  * •

Volatility Stabilized Markets:

$\frac{\textrm{d}S^{i}_{t}}{S^{i}_{t}}=\frac{1+\alpha}{2}\frac{1}{\mu_{t}^{i}}dt+\sqrt{\frac{1}{\mu_{t}^{i}}}dB_{t}^{i}\textrm{
for all }1\leq i\leq d$

and $\alpha\geq 0$. Volatility stabilized markets are of great interest in SPT
because they reflect the observation in real markets that smaller stocks tend
to have greater volatility than larger stocks, see for example [29]. See also
[15, 28] for further properties of volatility stabilized markets.

###  4.2 A Class of Non-Markovian Markets where the Growth-Optimal Portfolio
is a Signature Portfolio

Let us now turn to a class of (possibly) non-Markovian markets, where the
growth-optimal portfolio can not only be approximated by a signature portfolio
but _is_ a signature portfolio.

######  Theorem 4.7.

Let $X$ be a continuous semimartingale. For a market of $d$ stocks, consider
the class of _Sig-market_ models

$dS_{t}=\operatorname{diag}(S_{t})(a(\hat{X}_{[0,t]})dt+\Sigma dB_{t})$

where $a(\hat{X}_{[0,t]})_{i}=\sum_{0\leq|I|\leq N}\alpha^{(i)}_{I}\langle
e_{I},\hat{\mathbb{X}}_{t}\rangle$ for $N\geq 0$,
$\alpha^{(i)}_{I}\in\mathbb{R}$, $\Sigma$ is a constant $d\times m$ matrix and
$B_{t}$ is an $m$-dimensional Brownian motion for $m\geq d$. Then, the growth-
optimal portfolio is for any auxiliary portfolio a signature portfolio of type
$II$ and a signature portfolio of type $I$ for any constant and deterministic
auxiliary portfolio $\tau$.

###### Proof.

The integrability and measurability assumptions required in Definition 2.29
and Lemma 2.30 follow from the integrability and measurability of elements of
the signature. Applying Lemma 2.30, it follows that the growth-optimal weights
are linear functions on the signature of $\hat{X}$. This is for any auxiliary
portfolio a signature portfolio of type $II$, or a signature portfolio of type
$I$ for any constant and deterministic auxiliary portfolio $\tau$. ∎

######  Remark 4.8 (Existence of Solutions to Sig-Markets).

As addressed in Remark 4.4, the existence of solutions is non-trivial for Sig-
markets if $X=h(S)$, i.e. if the semimartingale of which we construct the
signature is a function of the price process itself. In particular, if $h$ is
a real analytic function, then we deal with signature-SDEs in the spirit of
[23]. The study of existence of solutions to such equations goes beyond the
subject of this paper, however, we give a simple example below, where the
existence is guaranteed. To this end, let us denote by

$\mathcal{I}_{1}:=\left\\{I|I=(i_{1},...,i_{m})\textrm{ for }0\leq n\leq
N\textrm{ and }i_{2}=i_{3}=\dots=i_{n}=1\right\\}$

Let $X=\log{S_{t}}$. If $\alpha_{I}^{(i)}=0$ for all $I\notin\mathcal{I}_{1}$,
then

${dS_{t}}=\operatorname{diag}(S_{t})\cdot\left(a((\widehat{\log{S}})_{[0,t]})dt+\Sigma
dB_{t}\right)$ (4.5)

admits a unique strong solution. To see this, note that (4.5) can be seen a
part of a (larger) system of linear equations. Setting $Y=\log{S}$,
$\hat{Y}=(t,Y)$ and denoting by
$\hat{\mathbb{Y}}^{N}\lvert_{\mathcal{I}_{1}}=(\langle
e_{I},\hat{\mathbb{Y}}\rangle)_{I\in\mathcal{I}_{1}}$ the restriction to
elements of the signature corresponding to multiindices in $\mathcal{I}_{1}$.
Let us denote the vectorization of the truncated signature denoted by
$\mathbf{vec}(\cdot)$ and fix a labelling function
$\mathscr{L}:\mathcal{I}_{1}\rightarrow\\{1,\dots,|\mathcal{I}_{1}|\\}$ such
that
$\mathbf{vec}(\hat{\mathbb{Y}}^{N}\lvert_{\mathcal{I}_{1}})_{\mathscr{L}(I)}=\langle
e_{I},\hat{\mathbb{Y}}^{N}\lvert_{\mathcal{I}_{1}}\rangle$. Moreover, for a
multiindex $I=(i_{1},\dots,i_{|I|})$, we denote by
$I^{\prime}:=(i_{1},\dots,i_{|I|-1})$ corresponding multiindex shortened by
the last letter. Then said linear system of equations is

$d\left(\mathbf{vec}(\hat{\mathbb{Y}}^{N}\lvert_{\mathcal{I}_{1}})_{t}\right)=\left(b+A\cdot\mathbf{vec}(\hat{\mathbb{Y}}^{N}\lvert_{\mathcal{I}_{1}})_{t}\right)dt+\tilde{\Sigma}dB_{t}.$

where $\tilde{\Sigma}_{j,r}=\Sigma_{s,r}\delta_{j,\mathscr{L}(s)}$ for $1\leq
j\leq|\mathcal{I}_{1}|$, $1\leq r\leq m$ and for
$k,l\in\\{1,\dots,|\mathcal{I}_{1}\\}$

$\displaystyle b_{k}=$
$\displaystyle\begin{cases}-\frac{1}{2}\Sigma^{\mathsf{T}}\Sigma+a_{\emptyset}^{(i)}&\textrm{
if }k=\mathscr{L}(i),\,i\neq 1\\\ 1&\textrm{ if }k=\mathscr{L}(1)\\\
0&\textrm{ else}\end{cases}$ $\displaystyle A_{k,l}=$
$\displaystyle\begin{cases}a^{(i)}_{\mathscr{L}(J)}&\textrm{ if
}k=\mathscr{L}(i),\,l=\mathscr{L}(J),\,i\neq 1\\\ 1&\textrm{ if }|I|\geq
2,\,k=\mathscr{L}(I),\,l=\mathscr{L}(I^{\prime})\\\ 0&\textrm{
else.}\end{cases}$

######  Remark 4.9 (NUPBR in Sig-Markets).

The NUPBR condition in the above Sig-markets holds. This follows from the
existence of the growth-optimal portfolio by Lemma 2.30, Lemma 2.31 and
Theorem 2.25.

##  5 Optimization Tasks for Linear Path-Functional Portfolios

We now want to study some tasks for portfolio optimization and their form for
linear path-functional portfolios. To formulate them in the most general way,
let us introduce what we call a _universe_ of stocks.

######  Definition 5.1 (Universe).

Consider a market of $d$ stocks with capitalization process
$S=(S^{1},...,S^{d})$. We call $\mathcal{U}\subseteq\\{1,...,d\\}$ a
_universe_ of stocks with capitalization process
$S^{\mathcal{U}}=(S^{u})_{u\in\mathcal{U}}$. Moreover, we define the universe
weights (resp. universe portfolio) $\mu^{\mathcal{U}}$ to be

$\mu^{\mathcal{U},i}_{t}=\begin{cases}\frac{S^{\mathcal{U},i}_{t}}{\sum_{j\in\mathcal{U}}S^{\mathcal{U},j}_{t}}&\textrm{if
}i\in\mathcal{U}\\\ 0&\textrm{otherwise}\end{cases}.$

Moreover, denote by $(W_{t}^{\mathcal{U}})_{t\in[0,T]}$ the wealth process of
universe $\mathcal{U}$, which is given by

$W_{t}^{\mathcal{U}}=\frac{\sum_{i\in\mathcal{U}}S_{t}^{i}}{\sum_{i\in\mathcal{U}}S_{t_{0}}^{i}}.$

This notion of a universe of stocks is useful if one does not necessarily want
to invest in all the stocks in the market but just in a subset of stocks. Of
course, this can always be achieved by fixing certain weights of a portfolio
to be zero, however using our notion of a universe of stocks is particularly
useful, if one wants to compare the wealth process of a portfolio to the
wealth process of the universe. Let us make this precise:

######  Corollary 5.2.

Take a universe $\mathcal{U}\subseteq\\{1,...,d\\}$ and consider a portfolio
$\pi$ where $\pi^{i}\equiv 0$ for $i\not\in\mathcal{U}$. Then, the relative
wealth process of $\pi$ with respect to $\mu^{\mathcal{U}}$ is given by

$\textrm{d}\left(\frac{W^{\pi}_{t}}{W^{{\mathcal{U}}}_{t}}\right)=\left(\frac{W^{\pi}_{t}}{W^{{\mathcal{U}}}_{t}}\right)\sum_{i\in\mathcal{U}}\pi_{t}^{i}\frac{\textrm{d}\mu_{t}^{\mathcal{U},i}}{\mu_{t}^{\mathcal{U},i}}.$
(5.1)

Or, equivalently

$\textrm{d}\log\left(\frac{W^{\pi}_{t}}{W^{{\mathcal{U}}}_{t}}\right)=\sum_{i\in\mathcal{U}}\pi_{t}^{i}\frac{\textrm{d}\mu_{t}^{\mathcal{U},i}}{\mu_{t}^{\mathcal{U},i}}-\frac{1}{2}\sum_{i,j\in\mathcal{U}}\frac{\pi^{i}_{t}}{\mu^{\mathcal{U},i}_{t}}\frac{\pi^{j}_{t}}{\mu^{\mathcal{U},j}_{t}}\textrm{d}[\mu^{\mathcal{U},i},\mu^{\mathcal{U},j}]_{t}.$
(5.2)

###### Proof.

Using the definition of $\pi$ and $W^{\mathcal{U}}$, the statement follows
directly from (2.3) and applying Itô’s formula. ∎

######  Remark 5.3.

When we consider path-functional portfolios investing in a universe
$\mathcal{U}$, it is convenient to construct them via the universe portfolio
$\mu^{\mathcal{U}}$. Note that for path-functional portfolios $\pi^{(I)}$ of
type $I$ it directly follows that $\pi^{(I),i}(\mu^{\mathcal{U}},\cdot)=0$ for
$i\not\in\mathcal{U}$. However, for path-functional portfolios $\pi^{(II)}$ of
type $II$, we have to require additionally that
$\pi^{(II),i}(\mu^{\mathcal{U}},\cdot)=0$ for $i\not\in\mathcal{U}$ by setting
$f^{i}\equiv 0$ for $i\not\in\mathcal{U}$.

######  Remark 5.4 (Investments in Ranked Markets).

Recall that our very basic assumptions on the financial market were that the
stocks’ capitalizations are positive continuous semimartingales. Considering a
market $M$ which fulfils these assumptions, we know that there exists another
market which also fulfils these assumptions and for which the stocks’
capitalizations are the same as the ranked capitalizations of the market $M$.
This follows simply from the fact that the ranked capitalizations are again
positive continuous semimartingales. Therefore, the optimization problems we
present in the following hold also for investments in the ranked market.

It is sometimes useful to write optimization tasks in a vectorized form. To
this end, let us introduce what we call a _labelling function_.

######  Definition 5.5.

For a given linear path-functional portfolio $\pi$ investing in a universe of
stocks $\mathcal{U}$, let $\mathcal{V}$ be the set of features,
$|\mathcal{V}|$ the number of features and $|\mathcal{U}|$ the number of
stocks in a universe. A bijective function $\mathscr{L}$ of the form

$\displaystyle\mathscr{L}:\mathcal{U}\times\mathcal{V}$
$\displaystyle\longrightarrow\\{1,...,|\mathcal{U}|\cdot|\mathcal{V}|\\}$
$\displaystyle(i,\nu)$ $\displaystyle\longmapsto\mathscr{L}(i,\nu)$

is called _labelling function._

Before studying two concrete optimization tasks for linear path-functional
portfolios, let us make the following statement about a more general class of
optimization tasks, which turn out to be _quadratic optimization problems_ for
linear path-functional portfolios:

######  Proposition 5.6.

Let us denote by $\mathbb{S}^{d}$ the space of symmetric matrices of dimension
$d\times d$. For a universe $\mathcal{U}$, consider a class of linear path-
functional portfolios $\Pi$ (of either type) given by a feature space
$\mathcal{V}$, a collection of feature maps
$\\{\phi^{\nu}\\}_{\nu\in\mathcal{V}}$ consisting of non-anticipative path-
functionals, a continuous semimartingale $X$ being the underlying process and
an auxiliary portfolio $\tau$. Recall that the portfolio controlling
functionals $\\{f^{i}\\}_{i\in\mathcal{U}}$ of such portfolios are given by
$f^{i}(t,X_{[0,t]})=\sum_{\nu\in\mathcal{V}}l^{i}_{\nu}\phi^{\nu}(t,X_{[0,t]})$
for $i\in\mathcal{U}$. Then, any optimization problem of the form

$\inf_{\pi\in\Pi}\mathbb{E}\left[\int_{t_{0}}^{t}\pi^{\mathsf{T}}_{s}C_{s}\pi_{s}\lambda_{1}(ds)-\int_{t_{0}}^{t}b_{s}^{\mathsf{T}}\pi_{s}\lambda_{2}(ds)\right]$

where $b$, $C$ are stochastic processes with values in $\mathbb{R}^{d}$,
$\mathbb{S}^{d}$ respectively and $\lambda^{i}$ are signed measures on
$[t_{0},t]$, is a quadratic optimization problem in the linear coefficients
$\\{l_{\nu}^{i}\\}_{\nu\in\mathcal{V},1\leq i\leq d}$.

###### Proof.

Recall the form of linear path-functional portfolios of type $I$ and $II$
respectively. Therefore, it is clear that the parameters $\\{l_{\nu}^{i}\\}$
appear at most in quadratic terms. Moreover, since they are constant in time
and deterministic, they can be pulled outside of the integrals.

∎

Let us now, present two prominent optimization tasks which fall into the above
class, namely _mean-variance and log-wealth optimization_. We assume that the
following optimization problems are well-posed and that the expectations are
finite.

###  5.1 Mean-Variance Optimization

In this optimization problem, we consider a strategy, where we invest at time
$t$ with portfolio weights $\pi_{t}$ of a linear path-functional portfolio
into a universe $\mathcal{U}$ and follow a buy-and-hold strategy over the
time-span $\Delta$. The (relative) return of this strategy over the time-
horizon $[t,t+\Delta]$ is given by

$\mathcal{R}^{W,\pi}_{t,t+\Delta}=\frac{W^{\pi}_{t+\Delta}}{W^{\pi}_{t}}-1=\sum_{j\in\mathcal{U}}\pi_{t}^{j}\frac{S_{t+\Delta}^{j}}{S_{t}^{j}}-1$
(5.3)

and respectively the relative return (relative to the universe portfolio
$\mu^{\mathcal{U}}$)

$\mathcal{R}^{V,\pi}_{t,t+\Delta}=\frac{V^{\pi}_{t+\Delta}}{V^{\pi}_{t}}-1=\sum_{j\in\mathcal{U}}\pi_{t}^{j}\frac{\mu_{t+\Delta}^{\mathcal{U},j}}{\mu_{t}^{\mathcal{U},j}}-1.$
(5.4)

In this setting, we can state the following corollary about the mean-variance
optimization problem:

######  Corollary 5.7.

Let $\lambda\in\mathbb{R}_{+}$ be a risk-factor, specifying the investor’s
risk-preferences. Then, for a linear path-functional portfolio $\pi(\tau,X)$
the following two optimization problems are convex quadratic optimization
problems in the parameters
$\\{l_{\nu}^{i}\\}_{i\in\mathcal{U},\nu\in\mathcal{V}}$. More precisely, for
the

  1. 1.

mean-variance optimization of returns we have

$\displaystyle\min_{\pi}\hskip
2.84544pt\mathrm{Var}\left(\mathcal{R}^{W,\pi}_{t,t+\Delta}\right)-\lambda\mathbb{E}\left(\mathcal{R}^{W,\pi}_{t,t+\Delta}\right)$
$\displaystyle\Leftrightarrow$ $\displaystyle\min_{\\{l_{\nu}^{i}\\}}\hskip
2.84544pt\mathbf{l}^{\mathsf{T}}\mathrm{Var}\left(\mathbf{Y}^{W}\right)\mathbf{l}-\mathbf{l}^{\mathsf{T}}\left(\lambda\mathbb{E}\left(\mathbf{Y}^{W}\right)-\boldsymbol{\sigma}^{W}\right)$

for $\mathbf{l}_{\mathscr{L}(i,\nu)}=l_{\nu}^{i}$ ,
$\boldsymbol{\sigma}^{W}_{\mathscr{L}(i,\nu)}=2\mathrm{Cov}(\mathbf{Y}^{W}_{\mathscr{L}(i,\nu)},\mathcal{R}^{W,\tau}_{t,t+\Delta})$
and

$\mathbf{Y}^{W}_{\mathscr{L}(i,\nu)}=\begin{cases}\tau_{t}^{i}\phi^{\nu}(t,X_{[0,t]})\left(\frac{S^{i}_{t+\Delta}}{S^{i}_{t}}-\mathcal{R}^{W,\tau}_{t,t+\Delta}-1\right)&\textrm{
if $\pi$ is of type $I$}\\\
\phi^{\nu}(t,X_{[0,t]})\left(\frac{S^{i}_{t+\Delta}}{S^{i}_{t}}-\mathcal{R}^{W,\tau}_{t,t+\Delta}-1\right)&\textrm{
if $\pi$ is of type $II$}.\end{cases}$

  2. 2.

mean-variance optimization of relative return we have

$\displaystyle\min_{\pi}\hskip
2.84544pt\mathrm{Var}\left(\mathcal{R}^{V,\pi}_{t,t+\Delta}\right)-\lambda\mathbb{E}\left(\mathcal{R}^{V,\pi}_{t,t+\Delta}\right)$
$\displaystyle\Leftrightarrow$ $\displaystyle\min_{\\{l_{\nu}^{i}\\}}\hskip
2.84544pt\mathbf{l}^{\mathsf{T}}\mathrm{Var}\left(\mathbf{Y}^{V}\right)\mathbf{l}-\mathbf{l}^{\mathsf{T}}\left(\lambda\mathbb{E}\left(\mathbf{Y}^{V}\right)-\boldsymbol{\sigma}^{V}\right)$

for $\mathbf{l}_{\mathscr{L}(i,\nu)}=l_{\nu}^{i}$,
$\boldsymbol{\sigma}^{V}_{\mathscr{L}(i,\nu)}=2\mathrm{Cov}(\mathbf{Y}^{V}_{\mathscr{L}(i,\nu)},\mathcal{R}^{V,\tau}_{t,t+\Delta})$
and

$\mathbf{Y}^{V}_{\mathscr{L}(i,\nu)}=\begin{cases}\tau_{t}^{i}\phi^{\nu}(t,X_{[0,t]})\left(\frac{\mu^{\mathcal{U},i}_{t+\Delta}}{\mu^{\mathcal{U},i}_{t}}-\mathcal{R}^{V,\tau}_{t,t+\Delta}-1\right)&\textrm{
if $\pi$ is of type $I$}\\\
\phi^{\nu}(t,X_{[0,t]})\left(\frac{\mu^{\mathcal{U},i}_{t+\Delta}}{\mu^{\mathcal{U},i}_{t}}-\mathcal{R}^{V,\tau}_{t,t+\Delta}-1\right)&\textrm{
if $\pi$ is of type $II$}.\end{cases}$

###### Proof.

We proof the above statement for the case of $\mathcal{R}^{W,\pi}$ and for
$\pi$ of type $I$. The proof of the other cases is completely analogous.

$\displaystyle\mathcal{R}^{W,\pi}_{t,t+\Delta}$
$\displaystyle=\sum_{i\in\mathcal{U}}\tau_{t}^{i}\left(f^{i}(t,X_{[0,t]})+1-\sum_{j\in\mathcal{U}}\tau_{t}^{j}f^{j}(t,X_{[0,t]})\right)\left(\frac{S_{t+\Delta}^{i}}{S_{t}^{i}}\right)-1$
$\displaystyle=\sum_{i\in\mathcal{U}}\tau^{i}_{t}f^{i}(t,X_{[0,t]})\left(\frac{S^{i}_{t+\Delta}}{S_{t}^{i}}-\mathcal{R}^{W,\tau}_{t,t+\Delta}-1\right)+\mathcal{R}^{W,\tau}_{t,t+\Delta}+1-1$
$\displaystyle=\mathbf{l}^{\mathsf{T}}\mathbf{Y}^{W}+\mathcal{R}^{W,\tau}_{t,t+\Delta}$

Hence,

$\displaystyle\mathbb{E}\left(\mathcal{R}^{W,\pi}_{t,t+\Delta}\right)=\mathbf{l}^{\mathsf{T}}\mathbb{E}\left(\mathbf{Y}^{W}\right)+\mathbb{E}\left(\mathcal{R}^{W,\tau}_{t,t+\Delta}\right).$
(5.5)

Note that the rightmost term of (5.5) do not contribute in the optimization
problem, as they are independent of $\mathbf{l}$. And,

$\displaystyle\mathrm{Var}\left(\mathcal{R}^{W,\pi}_{t,t+\Delta}\right)$
$\displaystyle=\mathrm{Var}\left(\mathbf{l}^{\mathsf{T}}\mathbf{Y}^{W}\right)+2\mathrm{Cov}\left(\mathbf{l}^{\mathsf{T}}\mathbf{Y}^{W},\mathcal{R}^{W,\tau}_{t,t+\Delta}\right)+\mathrm{Var}\left(\mathcal{R}^{W,\tau}_{t,t+\Delta}\right)$
$\displaystyle=\mathbf{l}^{\mathsf{T}}\mathrm{Var}\left(\mathbf{Y}^{W}\right)\mathbf{l}+\mathbf{l}^{\mathsf{T}}\boldsymbol{\sigma}^{W}+\mathrm{Var}\left(\mathcal{R}^{W,\tau}_{t,t+\Delta}\right),$
(5.6)

where the rightmost term of (5.6) does not contribute in the optimization
problem. The fact that the optimization problem is convex follows directly
from the fact that the covariance matrix
$\mathrm{Var}\left(\mathbf{Y}^{W}\right)$ is positive semi-definite. ∎

######  Remark 5.8.

Note that the expression of the optimization problem for the relative return
simplifies considerably if we use a linear path-functional portfolio of type
$I$ and choose the universe portfolio as the auxiliary portfolio, i.e.
$\tau=\mu^{\mathcal{U}}$. To be precise, we obtain

$\displaystyle\min_{\pi}\hskip
2.84544pt\mathrm{Var}\left(\mathcal{R}^{V,\pi}_{t,t+\Delta}\right)-\lambda\mathbb{E}\left(\mathcal{R}^{V,\pi}_{t,t+\Delta}\right)$
$\displaystyle\Leftrightarrow$ $\displaystyle\min_{\\{l_{\nu}^{i}\\}}\hskip
2.84544pt\mathbf{l}^{\mathsf{T}}\mathrm{Var}\left(\mathbf{Y}^{V}\right)\mathbf{l}-\lambda\mathbf{l}^{\mathsf{T}}\mathbb{E}\left(\mathbf{Y}^{V}\right)$

for $\mathbf{l}_{\mathscr{L}(i,\nu)}=l_{\nu}^{i}$ and
$\mathbf{Y}^{V}_{\mathscr{L}(i,\nu)}=\phi^{\nu}(t,X_{[0,t]})(\mu^{\mathcal{U},i}_{t+\Delta}-\mu^{\mathcal{U},i}_{t})$,
since $\mathcal{R}^{V,\mu^{\mathcal{U}}}_{t,t+\Delta}=0$.

######  Remark 5.9.

If $\textrm{Var}(\mathbf{Y}^{V})$ or $\textrm{Var}(\mathbf{Y}^{W})$
respectively are almost surely invertible, that is the optimization problem is
_strictly_ convex and hence admits a unique set of optimal parameters, then
the optimal vector $\mathbf{l}^{*}$ can be found via first-order conditions,
i.e.

$\mathbf{l}^{*}=\textrm{Var}(\mathbf{Y}^{V})^{-1}\left(\lambda\mathbb{E}(\mathbf{Y}^{V})-\mathbf{\sigma}^{V}\right)$

and likewise for the case of optimizing returns. Note that the problem is
never strictly convex for _any_ linear path-functional portfolio of type $I$,
because due to the form of such portfolios, the weights are invariant under
scalar shifts of the parameters $\\{l_{\nu}^{i}\\}_{\nu\in\mathcal{V},1\leq
i\leq d}$. More precisely, for linear path-functional portfolios of type $I$,
let $\pi^{*}(\tau,X)$ be optimal portfolio weights corresponding to an optimal
set $\\{l^{*,i}_{\nu}\\}_{\nu\in\mathcal{V},1\leq i\leq d}$, then the set
$\\{l^{*,i}_{\nu}+\alpha\\}_{\nu\in\mathcal{V},1\leq i\leq d}$ for
$\alpha\in\mathbb{R}$ results in _the same_ portfolio weights
$\pi^{*}(\tau,X)$.

######  Remark 5.10.

Let us here outline how the mean-variance optimization problem (of the
relative returns) of Corollary 5.7 can be related to a (model-free) real
market situation where only the observation of one trajectory is available
(see also Section 7.2). Tailored to Corollary 5.7 we assume a discrete time
situation with time steps of length $\Delta$ and suppose that the conditional
law of $\mu_{i\Delta}$ is time-homogeneous and depends on the last $n\wedge i$
values of $\mu$ for some $n\in\mathbb{N}$. More precisely, given
$(\mu_{(i-n)\Delta\wedge
0},\ldots,\mu_{(i-1)\Delta})=(x_{0},\ldots,x_{(n-1)\wedge(i-1)})\in(\Delta^{d})^{n\wedge
i}$, we know the conditional law
$\rho(x_{0},\ldots,x_{(n-1)\wedge(i-1)},\cdot)$ of $\mu_{i\Delta}$. Consider
now a portfolio $\pi$ that is a non-anticipative path functional depending
also only on the past $n$ values of $\mu$, i.e. a portfolio with rolling
window in the spirit of Remark 3.8. Then for $i\geq n$

$\displaystyle\mathbb{E}\left[\frac{V^{\pi}_{i\Delta}}{V^{\pi}_{(i-1)\Delta}}\bigg{|}\mathcal{F}_{(i-1)\Delta}\right]$
$\displaystyle=\mathbb{E}[(R_{(i-1)\Delta,i\Delta}^{V,\pi}+1)|\mathcal{F}_{(i-1)\Delta}]$
$\displaystyle=\int_{\Delta^{d}}\sum_{j=1}^{d}\pi^{j}(x_{0},\ldots,x_{n-1})\frac{y^{j}}{x^{j}_{n-1}}\rho(x_{0},\ldots,x_{n-1},dy)$

and similarly for $i<n$, taking just the history starting from $0$ into
account. Assume now further that we have an invariant measure for the
distribution of the rolling path segments of length $n$, which we denote by
$\rho$. Then under this invariant measure we have for each $i\geq n$

$\displaystyle\mathbb{E}\left[\frac{V^{\pi}_{i\Delta}}{V^{\pi}_{(i-1)\Delta}}\right]$
$\displaystyle=\int_{(\Delta^{d})^{n}}\int_{\Delta^{d}}\sum_{j=1}^{d}\pi^{j}(x_{0},\ldots,x_{n-1})\frac{y^{j}}{x^{j}_{n-1}}\rho(x_{0},\ldots,x_{n-1},dy)\rho(dx_{0},\ldots
dx_{n-1})$ $\displaystyle=:M^{\pi}$

and accordingly for $i<n$ assuming compatible invariant measures for the paths
of shorter lengths. Under these assumptions we then obtain the following
ergodicity result

$\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\frac{V^{\pi}_{i\Delta}}{V^{\pi}_{(i-1)\Delta}}=1+\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}R_{(i-1)\Delta,i\Delta}^{V,\pi}=M^{\pi}.$

This can be deduced from Birkhoff’s ergodic theorem for discrete time Markov
processes (see e.g. [Theorem 2.2, Section 2.1.4][26] and also [21, Section 3]
for the standard Markov situation with $n=1$). Indeed, this becomes applicable
by identifying the states of the Markov process as the past path segments of
length $n$, i.e. a state $z_{i\Delta}$ is given by
$z_{i\Delta}=(\mu_{(i-n)\Delta},\ldots,\mu_{i\Delta})$. The transition
probabilities from $z_{i\Delta}$ to $z_{(i+1)\Delta}$ must then be chosen such
that they assign $0$ probability to elements where there exists some
$j\in\\{1,\ldots,n-1\\}$ with $z^{j}_{(i+1)\Delta}\neq z^{j+1}_{i\Delta}$.

Clearly the above reasoning also holds true for
$\textrm{Var}\left[V^{\pi}_{i\Delta}/V^{\pi}_{(i-1)\Delta}\right]=\textrm{Var}[R_{(i-1)\Delta,i\Delta}^{V,\pi}].$
Therefore, under such ergodicity assumptions all the expected values in
Corollary 5.7 can be approximated by time averages, which will be important in
our numerical implementations as outlined in Section 7.2.

###  5.2 Optimizing the (Expected) Log-(Relative)-Wealth

Similarly as the mean-variance portfolio optimization problem also the log-
optimal portfolio over linear path functional portfolios can be found by
solving a convex quadratic optimization task. We state it here in form of the
relative log-utility optimization problem, i.e. the goal is to solve

$\sup_{\pi}\mathbb{E}\left[\log\left(\frac{W^{\pi}_{t}}{W^{{\mathcal{U}}}_{t}}\right)\right]=\sup_{\pi}\mathbb{E}[\log(V_{t}^{\pi})],$

where $V_{t}^{\pi}$ denotes again the relative wealth with respect to the
universe $\mathcal{U}$. Note however that under similar ergodicity conditions
as in Remark 5.10 the use of time averages

$\frac{1}{N}\log(V^{\pi}_{N\Delta})=\frac{1}{N}\sum_{i=1}^{N}\log\left(\frac{V^{\pi}_{i\Delta}}{V^{\pi}_{(i-1)\Delta}}\right)$

is justified to approximate $\frac{1}{N}\mathbb{E}[\log(V_{N\Delta}^{\pi})]$.
This can also be translated to continuous time, where under appropriate
assumptions (see e.g. [21, Section 4.2.3 and Theorem 4.9]) we also have that

$\lim_{t\to\infty}\frac{1}{t}\log(V_{t}^{\pi})=\lim_{t\to\infty}\frac{1}{t}\mathbb{E}[\log(V_{t}^{\pi})],$

whence

$\frac{1}{t}\log(V_{t}^{\pi})\approx\frac{1}{t}\mathbb{E}[\log(V_{t}^{\pi})].$

In this case the expected values in the subsequent theorem, i.e.
$\mathbb{E}\left[\mathbf{Q}(t)\right]$ and
$\mathbb{E}\left[\mathbf{c}(t)\right]$ can therefore be just replaced by
$\mathbf{Q}(t)$ and $\mathbf{c}(t)$ computed along the observed trajectory.
This will play again an important role in our empirical implementations.

######  Theorem 5.11 (Relative Log-Utility Optimization).

Consider a universe $\mathcal{U}\subseteq\\{1,...,d\\}$ and a linear path
functional portfolio $\pi(\mu^{\mathcal{U}},X)$. Let $t_{0}\geq 0$ be the time
at which we start investing. Denote by $\mathscr{L}$ an arbitrary but fixed
labelling function. Then, for the (relative) log-utility optimization problem
we have

$\sup_{\\{l_{\nu}^{(i)}\\}_{{i\in\mathcal{U}},{\nu}\in\mathcal{V}}}\mathbb{E}\left[\log\left(\frac{W^{\pi}_{t}}{W^{{\mathcal{U}}}_{t}}\right)\right]\Leftrightarrow\inf_{\mathbf{l}\in\mathbb{R}^{|\mathcal{U}|\cdot|\mathcal{V}|}}\frac{1}{2}\mathbf{l}^{\mathsf{T}}\mathbb{E}\left[\mathbf{Q}(t)\right]\mathbf{l}-\mathbb{E}\left[\mathbf{c}(t)^{\mathsf{T}}\right]\mathbf{l}.$
(5.7)

where $\mathbf{l}_{\mathscr{L}(i,\nu)}=l_{\nu}^{i}$ for $i\in\mathcal{U}$,
$\nu\in\mathcal{V}$ and

$(\mathbf{c}(t))_{\mathscr{L}(i,\nu)}=\int_{t_{0}}^{t}\gamma_{s}^{i}\phi^{\nu}(s,X_{[0,s]})\textrm{d}\mu^{\mathcal{U},i}_{s}$

$(\mathbf{Q}(t))_{\mathscr{L}(i,\nu),\mathscr{L}(j,\rho)}=\int_{t_{0}}^{t}\gamma_{s}^{i}\gamma_{s}^{j}\phi^{\nu}(s,X_{[0,s]})\phi^{\rho}(s,X_{[0,s]})\textrm{d}[\mu^{\mathcal{U},i},\mu^{\mathcal{U},j}]_{s}$

for

$\gamma_{s}^{i}=\begin{cases}1&\textrm{ if }\pi\textrm{ is of type $I$}\\\
\left(\mu^{\mathcal{U},i}_{s}\right)^{-1}&\textrm{ if }\pi\textrm{ is of type
$II$}\end{cases}.$

Moreover, it is a convex quadratic optimization problem.

######  Remark 5.12.

Let us highlight how the expression of $\mathbf{c}$, $\mathbf{Q}$ simplify in
the case of a signature portfolio of type $I$, $t_{0}=0$,
$\mathcal{U}=\\{1,\dots,d\\}$, $X=\mu$ and $\mu$ is such that it holds for
some $N_{0}>0$

$\textrm{d}[\mu^{i},\mu^{j}]_{t}=\sum_{0\leq|I|\leq
N_{0}}\alpha^{i,j}_{I}\langle e_{I},\hat{\bbmu}_{t}\rangle,$

where $\alpha^{i,j}_{I}\in\mathbb{R}$ for all $i,j\in\\{1,\dots,d\\}$,
$0\leq|I|\leq N_{0}$. Then we obtain

$(\mathbf{c}(t))_{\mathscr{L}(i,I)}=\langle e_{I}\otimes
e_{i},\hat{\bbmu}_{t}\rangle$
$(\mathbf{Q}(t))_{\mathscr{L}(i,I),\mathscr{L}(j,J)}=\sum_{0\leq|K|\leq
N_{0}}\alpha^{i,j}_{K}\langle(e_{I}\shuffle e_{J})\otimes
e_{K},\hat{\bbmu}_{t}\rangle\,.$

More generally, consider a signature portfolio $\pi(\mu,\breve{\mu})$ of type
$I$ where $\breve{\mu}_{t}:=(t,\mu_{t},[\mu,\mu]_{t})$ for all $t\in[0,T]$ in
a vectorized form, i.e. fix a labelling function $\kappa(\cdot,\cdot)$ such
that $\breve{\mu}^{\kappa(i,j)}=[\mu^{i},\mu^{j}]$ for $1\leq i,j\leq d$ and
set $t_{0}=0$, $\mathcal{U}=\\{1,\dots,d\\}$. Then the elements of
$\mathbf{c}$ and $\mathbf{Q}$ are linear functions on the signature of
$\breve{\mu}$, namely

$(\mathbf{c}(t))_{\mathscr{L}(i,I)}=\langle e_{I}\otimes
e_{i},\breve{\bbmu}_{t}\rangle$
$(\mathbf{Q}(t))_{\mathscr{L}(i,I),\mathscr{L}(j,J)}=\langle(e_{I}\shuffle
e_{J})\otimes e_{\kappa(i,j)},\breve{\bbmu}_{t}\rangle\,.$

######  Remark 5.13.

Note that although we start trading at time $t_{0}\geq 0$ we feed to the
feature maps $\phi^{\nu}$ the semimartingale $X$ starting at time $0$. The
interpretation of this is that we start “observing” the market at time $0$ and
but only start trading at time $t_{0}\geq 0$. This time-difference between
observing and starting to invest can be important in non-Markovian settings.

Before we prove Theorem 5.11, let us state the following lemma.

######  Lemma 5.14.

For a path-functional portfolio $\pi(\mu^{\mathcal{U}},{X})$ it holds that

$\displaystyle\begin{split}\log\left(\frac{W^{\pi}_{t}}{W^{{\mathcal{U}}}_{t}}\right)=&\sum_{i\in\mathcal{U}}\int_{t_{0}}^{t}\gamma_{s}^{i}f^{i}(s,{X}_{[0,s]})d\mu_{s}^{\mathcal{U},i}\\\
&-\frac{1}{2}\sum_{i,j\in\mathcal{U}}\int_{t_{0}}^{t}\gamma_{s}^{i}\gamma_{s}^{j}f^{i}(s,{X}_{[0,s]})f^{j}(s,{X}_{[0,s]})d[\mu^{\mathcal{U},i},\mu^{\mathcal{U},j}]_{s},\end{split}$
(5.8)

where

$\gamma_{s}^{i}=\begin{cases}1&\textrm{ if }\pi\textrm{ is of type $I$}\\\
\left(\mu^{\mathcal{U},i}_{s}\right)^{-1}&\textrm{ if }\pi\textrm{ is of type
$II$}\end{cases}.$

We refer the reader to Appendix C for a proof of Lemma 5.14.

###### Proof of Theorem 5.11.

To show the equivalence in (5.7), we use Lemma 5.14 and make use of the fact
that we are using linear path-functional portfolios.

$\displaystyle\log\left(\frac{W^{\pi}_{t}}{W^{{\mathcal{U}}}_{t}}\right)=$
$\displaystyle\sum_{i\in\mathcal{U}}\sum_{\nu\in\mathcal{V}}l_{\nu}^{i}\int_{t_{0}}^{t}\gamma_{s}^{i}\phi^{\nu}(s,X_{[0,s]})\textrm{d}\mu_{s}^{\mathcal{U},i}$
(5.9)
$\displaystyle-\frac{1}{2}\sum_{i,j\in\mathcal{U}}\sum_{\nu,\rho\in\mathcal{V}}l_{\rho}^{j}l_{\nu}^{i}\int_{t_{0}}^{t}\gamma_{s}^{i}\gamma_{s}^{j}\phi^{\nu}(s,X_{[0,s]})\phi^{\rho}(s,X_{[0,s]})\textrm{d}[\mu^{\mathcal{U},i},\mu^{\mathcal{U},j}]_{s}$
(5.10)

Hence, we have shown that

$\log\left(\frac{W^{\pi}_{t}}{W^{{\mathcal{U}}}_{t}}\right)=\mathbf{c}(t)^{\mathsf{T}}\mathbf{l}-\frac{1}{2}\mathbf{l}^{\mathsf{T}}\mathbf{Q}(t)\mathbf{l}$
(5.11)

from which the equivalence in (5.7) follows directly.

To show that the optimization problem given in (5.7) is convex, we show that
the matrix $\mathbf{Q}(t)$ is positive semidefinite. Consider the stochastic
process
$Y_{t}=\sum_{j=1}^{d}\int_{t_{0}}^{t}\gamma_{s}^{j}f^{j}(s,X_{[0,s]})d\mu^{\mathcal{U},j}_{s}$
whose quadratic variation $[Y,Y]_{t}$ is given by

$\displaystyle[Y,Y]_{t}=\sum_{i=1}^{d}\sum_{j=1}^{d}\int_{t_{0}}^{t}\gamma_{s}^{i}\gamma_{s}^{j}f^{i}(s,X_{[0,s]})f^{j}(s,X_{[0,s]})d[\mu^{\mathcal{U},i},\mu^{\mathcal{U},j}]_{s}.$

On the other hand

$\mathbf{l}^{T}\mathbf{Q}(t)\mathbf{l}=\sum_{i=1}^{d}\sum_{j=1}^{d}\int_{t_{0}}^{t}\gamma_{s}^{i}\gamma_{s}^{j}f^{i}(s,X_{[0,s]})f^{j}(s,X_{[0,s]})d[\mu^{\mathcal{U},i},\mu^{\mathcal{U},j}]_{s}.$

Hence, $\mathbf{l}^{T}\mathbf{Q}(t)\mathbf{l}=[Y,Y]_{t}\geq 0\hskip
5.69046pt\textrm{for all }t\in[t_{0},T]\textrm{ and for all
}\mathbf{l}\in\mathbb{R}^{(|\mathcal{U}|\cdot|\mathcal{V}|)}$, proving that
$\mathbf{Q}(t)$ is positive semidefinite for every $t\in[t_{0},T]$. ∎

######  Remark 5.15.

If the log-utility maximization problem is finite, it holds that

$\sup_{\\{l_{\nu}^{(i)}\\}_{{i\in\mathcal{U}},{\nu}\in\mathcal{V}}}\mathbb{E}\left[\log\left({W^{\pi}_{t}}\right)\right]\Leftrightarrow\sup_{\\{l_{\nu}^{(i)}\\}_{{i\in\mathcal{U}},{\nu}\in\mathcal{V}}}\mathbb{E}\left[\log\left(\frac{W^{\pi}_{t}}{W^{{\mathcal{U}}}_{t}}\right)\right].$
(5.12)

This simply follows from the following equivalences

$\displaystyle\sup_{\\{l_{\nu}^{(i)}\\}_{{i\in\mathcal{U}},{\nu}\in\mathcal{V}}}\mathbb{E}\left[\log\left({W^{\pi}_{t}}\right)\right]$
$\displaystyle\Leftrightarrow\sup_{\\{l_{\nu}^{(i)}\\}_{{i\in\mathcal{U}},{\nu}\in\mathcal{V}}}\mathbb{E}\left[\log\left({W^{\pi}_{t}}\right)-\log\left({W^{{\mathcal{U}}}_{t}}\right)\right]$
$\displaystyle\Leftrightarrow\sup_{\\{l_{\nu}^{(i)}\\}_{{i\in\mathcal{U}},{\nu}\in\mathcal{V}}}\mathbb{E}\left[\log\left(\frac{W^{\pi}_{t}}{W^{{\mathcal{U}}}_{t}}\right)\right].$

######  Remark 5.16.

Similar to Remark 5.9, the optimal vector $\mathbf{l}^{*}$ is given by

$\mathbf{l}^{*}=2\mathbb{E}[\mathbf{Q}(t)]^{-1}\mathbb{E}[\mathbf{c}(t)]\,,$

if $\mathbb{E}[\mathbf{Q}(t)]$ is almost surely invertible. As mentioned this
can never hold for linear path-functional portfolios of type $I$.

###  5.3 Constraints and Regularization

Given an optimization task involving a linear path-functional portfolio which
is convex and quadratic, we can make the following statements about adding
constraints or regularization:

######  Proposition 5.17.

Given an optimization problem

$\min_{\mathbf{l}}\hskip
5.69046pt-\mathbf{l}^{T}\mathbf{c}(t)+\frac{1}{2}\mathbf{l}^{T}\mathbf{Q}(t)\mathbf{l}$

which is convex quadratic, it remains convex quadratic under the following
modifications.

  1. 1.

Bounds Constraints: $|l_{\nu}^{i}|^{2}\leq b_{\mathscr{L}(i,\nu)}$ for
$b_{\mathscr{L}(i,\nu)}\geq 0$ for all $i\in\mathcal{U}$, $\nu\in\mathcal{V}$.

  2. 2.

L2-regularization:

we can add an $L^{2}$-regularization term of the form
$\gamma\mathbf{l}^{T}\mathbf{l}$ such that the optimization problem becomes

$\displaystyle\min_{\mathbf{l}}\hskip
5.69046pt-\mathbf{l}^{T}\mathbf{c}(t)+\frac{1}{2}\mathbf{l}^{T}\mathbf{Q}(t)\mathbf{l}+\gamma\mathbf{l}^{T}\mathbf{l}.$
(5.13)

Note that, this corresponds to a new matrix
$\tilde{\mathbf{Q}}(t)=\mathbf{Q}(t)+\gamma\mathbb{I}$, where $\mathbb{I}$ is
the identity matrix.

Moreover, the above modifications can be combined and the optimization task
remains convex.

###### Proof.

  1. 1.

$|l_{\nu}^{i}|\leq b_{\mathscr{L}(i,\nu)}$ for $b_{\mathscr{L}(i,\nu)}\geq 0$
corresponds to the constraint

$\mathbf{l}^{\mathsf{T}}\delta_{\mathscr{L}(i,\nu),\mathscr{L}(i,\nu)}\mathbf{l}-b^{2}_{\mathscr{L}(i,\nu)}\leq
0,$

where $\delta_{\mathscr{L}(i,\nu),\mathscr{L}(i,\nu)}$ is the matrix

$\delta_{jk}=\begin{cases}1&\textrm{if }j=k=\mathscr{L}(i,\nu)\\\
0&\textrm{otherwise}\end{cases},$

which is obviously positive semi-definite.

  2. 2.

Since $\mathbf{Q}(t)$ and $\mathbb{I}$ are positive semi-definite and the sum
of positive semi-definite matrices is positive semi-definite,
$\tilde{\mathbf{Q}}(t)$ is again positive semi-definite.

∎

##  6 Transaction Costs for Portfolios with Short-Selling and Without Bank
Account

When trading in the presence of (proportional) transaction costs, we need to
consider re-balancing at discrete times. Let $t$ be an arbitrary but fixed
time at which we re-balance. At time $t^{-}$ (just before re-balancing) our
portfolio has weights $\pi_{t^{-}}$ and a wealth of $W^{\pi}_{t^{-}}$. We want
to re-balance to some fixed target weights $\pi_{t}$. This situation has been
studied in [59] for the case of long-only weights. However, we allow for
short-selling in our portfolios which makes the problem significantly more
involved, as we illustrate below.

As the transaction costs are not paid on the portfolio weights, but on the
dollar amounts invested in a stock, we define $\psi^{i}_{t^{-}}$,
$\psi^{i}_{t}$ to be the dollar amounts invested in stock $i$ directly before
and after re-balancing. Note, that of course it holds

$\psi^{i}_{t}=\pi^{i}_{t}\cdot W^{\pi}_{t}$

and the transaction costs to be paid are

$TC_{t}=c\sum_{i=1}^{d}|\psi^{i}_{t}-\psi^{i}_{t^{-}}|$

where $c$ is the percentage of transaction costs to be paid. The difficulty
lies in solving for $\psi^{i}_{t}$, since the amount which we can invest in
stock $i$ depends on the amount of transaction costs to be paid _and vice-
versa_. This difficulty arises from the fact, that we do not have a bank-
account. Since we know the portfolio weights at each time, we need to solve
for $W^{\pi}_{t}$. We can use the self-financing identity which requires

$W^{\pi}_{t}=W^{\pi}_{t^{-}}-TC_{t}=W^{\pi}_{t^{-}}-c\sum_{i=1}^{d}|\psi^{i}_{t}-\psi^{i}_{t^{-}}|.$
(6.1)

Moreover, it is convenient to define
$\alpha(t):=\frac{W^{\pi}_{t}}{W^{\pi}_{t^{-}}}$ and rewrite (6.1) as

$1-\alpha_{t}=c\sum_{i=1}^{d}|\alpha_{t}\pi^{i}_{t}-\pi^{i}_{t^{-}}|.$ (6.2)

Hence, we reduced the problem to solving for $\alpha_{t}$. Since we consider
an arbitrary but fixed re-balancing time $t$, let us set $\alpha:=\alpha_{t}$
in the following. We can make the following statement about solutions for
$\alpha$:

######  Proposition 6.1.

Let $\alpha^{*}$ be a solution to the equation

$1-\alpha=c\sum_{i=1}^{d}|\alpha\pi^{i}_{t}-\pi^{i}_{t^{-}}|$

and denote by $L(\alpha)=1-\alpha$ and
$R(\alpha)=c\sum_{i=1}^{d}|\alpha\pi^{i}_{t}-\pi^{i}_{t^{-}}|$. We can make
the following statements about $\alpha^{*}$:

  1. 1.

If $\sum_{i=1}^{d}|\pi^{i}_{t^{-}}|<\frac{1}{c}$ a unique solution
$\alpha^{*}\in[0,1]$ exists.

  2. 2.

If $\sum_{i=1}^{d}|\pi^{i}_{t^{-}}|\geq\frac{1}{c}$ there _may not exist a
solution_ , or at least none for $\alpha\in[0,1]$. If a solution exists,
$\alpha^{*}\leq 1$ and it _may not be unique_.

######  Remark 6.2.

If we have long-only portfolios (and if $c<1$ i.e. less than 100% transaction
costs) only the first case of Proposition 6.1 is relevant. Hence, in that case
one always has a unique solution $\alpha^{*}\in[0,1]$. However, when we allow
for short-selling, the second case of Proposition 6.1 becomes important.

We give a proof of Proposition 6.1 in Appendix D and move directly to the
interpretation of situation in the second case of Proposition 6.1.

######  Remark 6.3 (Interpretation of No or Non-Unique Solutions for
$\alpha$).

We want to give some interpretation of the three possible outcomes of the
second case of Proposition 6.1.

  1. (a)

_no solution_ : The trade is infeasible. In order to pay the transaction
costs, we have to reduce the dollar amounts we invest. However, reducing the
dollar amounts $\psi_{t}$ leads to more transaction costs (because the
difference to the previous investment amount grows, i.e.
$|\psi_{t}-\psi_{t^{-}}|$ increases). If the transaction costs grow faster
than the money we gain from reducing the dollar amount $\psi_{t}$, the trade
is infeasible.

  2. (b)

_no solution in $[0,1]$_: Again, if we need to pay transaction costs without a
bank-account we have to free-up the money by selling stocks. Unlike case a)
this might be feasible, but the amount of transaction costs to be paid is more
than the total wealth of the portfolio. This leads to a solution with negative
$\alpha$.

  3. (c)

_no unique solution in $[0,1]$_: Here, we are in the case where the
transaction costs do not exceed our total wealth. Since reducing the dollar
amount leads to an increase in transaction costs, there might be several
strategies with respect to dollar amounts invested leading to the desired
portfolio weights after transaction costs. For example, one could do a more
expensive trade (smaller $\psi_{t}$) and pay more transaction costs or a less
expensive trade (larger $\psi_{t}$) where one pays less transaction costs.

In practice, we treat the above cases as follows. We consider cases a) and b)
to lead to ruin, because both are caused by strategies which are infeasible
given our wealth. If one of those cases occurs, we set our wealth to zero and
terminate the investment. In case c) we choose the solution with highest
$\alpha$. This is what any reasonable investor would do: choose the trade with
the least transaction costs.

##  7 Numerical Results

In this section we present our numerical results using simulated and real
market data.555 The code corresponding to this section is available at
https://github.com/janka-moeller/Sig-SPT

###  7.1 Simulated Data

In this subsection, we aim to learn the signature portfolios which maximize
$\mathbb{E}[\log\frac{W_{T}^{\pi}}{W^{\mu}_{T}}]$ in three financial market
models respectively, by using simulations of those market models. This
corresponds to the optimization task described in Theorem 5.11. To apply this
in a numerical setting, we use the following Monte-Carlo optimization to
approximate the expected values:

######  Corollary 7.1.

Let $\tilde{\Omega}$ be a probability-one set on which $\mathbf{Q}(t)$ is
positive-semidefinite. Consider a Monte-Carlo type optimization of the
expected log-relative wealth. Recall that
$V_{t}^{\pi}=\frac{W_{t}^{\pi}}{W^{\mu}_{t}}$, hence we optimize for

$\max_{\pi}\hskip
5.69046pt\frac{1}{M}\sum_{m=1}^{M}\log\left(V_{t}^{\pi}\right)(\omega_{m})$

for $\omega_{1},\ldots,\omega_{m}\in\tilde{\Omega}$. Again, this can be
formulated as a convex quadratic optimization problem, namely

$\displaystyle\min_{\mathbf{l}}\hskip
5.69046pt\frac{1}{2}\mathbf{l}^{T}\hat{\mathbf{Q}}(t)\mathbf{l}-\mathbf{l}^{T}\hat{\mathbf{c}}(t),$
(7.1)

where $\hat{\mathbf{Q}}(t)=\frac{1}{M}\sum_{m=1}^{M}\mathbf{Q}(t)(\omega_{m})$
and $\hat{\mathbf{c}}(t)=\frac{1}{M}\sum_{m=1}^{M}\mathbf{c}(t)(\omega_{m})$.

###### Proof.

We have shown in the proof of Theorem 5.11 that $\mathbf{Q}(t)$ is almost
surely positive semi-definite for all $t\in[t_{0},T]$, hence such a set
$\tilde{\Omega}$ exists. Since the sum of positive semi-definite matrices is
positive semi-definite, $\hat{\mathbf{Q}}(t)$ is positive semi-definite and
the statement follows. ∎

Moreover, we will compare the performance of the learned portfolio with the
growth-optimal portfolio of the respective markets. This is justified by Lemma
2.31, which states that the growth-optimal portfolio (which is also the
numeraire portfolio) solves the relative log-utility optimization problem.

For each market, we simulate $M_{train}$ trajectories over the time-horizon
$[0,1]$ of a given market model and consider investments over the whole time-
horizon, i.e. $t_{0}=0$, $t=1$. We train a signature portfolio using the
Monte-Carlo type optimization and the formulas for $\mathbf{Q}(t)(\omega)$ and
the vector $\mathbf{c}(t)(\omega)$ provided in Theorem 5.11, where we use
signature portfolios $\pi(\mu,\hat{\mu})$, i.e. choose $X=\mu$ to construct
the portfolios. Note, that calculating the matrix $\mathbf{Q}(t)(\omega)$ and
vector $c(t)(\omega)$ for each training-sample can be parallelized and
computed offline. Having obtained the respective $\hat{\mathbf{Q}}(t)$ and
$\hat{c}(t)$, we use the gurobipy.model.optimize()666for more information, see
https://www.gurobi.com/documentation/9.1/refman/py_python_api_overview.html#sec:Python.
method to solve the convex quadratic optimization problem. We want to
emphasize again, that we only need to solve the convex quadratic optimization
task once and we never have to update $\hat{\mathbf{Q}}(t)$ and $\hat{c}(t)$,
which is very beneficial for the computational tractability of the
optimization task.

After the optimization, we compare the weights and performance of the trained
signature portfolio and the theoretical growth-optimal portfolio on $M_{test}$
out-of-sample trajectories.

More concretely, we consider the following three market models and signature
portfolios:

  1. 1.

Correlated Black-Scholes Model:

$\textrm{d}S_{t}=\operatorname{diag}(S_{t})(adt+\Sigma dB_{t}),$

where $a\in\mathbb{R}^{d}$ and $\Sigma\in\mathbb{R}^{d\times d}$, $B_{t}$ is a
$d$-dimensional Brownian motion. We here take $d=3$ and train a signature
portfolio $\pi(\mu,\hat{\mu})$ of type $I$ of degree three. We simulate 1’000
time-steps.

  2. 2.

Volatility Stabilized Market:

$\frac{\textrm{d}S^{i}_{t}}{S^{i}_{t}}=\frac{1+\alpha}{2}\frac{1}{\mu_{t}^{i}}dt+\sqrt{\frac{1}{\mu_{t}^{i}}}dB_{t}^{i},\quad
1\leq i\leq d.$

We choose again $d=3$, so that $B$ is a three-dimensional Brownian motion. We
take $\alpha=10$ and train a signature portfolio $\pi(\mu,\hat{\mu})$ of type
$I$ of degree three. We simulate 10’000 time-steps.

  3. 3.

Signature Market:

$dS_{t}=\operatorname{diag}(S_{t})(\mathbf{a}_{t}dt+\Sigma\,dB_{t}).$

We choose again $d=3$, so that $B$ is a three-dimensional Brownian motion.
Moreover, $(\mathbf{a}_{t})_{i}=\sum_{0\leq|I|\leq 2}\alpha_{I}^{(i)}\langle
e_{I},\hat{\bbmu}\rangle_{t}$, $\alpha_{I}^{(i)}\in\mathbb{R}$ with the
restriction that $\alpha_{I}^{(i)}=0$ if $I=(i_{1},i_{2})$ and $i_{2}\neq 1$
and $\Sigma$ is a constant $3\times 3$ matrix. Here we train a signature
portfolio $\pi(\mu,\hat{\mu})$ of type $II$ of degree two and simulate 1’000
time-steps.

We use $M_{train}=100^{\prime}000$ in-sample trajectories for training and
$M_{test}=100^{\prime}000$ out-of-sample trajectories for evaluation, i.e.
computing the average logarithmic relative wealth of the respective portfolios
on the test-samples. Furthermore, we used the bound constraints
$|l_{I}^{i}|\leq 10^{\prime}000$ for all $I$ and for all $1\leq i\leq d$ for
each signature portfolio.

######  Remark 7.2.

In a volatility stabilized market, signature portfolios $\pi(\mu,\hat{\mu})$
of type $I$, with $\hat{\mu}=(t,\mu)$ fall into the first case of Remark 5.12,
where

$d\left[\mu^{i},\mu^{j}\right]_{t}=\begin{cases}\mu_{t}^{i}(1-\mu_{t}^{i})dt\textrm{
for }i=j\\\ -\mu_{t}^{i}\mu_{t}^{j}dt\textrm{ for }i\neq j.\end{cases}$

######  Remark 7.3.

In Lemma 2.31 we have already established that the growth-optimal portfolio
solves the relative log-utility optimization problem. We now want to outline
that the growth-optimal portfolio exists _and_ that the relative log-utility
optimization for signature portfolios is well-posed, i.e. that the expectation
of this optimization task is finite for all signature portfolios. Let us
denote the growth-optimal portfolio by $\pi^{(g)}$, recall its form given by
Lemma 2.30 and note that a sufficient condition for both of the above to be
satisfied is if $\rvert\mathbb{E}[\log W_{T}^{\pi^{(g)}}]\rvert<\infty$. We
will now present the arguments for the existence of $\pi^{(g)}$ and the well-
posedness of the optimization problem for each of the markets.

  1. 1.

Correlated Black-Scholes Model: It can easily be checked that for the growth-
optimal portfolio $\pi^{(g)}$ it holds that $\rvert\mathbb{E}[\log
W_{T}^{\pi^{(g)}}]\rvert<\infty$.

  2. 2.

Volatility Stabilized Market: Our optimization problem is well-posed for
signature portfolios, because volatility stabilized market models are
polynomial processes, see [15] and all coefficients of $\mathbf{c}$ and
$\mathbf{Q}$ are linear functions on the signature of $\hat{\mu}$, as
established in Remarks 5.12 and 7.2. Moreover, we know that the truncated
signature of a polynomial process is again a polynomial process (see [16]) and
hence their expectation is finite. Therefore, the expectation of any
coefficient of $\mathbf{c}$ and $\mathbf{Q}$ is finite and
$|\mathbb{E}\left[\frac{W^{\pi}}{W^{\mu}}\right]|<\infty$ holds for any
signature portfolio $\pi$ of type $I$.

  3. 3.

Signature Market: Assuming that a solution exists, the coefficients
$\mathbf{a}_{i}$ are all uniformly bounded on $[0,1]$. This follows form the
fact that $\mu^{i}$ are positive and uniformly bounded for all $i=1,\dots,d$,
hence so is $\int_{0}^{T}\mu^{i}_{t}dt$. It thereby follows
$\rvert\mathbb{E}[\log W_{T}^{\pi^{(g)}}]\rvert<\infty$ for the growth-optimal
portfolio $\rho$.

######  Remark 7.4.

Note that the growth-optimal portfolios of the Black-Scholes and volatility
stabilized markets are also functionally generated portfolios in the classical
sense. Indeed, in the Black-Scholes case the portfolio generating function is

$G^{(BS)}(\mu_{t})=\prod_{i=1}^{d}(\mu_{t}^{i})^{c_{i}},$

with portfolio weights $\pi^{(BS),i}_{t}=c_{i}$, where

$c_{i}=((\Sigma^{\mathsf{T}}\Sigma)^{-1}a)_{i}-\sum_{j=1}^{d}((\Sigma^{\mathsf{T}}\Sigma)^{-1})_{ij}\left(\frac{\sum_{k=1}^{d}(\Sigma^{\mathsf{T}}\Sigma)^{-1}a)_{k}-1}{\sum_{j,k=1}^{d}((\Sigma^{\mathsf{T}}\Sigma)^{-1})_{jk}}\right).$

In the volatility stabilized market case the generating function of the growth
optimal portfolio is given by

$G^{(Vol)}(\mu_{t})=\prod_{i=1}^{d}(\mu_{t}^{i})^{\frac{1+\alpha}{2}}\exp{\left(\mu^{i}_{t}\frac{d}{2}(\alpha-1)\right)},$

with portfolio weights

$\pi^{(Vol),i}_{t}=\frac{1+\alpha}{2}+\mu^{i}_{t}\frac{d}{2}(\alpha-1).$

Therefore, these two examples are not only examples of signature portfolios
approximating the respective growth-optimal portfolios but also examples for
signature portfolios approximating functionally generated portfolios.

Figure 1 shows the trained signature portfolios (right) and the growth-optimal
portfolios (left) evaluated at one out-of-sample trajectory in each market
respectively (rows). As expected from the theoretical results, the signature
weights are very similar to the theoretical growth-optimal weights in all
markets. We want to emphasize, that the growth-optimal weights were never
shown to the signature portfolio during training, but the signature portfolio
was trained to maximize the expected logarithmic relative wealth.

(a) Black-Scholes market: growth-optimal weights

(b) Black-Scholes market: signature weights

(c) Vol. stabilized market: growth-optimal weights

(d) Vol. stabilized Market: signature weights

(e) Signature market: growth-optimal weights

(f) Signature market: signature weights

Figure 1:  The theoretical growth-optimal weights (left) and the signature
portfolio’s weights (right) for the Black-Scholes, volatility stabilized and
signature market respectively, evaluated at one test sample.

Furthermore, we want to highlight, that although the growth-optimal portfolio
in the Black-Scholes market has constant weights, this approximation task is
far from trivial for a signature portfolio of type $I$. Because signature
portfolios of type $I$ approximate the controlling functions of the growth-
optimal portfolio, the approximation task was actually

$f^{(BS),i}(t,\mu_{[0,t]})\approx\frac{c_{i}}{\mu_{t}^{i}}.$

And likewise, in the volatility stabilized market, the approximation task was

$f^{(Vol),i}(t,\mu_{[0,t]})\approx\frac{\alpha+1}{2\mu_{t}^{i}}+\frac{d}{2}(\alpha-1).$

We quantify the performance by comparing the average logarithmic relative
wealth over the $M_{test}$ test samples of the growth-optimal and signature
portfolios, where in both cases we compute the average logarithmic relative
wealth numerically on the test samples. This explains why the signature
portfolio sometimes even leads to higher values. We present these results in
Table 1.

Mean log-relative wealth of | growth-optimal portfolio | signature portfolio
---|---|---
Black-Scholes market: | 9.0115 | 9.0122
Volatility stabilized market: | 8.7619 | 8.7417
Signature market: | 0.4399 | 0.4398
Table 1:  Mean logarithmic relative wealth of the theoretical growth-optimal
and signature portfolios, evaluated on 100’000 test-samples, in each market
respectively.

###  7.2 Real Market Data

####  7.2.1 Details on the Optimization and Investment Procedure

Before we present the performance of signature portfolios, as well as, JL- and
randomized-signature portfolios in real markets, we want to give some details
on the optimization and investment procedure.

##### Approximating Expectations by Time-Averages

When working with real market data, we only have one realization available.
Therefore, as already discussed in Remark 5.10 and at the beginning of Section
5.2 we replace the expectation by a time-average. More precisely, consider
$[0,\Delta,2\Delta,\dots,N\Delta=t]$ to be a equally spaced grid of $[0,t]$
with distance $\Delta$. Then, for $s\in[0,t-\Delta]$ and analogously to Remark
5.10 we consider the following approximations:

$\mathbb{E}\left[\log\left(\frac{V^{\pi}_{s+\Delta}}{V^{\pi}_{s}}\right)\right]\approx\frac{1}{N}\sum_{i=1}^{N}\log\left(\frac{V^{\pi}_{i\Delta}}{V^{\pi}_{(i-1)\Delta}}\right)=\frac{1}{N}\log(V_{t}^{\pi})$

$\mathbb{E}[\mathcal{R}^{V,\pi}_{s,s+\Delta}]\approx\frac{1}{N}\sum_{i=1}^{N}\mathcal{R}^{V,\pi}_{(i-1)\Delta,i\Delta}=:\overline{\mathcal{R}}^{V,\pi},$

$\textrm{Var}[\mathcal{R}^{V,\pi}_{s,s+\Delta}]\approx\frac{1}{N-1}\sum_{i=1}^{N}(\mathcal{R}^{V,\pi}_{(i-1)\Delta,i\Delta}-\overline{\mathcal{R}}^{V,\pi})^{2}.$

Recall that sufficient ergodicity/stationarity conditions guaranteeing that
these approximations are justified have been discussed in Remark 5.10.

Note however that we cannot expect that stationarity of the relative returns
and log-relative returns is satisfied for _every_ portfolio in our
optimization class. Consider for example signature portfolios of type $I$ with
$\tau=(\frac{1}{d},\dots\frac{1}{d})^{\mathsf{T}}$, $\hat{X}_{t}=(t,X_{t})$
and $\mathcal{U}=\\{1,\dots,d\\}$. It is easy to see that the stationarity
assumption can not hold for all portfolios in this class, just by considering
the following portfolio $\pi^{1}$ where the only parameters that are non-zero
are chosen to be $l^{d}_{(1)}$. Then

$\displaystyle\mathbb{E}\left[\mathcal{R}^{V,\pi^{1}}_{t,t+\Delta}\right]=\frac{1}{d}\mathbb{E}\left[t\cdot\frac{\mu^{d}_{t+\Delta}}{\mu^{d}_{t}}\right]+(\frac{1}{d}-\frac{t}{d^{2}})\mathbb{E}\left[\sum_{j=1}^{d}\frac{\mu^{j}_{t+\Delta}}{\mu^{j}_{t}}\right]-\frac{t+1}{d}.$

However, there are of course classes of linear path-functional portfolios
where the stationarity assumptions holds for all portfolios in the class.
Indeed, the simplest example is the class of Markowitz-type portfolios, i.e.
with $\tau$ being constant and constant portfolio maps. A far-reaching
generalization thereof are signature portfolios with _with rolling windows_ ,
as defined in Remark 3.8 and for which stationarity holds under the conditions
of Remark 5.10. Computing these portfolios is slightly more expensive, since
one needs to compute the increments of the signature. Therefore we did not
consider them in our implementations.

Finally, let us emphasize that the optimization using time-averages does make
sense in practice, even if the stationarity of returns may not hold for every
portfolio. For example consider $\Delta=1\textrm{ day}$, then the mean-
variance optimization using time-averages can be seen as looking for a
portfolio with high average daily returns (in time) but without the daily
returns varying too much over time and likewise for the log-relative wealth
optimization. Moreover, we argue that the relative returns and log-relative
returns of our optimized portfolios do exhibit some ”stability” in time, as
almost all the learnt portfolios perform very well in the out-of-sample
period, as we will present in the following. Note, that all of the above
optimization problems remain convex quadratic optimization problems if we
replace expectations by time-averages.

##### Hyperparameters and Cross-Validation

The main hyperparameters of our optimization procedure are:

  * •

$t_{0}$: this is the time at which we start to invest, with respect to the
time when we start calculating the signature. The interpretation of this is
that we can observe the market for some time before we start to invest, which
is particularly relevant in the non-Markovian setting. In the following we
choose $t_{0}=100$, hence, we always start calculating the signature 100 days
before the respective investment period starts.

  * •

$\gamma$: this is the parameter of the $L2$-regularization described in
Proposition 5.17. For the case of maximizing the (expected) log-relative
wealth, we choose our $\gamma$ during cross-validation. We do not train this
regularization parameter in the mean-variance optimization but fix it a priori
to a small value, as the minimization of the variance itself can already be
regarded as regularizing.

For performing the cross-validation, we first split our data into a in-sample-
period of $T_{ins}$ days, a cross-validation of $T_{cv}$ (consecutive to the
in-sample period) and a testing period of $T_{test}$ days, starting at the end