# Fractal properties, information theory, and market efficiency

Xavier Brouty${}^{\text{a}}$, Matthieu Garcin${}^{\text{b,}}$ Corresponding
author<EMAIL_ADDRESS>
${}^{\text{a}}$ ESILV, 92916 Paris La Défense, France.
${}^{\text{b}}$ Léonard de Vinci Pôle Universitaire, Research center, 92916
Paris La Défense, France.

###### Abstract

Considering that both the entropy-based market information and the Hurst
exponent are useful tools for determining whether the efficient market
hypothesis holds for a given asset, we study the link between the two
approaches. We thus provide a theoretical expression for the market
information when log-prices follow either a fractional Brownian motion or its
stationary extension using the Lamperti transform. In the latter model, we
show that a Hurst exponent close to $1/2$ can lead to a very high
informativeness of the time series, because of the stationarity mechanism. In
addition, we introduce a multiscale method to get a deeper interpretation of
the entropy and of the market information, depending on the size of the
information set. Applications to Bitcoin, CAC 40 index, Nikkei 225 index, and
EUR/USD FX rate, using daily or intraday data, illustrate the methodological
content.

Keywords – fractional Brownian motion, Hurst exponent, market information,
multiscale entropy, Shannon entropy, stationary process

##  1 Introduction

One of the main challenges for quantitative asset managers is to find
statistical models able to predict future price evolutions of financial
assets. In this perspective, the classical financial theory is discouraging.
Indeed, the efficient market hypothesis (EMH) states that current prices
reflect all the available information, so that statistical arbitrage is
impossible. The repeated profits of many actors in this industry of course
disprove the EMH, at least as a general statement. In practice, the difficulty
of predicting future prices is not shared equally by all financial assets. It
is thus important to have statistical methods able to identify, among several
time series of prices, which can successfully be forecast and which cannot. A
rich and varied literature, mainly in econophysics, therefore addresses this
question. Two of the most widespread approaches are based on information
theory and on fractals.

The main tool of information theory is Shannon’s entropy. It measures the
uncertainty inherent to a specific probability distribution. The distribution
with the highest uncertainty, that is the one leading to the biggest
difficulty of predicting the value of a random variable following this
distribution, is the uniform distribution. The usefulness of the entropy
notion in the analysis of the EMH thus appears when one is able to define
probability distributions that are uniform if and only if the EMH holds. It is
for instance the case of ordinal patterns of a time series of prices, which
lead to the permutation entropy [7]. A more natural method focuses on the
conditional distribution of the sign of future price returns [58, 14]. We will
study the latter entropy, from which we build the market information. This
information consists of the difference between the measured entropy and an
ideal entropy consistent with the EMH [14, 33].

The fractal analysis is an alternative approach used for determining the
degree of market efficiency of a financial asset. Mainly based on the Hurst
exponent, this method studies the impact of changing the time scale of the
observations on some statistic of a time series. It asserts that the EMH
corresponds to a Hurst exponent equal to $1/2$ [57]. When one uses a moment-
based estimator of the Hurst exponent, this approach implicitly relies on the
assumption that log-prices follow a fractional Brownian motion (fBm). This
model, which uses fractional calculus to reproduce specific fractal properties
[48], is consistent with the interpretation of the Hurst exponent regarding
the EMH and given above, but it relies on strong assumptions, like Gaussian
price returns.

It is tempting to compare the outcome of the two competing approaches cited
above, namely entropy-based and fractal methods. Are they redundant, do they
complement each other? With the aim of combining the two approaches, the
multiscale entropy (MSE) makes it possible to study the entropy of a system
observed at various time scales [20, 19]. It has been used in particular to
analyse the scale-dependent market efficiency [49, 2]. Beyond this attempt to
build a synthetic method, some rare articles examine the theoretical link
between entropy and fractal dimension [66], or the link, obtained by
simulations, between the MSE and the Hurst exponent of an fBm [52]. It is also
possible to obtain a theoretical expression for the permutation entropy in the
case where the data are generated by some specific stochastic process,
including the fBm [8, 68].

In this article, we intend to study the link between information theory and
fractal properties, in the perspective of investigating the EMH, in three
different and new directions. First, in addition to the MSE, we study
theoretically and empirically the dependence of both the entropy and the
market information on the size of the information set. Second, we explore the
theoretical link between the Hurst exponent of an fBm and its market
information, using the definition of entropy based on the conditional
distribution of the sign of future price returns, instead of the sole
permutation entropy. Third, noting that the fBm is not the only fractional
model having fractal properties, we derive similar results for a stationary
extension of the fBm, namely its inverse Lamperti transform. This kind of
result is useful for instance for modelling fixed-income instruments and it
shows new possible interpretations of the Hurst exponent regarding the
information of time series.

Overall, this article sheds new lights on the informational content of
fractional models. It confirms in a new manner previous results on the
forecast of fBm in finance [39, 38, 32]. It also deepens the understanding of
the stationary inverse Lamperti transform of the fBm [29]. In this stationary
framework, we get a counter-intuitive result, because a Hurst exponent equal
to $1/2$ can depict a very high information in the series, depending on the
intensity of the mean reversion. All these results may be instructive for
quantitative traders, risk managers, and every person interested in
forecasting time series in finance or in another context.

The rest of the paper is organized as follows. Section 2 presents the entropy-
based concept of market information. In Section 3, we derive theoretical
results about the market information of two fractional processes: the fBm and
its stationary extension with the Lamperti transform. We present in Section 4
the multiscale approach applied to market information along with numerical
results for both simulated time series and real financial data, including
stock indices, Bitcoin, and an FX rate. Section 5 concludes.

##  2 Market information

We are given a discrete random vector $(X_{1},...,X_{L})^{\prime}$. We assume
that each component of this vector has its value in $\\{0,1\\}$, so that the
vector can be in one of the $2^{L}$ possible states noted
$\textbf{s}^{L}_{i}$, for $i\in\llbracket 1,2^{L}\rrbracket$. In information
theory, Shannon’s entropy is a basic concept which quantifies the amount of
uncertainty in a probability distribution. This uncertainty can also be seen
as an aspect of the complexity of this distribution [46, 33]. The Shannon
entropy of $(X_{1},...,X_{L})^{\prime}$ is defined by [21]:

$H(X_{1},...,X_{L})=-\sum_{i=1}^{2^{L}}\mathbb{P}((X_{1},...,X_{L})^{\prime}=\textbf{s}^{L}_{i})\log_{2}\left(\mathbb{P}((X_{1},...,X_{L})^{\prime}=\textbf{s}^{L}_{i})\right).$

The least informative probability distribution, that is the one with the
highest uncertainty, is the uniform distribution. In this case, the entropy is
$L$. The lowest uncertainty corresponds to the degenerate distribution in
which all the probability is concentrated in a single state. Using the
convention that $0\log_{2}(0)=0$, which we easily get from the limit of
$x\log_{2}(x)$ when $x\rightarrow 0$, this degenerate case leads to a zero
entropy. All the probability distributions have an entropy in the interval
$[0,L]$, thus delimited by these two special cases.

Shannon’s entropy has been exploited in finance in several ways, in particular
to determine whether the market is efficient or not. We can identify two main
lines of research, depending on how one defines the random vector
$(X_{1},...,X_{L})^{\prime}$ and its corresponding probability. First, the
Bandt and Pompe method focuses on the distribution of the ordinal patterns of
a time series of prices, giving rise to what is known as the permutation
entropy [7, 69, 67, 9]. The second approach, which is the one we are
interested in, is the Risso method. It applies the entropy to the discrete
distribution of the sign of successive price returns [58, 51, 25, 14]. More
precisely, we consider a sequence of prices of a given financial asset,
$P_{0},...,P_{n}$, sampled at a daily frequency.111 The daily frequency is
chosen to fix the ideas, but the approach can naturally be extended to other
frequencies. From these prices, we compute price returns over $m$ days, for
each date $i\in\llbracket m,n\rrbracket$,

$R_{m,i}=\frac{P_{i}-P_{i-m}}{P_{i-m}},$

and we build a new sequence of random variables, $J_{m,i}$, which are
indicators of positive price return:

$J_{m,i}=\left\\{\begin{array}[]{ll}1&\text{if }R_{m,i}>0\\\
0&\text{else.}\end{array}\right.$

Assuming that the random vector
$\mathcal{J}_{i}=(J_{m,i},J_{m,i+m},J_{m,i+2m},...,J_{m,i+Lm})^{\prime}$, of
size $L+1$, is part of a stationary sequence
$\\{\mathcal{J}_{i}\\}_{i\in\llbracket m,n-Lm\rrbracket}$, we note the entropy
of this vector

$H^{L+1}_{m}=H(J_{m,.},...,J_{m,.+Lm}).$ (1)

If we know the distribution of a future increase indicator $J_{m,.+Lm}$,
conditionally on the previous $L$ indicators $J_{m,.},...,J_{m,.+(L-1)m}$, we
can also write this entropy

$H^{L+1}_{m}=-\sum_{i=1}^{2^{L}}\sum_{j=1}^{2}\pi_{m}^{L}(\textbf{s}^{1}_{j}|\textbf{s}^{L}_{i})p_{m}^{L}(\textbf{s}^{L}_{i})\log_{2}\left(\pi_{m}^{L}(\textbf{s}^{1}_{j}|\textbf{s}^{L}_{i})p_{m}^{L}(\textbf{s}^{L}_{i})\right),$
(2)

where
$p_{m}^{L}(\textbf{s}^{L}_{i})=\mathbb{P}((J_{m,.},...,J_{m,.+(L-1)m})^{\prime}=\textbf{s}^{L}_{i})$
and
$\pi_{m}^{L}(\textbf{s}^{1}_{j}|\textbf{s}^{L}_{i})=\mathbb{P}(J_{m,.+Lm}=\textbf{s}^{1}_{j}|(J_{m,.},...,J_{m,.+(L-1)m})^{\prime}=\textbf{s}^{L}_{i})$.
Using the notation of the conditional entropy

$H(J_{m,.+Lm}|J_{m,.},...,J_{m,.+(L-1)m})=-\sum_{i=1}^{2^{L}}p_{m}^{L}(\textbf{s}^{L}_{i})\sum_{j=1}^{2}\pi_{m}^{L}(\textbf{s}^{1}_{j}|\textbf{s}^{L}_{i})\log_{2}\left(\pi_{m}^{L}(\textbf{s}^{1}_{j}|\textbf{s}^{L}_{i})\right),$
(3)

we can decompose equation (2) thanks to the chain rule [21, Th.2.2.1]:

$H(J_{m,.},...,J_{m,.+Lm})=H(J_{m,.},...,J_{m,.+(L-1)m})+H(J_{m,.+Lm}|J_{m,.},...,J_{m,.+(L-1)m}),$

that is

$H^{L+1}_{m}=H^{L}_{m}+H(J_{m,.+Lm}|J_{m,.},...,J_{m,.+(L-1)m}).$ (4)

In the case where the market is efficient, we have
$\pi_{m}^{L}(1|\textbf{s}^{L}_{i})=\mathbb{P}(J_{m,.+Lm}=1)=1/2$: the
uncertainty of a future price increment is maximal, whether or not we
condition on past observations. We note $J^{\star}_{m,.+Lm}$ this theoretical
EMH-based future price increment indicator, which is independent from the
vector $(J_{m,.},...,J_{m,.+(L-1)m})^{\prime}$. The corresponding entropy is

$\begin{array}[]{ccl}H^{L+1,\star}_{m}&=&H(J_{m,.},...,J_{m,.+(L-1)m},J^{\star}_{m,.+Lm})\\\
&=&H(J_{m,.},...,J_{m,.+(L-1)m})+H(J^{\star}_{m,.+Lm}|J_{m,.},...,J_{m,.+(L-1)m})\\\
&=&H(J_{m,.},...,J_{m,.+(L-1)m})+H(J^{\star}_{m,.+Lm})\\\
&=&H^{L}_{m}+1.\end{array}$ (5)

where we successively used the chain rule of the conditional entropy, the
independence of $J^{\star}_{m,.+Lm}$ with respect to the vector
$(J_{m,.},...,J_{m,.+(L-1)m})^{\prime}$, and the 2-state uniformity of
$J^{\star}_{m,.+Lm}$.

Finally, the market information is the divergence between the uncertainty in
the real financial time series and the uncertainty in the ideal efficient time
series [14]:

$\begin{array}[]{ccl}I^{L+1}_{m}&=&H^{L+1,\star}_{m}-H^{L+1}_{m}\\\
&=&1-H(J_{m,.+Lm}|J_{m,.},...,J_{m,.+(L-1)m}),\end{array}$ (6)

according to equations (4) and (5). The market information is always positive,
being equal to zero if and only if $\forall i\in\llbracket 1,2^{L}\rrbracket$,
$\pi_{m}^{L}(1|\textbf{s}^{L}_{i})=1/2$. One can use this quantity to build
statistical tests of market efficiency [14, 60]. Estimating this quantity is
thus a major challenge. If we simply replace the conditional probabilities by
their empirical estimator in equations (6) and (3), we get a plug-in estimator
which converges toward the true market information when $n\rightarrow\infty$
[63]. However, this estimator has two drawbacks. First, it is non-
asymptotically biased. The bias is particularly striking when the market is
really efficient, that is with a theoretical information equal to zero,
because the estimated information is always positive. Second, when $L$ is
large, the estimation of the empirical probabilities requires a huge amount of
data, much larger than the number of states $2^{L+1}$. More advanced
estimators make it possible to bypass these two drawbacks [27, 63]. One can
for example cite estimators based on the Lempel-Ziv compression algorithm
[42], on the context-tree weighting algorithm [27], or on Bayesian context
trees [56].

We also note that, from the beginning, we have considered variables with only
two possible states, 0 and 1. The definitions above can be generalized to a
higher number of states [62, 44], but the interpretation regarding the market
efficiency is then less obvious [14, 33].

##  3 Market information of fractal dynamics

Fractal dynamics are often characterized by a selfsimilarity property. A
stochastic process $X_{t}$ is said to be $\mathcal{H}$-selfsimilar if,
whatever $c>0$ and $t\geq 0$, $X_{t}\overset{d}{=}c^{-\mathcal{H}}X_{ct}$,
where $\overset{d}{=}$ means equality in finite-dimensional distributions. The
parameter $\mathcal{H}$ is the Hurst exponent. The Brownian motion is an
example of a stochastic process having such a property, with
$\mathcal{H}=1/2$. This process is widespread in finance, but it is only a
particular case of the fBm, which offers more flexibility in the selection of
the Hurst exponent. The fBm of Hurst exponent $\mathcal{H}$ is a
$\mathcal{H}$-selfsimilar stochastic process which has some additional simple
properties: a Gaussian distribution, stationary increments. Some extensions
and transformations of the fBm make it possible to consider more general
dynamics with some fractal features or even with more complex multifractal
properties [59, 41, 22, 37]. One can cite for instance stationary
transformations [16, 40, 29, 31], non-Gaussian extensions [61, 64, 3], Hurst
exponents varying through time in a deterministic [18, 11, 28] or stochastic
fashion [12, 13, 30], and the multifractal random walk [6, 5, 53].

In what follows, we are interested in the entropy analysis for dynamics of
log-prices generated by an fBm or by the stationary delampertized fBm. We note
that the results below, in particular Theorems 1 and 2, are also valid if the
price, instead of being the exponential of a specific fractal process, is any
monotonic function of this process, as soon as this function remains the same
trough time. After giving some details on the models, we provide a theoretical
formula for the market information $I^{2}_{m}$, that is the ability of a
single lagged observation to predict future price evolutions.

###  3.1 Log-prices following an fBm

Exploiting the fractional derivative or the fractional integral of a standard
Brownian motion, Mandelbrot and Van Ness have proposed the fBm, a simple
Gaussian model with stationary increments which is able to fit the fractal
property of log-prices [48]. More precisely, an fBm of Hurst exponent
$\mathcal{H}\in(0,1)$ and volatility parameter $\sigma>0$ is a stochastic
Gaussian process $B^{\mathcal{H}}_{t}$ such that $B^{\mathcal{H}}_{0}=0$,
$\mathbb{E}\\{B^{\mathcal{H}}_{t}\\}=0$, and

$\mathbb{E}\left\\{\left(B^{\mathcal{H}}_{t}-B^{\mathcal{H}}_{s}\right)^{2}\right\\}=\sigma^{2}|t-s|^{2\mathcal{H}}$
(7)

for all $s,t\in\mathbb{R}$. It is worth noting that the case $\mathcal{H}=1/2$
corresponds to the standard Brownian motion. On the contrary, when
$\mathcal{H}<1/2$ (respectively $\mathcal{H}>1/2$), $B^{\mathcal{H}}_{t}$ is
the fractional derivative (resp. integral) of order $1/2-\mathcal{H}$ (resp.
$\mathcal{H}-1/2$) of a standard Brownian motion and is therefore not
Markovian. In other words, as soon as $\mathcal{H}\neq 1/2$, it is legitimate
to forecast future evolutions of an fBm [55]. In practice, one can do it by
exploiting the covariance of the process, which we easily get from equation
(7):

$\mathbb{E}\left\\{B^{\mathcal{H}}_{t}B^{\mathcal{H}}_{s}\right\\}=\frac{\sigma^{2}}{2}\left(|t|^{2\mathcal{H}}+|s|^{2\mathcal{H}}-|t-s|^{2\mathcal{H}}\right),$
(8)

for all $s,t\in\mathbb{R}$. In finance, the propensity of the fBm to be
forecast contradicts the EMH and is at the root of many statistical arbitrage
strategies [39, 38, 32]. It is thus relevant to quantify the market
information of such a dynamic. Using the entropy-based method exposed in
Section 2 we specify the theoretical link between the market information and
the Hurst exponent in the following theorem.

######  Theorem 1.

Let $B^{\mathcal{H}}_{t}$ be an fBm of Hurst exponent $\mathcal{H}$ and
volatility parameter $\sigma$. Let the price $P_{t}$ be equal to
$\log(B^{\mathcal{H}}_{t})$ for all $t\in\mathbb{R}$, and $m>0$. Then, the
market information $I_{m}^{2}$ introduced in equation (6) is equal to

$I_{m}^{2}=1+f\left(\frac{1}{2}-\frac{1}{\pi}\arctan\left(\frac{\rho}{\sqrt{1-\rho^{2}}}\right)\right)+f\left(\frac{1}{2}+\frac{1}{\pi}\arctan\left(\frac{\rho}{\sqrt{1-\rho^{2}}}\right)\right),$
(9)

where $f:x>0\mapsto x\log_{2}(x)$ and $\rho=2^{2\mathcal{H}-1}-1$.

The proof of Theorem 1 is postponed in Appendix A. We note that the
information and the entropy in this theorem are based on probabilities which
are the same as the ones appearing in the permutation entropy [8].

Figure 1 represents the theoretical market information $I^{2}_{m}$ with
respect to the Hurst exponent $\mathcal{H}$, according to Theorem 1.
Consistently with the above remark, the case where $\mathcal{H}=1/2$, that is
where the fBm is a standard (Markovian) Brownian motion, is the only case for
which the market information is equal to zero. When $\mathcal{H}$ moves away
from $1/2$, from above or from below, the market information increases. The
increase of the information is stronger for $\mathcal{H}>1/2$ than for
$\mathcal{H}<1/2$. In other words, having $\mathcal{H}\neq 1/2$ is enough for
predicting future price increments, but the accuracy of the forecast is lower
for $\mathcal{H}<1/2$ than for $1-\mathcal{H}$. It confirms previous findings
based on another accuracy metric, namely the mean squared error or the hit
ratio [55, 32].

Figure 1: Theoretical market information $I^{2}_{m}$ for a dynamic of log-
prices following an fBm, depending on its Hurst exponent, according to Theorem
1.

Theorem 1 also shows that the market information does not depend on the size
of the time steps $m$ if the dynamic is generated by an fBm. If this model
appropriately describes the market, predictions with daily price returns are
as accurate as predictions with intraday observations, as soon as the number
of lagged returns is the same. This is of course an idealized and somehow
unrealistic view and practitioners do not use in general the same model to
describe prices with a high- or a low-frequency sampling of time. Indeed,
despite its interest compared to simpler models, the fBm is only another model
and fails to describe everything. For this reason, people may use and stack
alternative models featuring also some fractal component, as detailed in the
preamble of Section 3. As a consequence, one can reasonably think that the
link between the selfsimilarity parameter $\mathcal{H}$ and the market
information is not always as obvious as depicted by Figure 1, which is
restricted to a specific model, the fBm.

###  3.2 Log-prices following a stationary delampertized fBm

Among the models with a fractal feature, the stationary adaptations of the fBm
are of particular interest in finance. They can indeed be useful for
describing time series reknown to be stationary, like interest rates [4, 50],
FX rates [29, 30], and volatilities [17, 10, 35, 34]. Several specifications
of such a stationary model exist, for example depending on the presence or not
of long memory [16, 29]. We focus on the delampertized fBm, which has been
shown to be a relevant empirical choice [29] and which leads to tractable
equations.

The delampertized fBm relies on the Lamperti transform, which makes it
possible to transform a stationary process $Y_{t}$ in a
$\mathcal{H}$-selfsimilar process,
$(\mathcal{L}_{\mathcal{H}}Y)_{t}=t^{\mathcal{H}}Y_{\ln(t)}$, and reciprocally
a $\mathcal{H}$-selfsimilar process $X_{t}$ in a stationary process,
$(\mathcal{L}^{-1}_{\mathcal{H}}X)_{t}=e^{-\mathcal{H}t}X_{\exp(t)}$ [26].
This inverse Lamperti transform has a stationarising effect like the mean-
reverting Ornstein-Uhlenbeck mechanism [29]. We place ourselves in the
particular case where the underlying selfsimilar process is an fBm
$B^{\mathcal{H}}_{t}$ and we add a parameter $\theta>0$ to adjust the strength
of the mean reversion. We thus introduce the stationary process
$\mathcal{X}^{\mathcal{H},\theta}$, defined by

$\mathcal{X}^{\mathcal{H},\theta}_{t}=\left(\mathcal{L}^{-1}_{\mathcal{H}}B^{\mathcal{H}}\right)_{\theta
t}.$ (10)

It has been shown that, in general, the delampertized fBm is not equal to the
fractional Ornstein-Uhlenbeck process [16]. However, in the case where
$\mathcal{H}=1/2$, the two processes are equal: a delampertized Brownian
motion $X^{\frac{1}{2},\theta}_{t}$ is a (non-fractional) Ornstein-Uhlenbeck
process and thus is defined by the following equation, which explicitly
incorporates the mean reversion mechanism around 0:

$d\mathcal{X}^{\frac{1}{2},\theta}_{t}=-\frac{\theta}{2}\mathcal{X}^{\frac{1}{2},\theta}_{t}dt+\sqrt{\theta}dB^{1/2}_{t},$

where $B^{1/2}_{t}$ is a Brownian motion with volatility parameter $\sigma$.

Like in the case of the fBm, Theorem 2 provides the market information when
the log-price follows a delampertized fBm. As mentioned in the preamble of
Section 3, this theorem remains valid if, instead of writing the price
$P_{t}=\log(\mathcal{X}^{\mathcal{H},\theta}_{t})$, we write
$P_{t}=g(\mathcal{X}^{\mathcal{H},\theta}_{t})$ for any monotonic function
$g$. This would legitimate describing an interest rate or a volatility, and
not only a log-price, by the delampertized fBm
$\mathcal{X}^{\mathcal{H},\theta}_{t}$.

######  Theorem 2.

Let $B^{\mathcal{H}}_{t}$ be an fBm of Hurst exponent $\mathcal{H}$ and
volatility parameter $\sigma$. Let $\mathcal{X}^{\mathcal{H},\theta}_{t}$ be
the delampertized fBm defined by equation (10), with parameter $\theta>0$. Let
the price $P_{t}$ be equal to $\log(\mathcal{X}^{\mathcal{H},\theta}_{t})$ for
all $t\in\mathbb{R}$, and $m>0$. Then, the market information $I_{m}^{2}$
introduced in equation (6) is equal to

$I_{m}^{2}=1+f\left(\frac{1}{2}-\frac{1}{\pi}\arctan\left(\frac{\rho}{\sqrt{1-\rho^{2}}}\right)\right)+f\left(\frac{1}{2}+\frac{1}{\pi}\arctan\left(\frac{\rho}{\sqrt{1-\rho^{2}}}\right)\right),$

where $f:x>0\mapsto x\log_{2}(x)$,
$\rho=\frac{2-h(2m\theta)}{4-2h(m\theta)}-1$, and $h:x\geq 0\mapsto
2\cosh(\mathcal{H}x)-\left(2\sinh\left(\frac{x}{2}\right)\right)^{2\mathcal{H}}$.

The proof of Theorem 2 is postponed in Appendix B.

We display in Figure 2 the theoretical market information in the case of a
delampertized fBm, following Theorem 2, for various values of the Hurst
exponent $\mathcal{H}$ and of the mean reversion strength $\theta$. We obtain
curves which have a shape which is in general very different from the fBm
case, displayed in Figure 1. In fact, the fBm case seems to be the limit of
the delampertized fBm when $\theta$ tends to 0, that is when the fractional
aspect is predominant compared to the mean reversion, or when $m$ tends to
zero, that is when one considers the market information at a very high-
frequency sampling. In fact, in Theorem 2, the combined parameter $m\theta$ is
more relevant than $\theta$ or $m$ alone and these two parameters play exactly
the same role. For this reason, we have fixed $m=1$ in Figure 2.

Figure 2: Theoretical market information $I^{2}_{1}$ for a dynamic of log-
prices following a delampertized fBm, $\mathcal{X}^{\mathcal{H},\theta}$,
depending on its Hurst exponent. Each curve corresponds to a different value
of $\theta$: between 0.1 and 1 by step of 0.1 (left graph, from bottom to top
at the abscissa $\mathcal{H}=0.5$) and between 1 and 15 by step of 1 (right
graph, from bottom to top at the abscissa $\mathcal{H}=0.5$).

Like for the fBm, we observe that the market information strongly depends on
the Hurst exponent, but, unlike the fBm case, the sampling $m$, or
equivalently $\theta$, also plays an important role. We can identify two
regimes, corresponding to the left and the right graphs in Figure 2. For
$\theta\in(0,1]$, increasing $\theta$ does not change much the market
information for low $\mathcal{H}$, but it shrinks the information for high
values of $\mathcal{H}$ and it translates the $\mathcal{H}$ minimizing the
information, from $1/2$ to a higher value. On the other hand, when $\theta>1$,
the minimum information is reached for $\mathcal{H}\rightarrow 1$ and, as soon
as $\theta$ is large enough, the maximum information corresponds both to
$\mathcal{H}\rightarrow 0$ and $\mathcal{H}=1/2$.

This apparent contradiction with the traditional interpretation of the Hurst
exponent is in fact clearly established in the literature devoted to this
model [29, 31]. The parameter $\mathcal{H}$ is related to the fractal
properties of the process $B^{\mathcal{H}}_{t}$, not to the fractal properties
of $\mathcal{X}^{\mathcal{H},\theta}_{t}$. In other words, $\mathcal{H}$ is
only the underlying Hurst exponent, not the Hurst exponent perceived when one
conducts a selfsimilarity analysis on $\mathcal{X}^{\mathcal{H},\theta}_{t}$.
In fact, in the delampertized fBm $\mathcal{X}^{\mathcal{H},\theta}_{t}$, two
mechanisms are at work: the fractional aspect and the stationary aspect. The
fractional aspect in the fBm, which generates its memory, leads to fractal
properties, which are statistically observable at small time scales. The
stationary aspect corresponds to a mean reversion statistically observable at
larger time scales [29, 34]. When one goes from a low time scale, that is a
low $m$ or a low $\theta$, to a higher time scale, one gradually modifies the
behaviour of $\mathcal{X}^{\mathcal{H},\theta}_{t}$, from the pure fBm to the
stationary process, and thus it modifies the way one predicts such a process,
exploiting either the fractional aspect or the mean reversion.

This example of the delampertized fBm shows that the link between the market
information and the Hurst exponent is not straightforward. These two
quantities are in fact complementary notions.

##  4 Multiscale market information

In Section 3, we have studied the market information for time series with
specific scaling properties. We are now interested in the scaling properties
of the market information itself, for very general time series. We are going
to present first the method of multiscale information analysis. An application
to simulations and to real financial data will then follow.

###  4.1 The method

The multiscale aspect of the market information $I^{L}_{m}$ may lie either in
its dependence on the number $L$ of lags considered, or in its dependence on
the time scale $m$ of the price returns. These two axis provide different
interpretations.

We first focus on the dependence of the market information on the parameter
$L$. Before addressing this question regarding the market information, we have
to understand the dependence of the entropy on $L$. This is the purpose of
Theorem 3.

######  Theorem 3.

Let $m>0$ and $\mathcal{J}_{i}$ be the random vector
$(J_{m,i},J_{m,i+m},J_{m,i+2m},...,J_{m,i+Lm})^{\prime}$, where we assume
that, whatever $L\in\mathbb{N}\setminus\\{0\\}$, the sequence
$\\{\mathcal{J}_{i}\\}_{i\in\llbracket m,n-Lm\rrbracket}$ is stationary. Then,
the mapping $L\mapsto H^{L}_{m}$, as defined in equation (1), is increasing
and concave in $L$.

The proof of Theorem 3 is postponed in Appendix C.

The intuition behind Theorem 3 is that adding a lagged observation in the
conditioning set diminishes the uncertainty about the future increase
indicator of the price. But the more lagged observations one adds, the less
informative each new observation. When, from a certain value of $L$, one adds
uninformative lagged observations, the conditional entropy is constant from
this $L$ [21, Th.2.6.5] and $H^{L}_{m}$ becomes linear in $L$.

In the many examples below, the entropy seems to be a linear function of $L$,
but a more accurate look at the first difference through the lens of the
market information $I_{m}^{L}$ shows something different: $L\mapsto I_{m}^{L}$
is not a constant mapping for real financial datasets. This has important
consequences. Indeed, since

$I^{L}_{m}=1+H^{L-1}_{m}-H^{L}_{m},$

the EMH holds whatever the size $L$ of the information set if and only if
$\forall L\in\mathbb{N}\setminus\\{0\\}$, $I^{L}_{m}=0$, that is if and only
if $L\mapsto H^{L}_{m}$ is linear of slope equal to 1. An inefficient market
must thus have a nonlinear entropy $L\mapsto H^{L}_{m}$ or a slope lower than
1.

When the mapping $L\mapsto H^{L}_{m}$ is linear, its slope is similar to the
entropy rate [19],

$h=\underset{L\rightarrow\infty}{\lim}\frac{H^{L}_{m}}{L},$

that is one minus the average amount of information carried by each vector of
lagged observations. It is however difficult to estimate the entropy rate
since it is based on a limit of the entropy when $L$ tends to infinity.
Indeed, when the number of states, $2^{L}$, is big, that is not much lower
than the number of observed returns $n-Lm$, there is a consequent statistical
error in the estimation of the probabilities $p^{L}_{m}$ and $\pi_{m}^{L}$,
which leads to underestimating the entropy by increasing artificially the
divergence with respect to the uniform probability. This underestimation of
the entropy is a challenge for estimating the entropy rate [19], but it may
also create spurious concavity of $L\mapsto H^{L}_{m}$ or lead to overestimate
market information if one considers too large values for $L$.

If one restricts oneself to small values of $L$, one can also depict the
marginal contribution to the information of each lagged observation thanks to
the partial market information defined as the difference of informations of
successive values of $L$: $\mathcal{I}^{L}_{m}=I^{L}_{m}-I^{L-1}_{m}$.

In the multiscale framework, the second dependence of interest for the market
information is with respect to the parameter $m$. This is an adaptation to the
information level of the MSE [20, 19]. This concept of MSE has been used to
study the complexity of financial time series [54, 47, 65] and even to analyse
the market efficiency at different time scales [49, 2], noting that,
consistently with the adaptive market hypothesis [45], a market can be
efficient at some time scales and not at others. Obviously, if the EMH is to
be verified whatever $m$, then $\forall m>0$, $I^{L}_{m}=0$. But we don’t have
a unique pattern of the mapping $m\mapsto I^{L}_{m}$, when the EMH is
violated. In particular, Section 3 has shown that various models contradicting
the EMH lead to a market information which may depend or not on the value of
$m$.

###  4.2 Simulations

We simulate an fBm of Hurst exponent 0.4 and of length 3,000. We represent in
Figure 3 several indicators presented in this article. First, the estimated
entropy $H^{L+1}_{m}$, as a function of the size $L$ of the vector considered,
has a linear aspect, whatever the value of the time scale $m$. As exposed
above, the linear aspect is not enough to assert the efficiency of a price
following this dynamic. The slope is also to be considered. Its value is 0.98.
The difference with the value consistent with the EMH, that is 1, is small and
may be partly explained by the statistical error, but the same simulation with
a Hurst exponent equal to 0.5 leads to a higher slope, equal to 0.99. More
convincingly, the market information and the partial market information show a
non-negligible contribution of lagged observations to the information of the
series, whatever the time scale, in particular for more recent observations,
that is small values of $L$. This is consistent with the propensity of such a
dynamic to be forecast [32]. The larger partial information one observes for
large values of $L$ illustrates the statistical artefacts described in Section
4.1. In order to disentangle the informative part from the statistical error,
we display in Figure 3 the $95\%$ confidence bound of a zero market
information for $m=1$,222 Since increasing $m$ to 2 or 3 only diminishes the
number of observations used in the estimation of the probabilities $p_{m}^{L}$
and $\pi_{m}^{L}$ by one or two units, the confidence bound for $m=2$ or 3 is
very close to the case $m=1$. using the asymptotic gamma distribution
$\Gamma(k,\theta)$ of the empirical market information, which depends on the
number of observations $n+1$ and on $L$, with $k=2^{L-1}$ and
$\theta=[(n-mL)\ln(2)]^{-1}$ [14]. Moreover, by differentiating this bound, we
get an idea of the lower bound for significant partial market information, as
displayed with a thick dotted line in the corresponding graph, Figure 3. It
shows that the three first lagged observations only seem to contribute
significantly to the predictability of the sign of the future price return.
Last, we also show the traditional log-log plot used to estimate the Hurst
exponent, which is half the slope of the linear regression.

Figure 3: Estimated entropy $H^{L+1}_{m}$ (top left), estimated market
information $I^{L+1}_{m}$ (top right), and estimated partial market
information $\mathcal{I}^{L+1}_{m}$ (bottom left) for $L\in\llbracket
1,7\rrbracket$ and $m$ equal to 1 (black), 2 (dark grey), and 3 (light grey).
The thick dotted lines are derived from the $95\%$ confidence bound of a zero
market information for $m=1$. Bottom right: Log-log plot (black) and linear
model fitted on the smallest scales (grey). The price series is a simulated
geometric fBm of Hurst exponent 0.4 and of length 3,000.

This simulation study of the fBm can be generalized to other values of
$\mathcal{H}$. The results are gathered in Figure 4 and confirm the
theoretical analysis made in Section 3.1: the information is low when
$\mathcal{H}$ is close to $1/2$, high when $\mathcal{H}$ is close to 0 and
even higher when it is close to 1. We note however that, whatever
$\mathcal{H}$, the mapping $L\mapsto H^{L}_{m}$ seems to be almost linear,
with a slope all the more far from 1 as $\mathcal{H}$ is far from $1/2$. But
it doesn’t mean that the mapping is universally linear, whatever the model of
dynamic.

Figure 4: Estimated entropy $H^{L+1}_{m}$ (top left), estimated market
information $I^{L+1}_{m}$ (top right), and estimated partial market
information $\mathcal{I}^{L+1}_{m}$ (bottom left) for $L\in\llbracket
1,7\rrbracket$, $m$ equal to 1, and for all the Hurst exponents in the
interval $[0.05,0.95]$ with a step of $0.05$. Bottom right: Slope of the
function $L\mapsto H^{L+1}$, depending on the Hurst exponent. Each price
series is a simulated geometric fBm of length 3,000.

Figure 5 displays the same quantities for a toy model consisting in a pseudo-
periodic time series, where price returns follow

$R_{1,i}=\beta R_{1,i-\tau}+\sqrt{1-\beta^{2}}\varepsilon_{i},$ (11)

where the $\varepsilon_{i}$ are i.i.d. standard Gaussian variables, and where
we consider $\tau=5$ and $\beta=-0.9$. For this model, the mapping $L\mapsto
H^{L}_{m}$ shows a clear concavity. The information is mainly provided by the
information lagged by 5 units of time, as one can see in the partial
information graph. When increasing the time unit $m$, the peak in the partial
market information appears for a lower number of lags $L$. Last, the log-log
plot is not linear, because of the pseudo-period, illustrating the
complementarity of the fractal and informational methods.

Figure 5: Estimated entropy $H^{L+1}_{m}$ (top left), estimated market
information $I^{L+1}_{m}$ (top right), and estimated partial market
information $\mathcal{I}^{L+1}_{m}$ (bottom left) for $L\in\llbracket
1,7\rrbracket$ and $m$ equal to 1 (black), 2 (dark grey), and 3 (light grey).
The thick dotted lines are derived from the $95\%$ confidence bound of a zero
market information for $m=1$. Bottom right: Log-log plot (black) and linear
model fitted on the smallest scales (grey). The price series is the simulated
pseudo-periodic process of length 3,000 introduced in equation (11).

###  4.3 Empirical analysis

We now apply the multiscale analysis of market information to real financial
data. We focus on four datasets corresponding to various asset classes: a
cryptocurrency, the Bitcoin (BTC-USD), between January 2015 and January 2023,
two stock indices, the French CAC 40 index and the Japanese Nikkei 225 index,
respectively between January 2012 and January 2023 and between January 2022
and June 2022, and an FX rate, the EUR/USD, between December 2006 and January
2016. Moreover, we consider various time samplings: daily observations for the
BTC-USD and the CAC 40 index, intraday prices with a 10-minute sampling for
Nikkei 225 index and a 1-minute sampling for EUR/USD.

We start with the study of the cryptocurrency market, namely the BTC-USD.
According to several contributions, the market efficiency of this asset
fluctuates through time. In particular, it became progressively less
inefficient before 2018 and was inefficient again in 2020 [23, 43, 14, 33].
Since the multiscale approach requires quite long datasets, we do not analyse
each year separately but we use this tool to describe the informativeness of
the series globally in the period 2015-2023. The Shannon entropy, the market
information, the partial market information, and the log-log plot are gathered
in Figure 6. They illustrate that mainly one lag, $L=1$, contributes to the
market information, whatever the horizon $m$ considered. Like for the
simulations in Section 4.2, the estimated market information strongly
increases for large values of $L$. But it is the result of the statistical
error and not of a true informativeness at these scales. The really relevant
quantity to be considered is instead the difference between the market
information and the confidence bound of the zero information introduced in
Section 4.2. Figure 6 thus illustrates the short-term memory of the series of
price increase indicators. Last, the log-log plot is not linear but convex,
thus confirming the multifractal feature of the time series of Bitcoin prices,
documented in other studies [24]. Precisely, a Hurst exponent of 0.48 seems to
lead the dynamic at small time scales, whereas a Hurst exponent with a
slightly higher value describes the dynamic at greater scales.

Figure 6: Estimated entropy $H^{L+1}_{m}$ (top left), estimated market
information $I^{L+1}_{m}$ (top right), and estimated partial market
information $\mathcal{I}^{L+1}_{m}$ (bottom left) for $L\in\llbracket
1,7\rrbracket$ and $m$ equal to 1 (black), 2 (dark grey), and 3 (light grey).
The thick dotted lines are derived from the $95\%$ confidence bound of a zero
market information for $m=1$. Bottom right: Log-log plot (black) and linear
model fitted on the smallest scales (grey). The price series is the BTC-USD,
observed at a daily frequency between January 2015 and January 2023, thus
representing 2,934 observations.

The variations of market efficiency through time for stock indices are also
well documented [58, 36, 15, 1, 3, 14]. Focusing on the CAC 40 index, the
first lag ($L=1$) seems to be the only significant one, in the information
perspective, as one can see in Figure 7. But, much more markedly than for the
BTC-USD, the information content seems greater for longer horizons
($m\in\\{2,3\\}$) than for $m=1$. It means that the presence of noise makes it
more difficult to forecast future price increments at short horizons. This
would be consistent with the hypothesis of a trend leading the long-term
dynamic of prices. The fractal analysis enriches the description of the price
dynamic. Indeed, the log-log plot is concave, depicting a Hurst exponent of
0.49 for the short-term dynamic and a kind of mean-reversion mechanism for
very long time scales.

Figure 7: Estimated entropy $H^{L+1}_{m}$ (top left), estimated market
information $I^{L+1}_{m}$ (top right), and estimated partial market
information $\mathcal{I}^{L+1}_{m}$ (bottom left) for $L\in\llbracket
1,7\rrbracket$ and $m$ equal to 1 (black), 2 (dark grey), and 3 (light grey).
The thick dotted lines are derived from the $95\%$ confidence bound of a zero
market information for $m=1$. Bottom right: Log-log plot (black) and linear
model fitted on the smallest scales (grey). The price series is the CAC 40
index, observed at a daily frequency between January 2012 and January 2023,
thus representing 2,822 observations.

It may also be interesting to apply the multiscale information analysis to
intraday data. Using a 10-minute sampling, the Nikkei 225 index seems to
behave quite efficiently at all scales at least for short horizon, that is for
$m=1$, according to Figure 8. We get a slightly more significant inefficiency
for $m=3$, for which microstructure noise is less noticeable, and for a small
number of lagged observations. Like for the CAC 40 index with daily
observations, we have a concave log-log plot, but with a higher slope. Indeed,
the Hurst exponent for the short-term dynamic is 0.60, indicating forecasting
opportunities for time horizons less than about 50 minutes.

Figure 8: Estimated entropy $H^{L+1}_{m}$ (top left), estimated market
information $I^{L+1}_{m}$ (top right), and estimated partial market
information $\mathcal{I}^{L+1}_{m}$ (bottom left) for $L\in\llbracket
1,7\rrbracket$ and $m$ equal to 1 (black), 2 (dark grey), and 3 (light grey).
The thick dotted lines are derived from the $95\%$ confidence bound of a zero
market information for $m=1$. Bottom right: Log-log plot (black) and linear
model fitted on the smallest scales (grey). The price series is the close for
Nikkei 225, observed each 10 minutes between January 2022 and June 2022, thus
representing 3,808 observations.

Each of the real datasets above contains less than 4,000 observations. It
limits the analysis to small values of $L$. Using a much longer dataset for
the EUR/USD FX rate, sampled every minute, we are able to increase the number
of lags considered. The results are gathered in Figure 9 and indicate the
presence of market inefficiency at all the time scales and all the numbers of
lags considered, even though the most recent lagged observations has a
predominant informativeness. The most significant market information is
obtained for $m=1$, with a clear difference with other time scales. Regarding
the log-log plot, it is linear, with an estimated Hurst exponent of 0.49, very
close to the case of the uninformative standard Brownian motion. It thus seems
to contradict the graphs on market information. However, a more deepened study
shows that the true dynamic is not Gaussian, so that the standard scaling is
the consequence of autocorrelation and fat tails, which compensate each other
in the fractal statistic, namely the estimated Hurst exponent [30].

Figure 9: Estimated entropy $H^{L+1}_{m}$ (top left), estimated market
information $I^{L+1}_{m}$ (top right), and estimated partial market
information $\mathcal{I}^{L+1}_{m}$ (bottom left) for $L\in\llbracket
1,14\rrbracket$ and $m$ equal to 1 (black), 2 (dark grey), and 3 (light grey).
The thick dotted lines are derived from the $95\%$ confidence bound of a zero
market information for $m=1$. Bottom right: Log-log plot (black) and linear
model fitted on the smallest scales (grey). The price series is the close for
EUR/USD, observed at each minute between December 2006 and January 2016, thus
representing 3,352,284 observations.

All the above results are based on series of close prices. However, it may be
difficult to build a trading strategy based on such an analysis. Indeed, the
close prices correspond to transaction prices and do not take into account
relevant liquidity constraints that a trader would face. If a trading strategy
based on the information analysis leads the trader to issue a bid order, it
cannot be at the close price if this one is an ask price. In order to get less
overoptimistic results, we propose in Figure 10 the same analysis as the one
conducted in Figure 9, in which we replace the close price of each 1-minute
interval by the average between the high and low prices in the same interval.
The resulting price is not tradable, but it tends to erase microstructure
noise. Surprisingly, the results indicate an even stronger informativeness for
$m=1$, whereas the market information slightly decreases for $m\in\\{2,3\\}$
with respect to the close price case. The log-log plot also shows a stronger
slope for small time scales, with the corresponding Hurst exponent estimated
at the high value of 0.69.

Figure 10: Estimated entropy $H^{L+1}_{m}$ (top left), estimated market
information $I^{L+1}_{m}$ (top right), and estimated partial market
information $\mathcal{I}^{L+1}_{m}$ (bottom left) for $L\in\llbracket
1,14\rrbracket$ and $m$ equal to 1 (black), 2 (dark grey), and 3 (light grey).
The thick dotted lines are derived from the $95\%$ confidence bound of a zero
market information for $m=1$. Bottom right: Log-log plot (black) and linear
model fitted on the smallest scales (grey). The price series is the average
between high and low for EUR/USD, observed in each one-minute interval between
December 2006 and January 2016, thus representing 3,352,284 observations.

##  5 Conclusion

Fractal statistics and information theory provide complementary analysis tools
about market efficiency. These two approaches are not unrelated and, in the
case where log-prices follow an fBm, we have provided a simple theoretical
expression of Shannon’s entropy and market information as functions of the
Hurst exponent. Since reality may be more complex than the fBm, we have also
provided an expression for the market information in the case of another
fractal model, namely the delampertized fBm. Combining the information
analysis at various time scales and for various sizes of the information set
is also a fruitful exercise. Several real-life examples, using daily or
intraday price series, have shown the usefulness of this method for
understanding the complexity of the price dynamics. Depending on the asset
class, one can for instance observe a stronger market information for small
time scales or for longer ones. This has some consequences for market
practitioners such as market makers, quantitative traders, and risk managers.

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##  Appendix A Proof of Theorem 1

###### Proof.

Using the notations of Section 2, we write
$J_{m,t}=\mathds{1}_{\\{P_{t}-P_{t-m}>0\\}}=\mathds{1}_{\\{B^{\mathcal{H}}_{t}-B^{\mathcal{H}}_{t-m}>0\\}}$.
Let $Y=B^{\mathcal{H}}_{2m}-B^{\mathcal{H}}_{m}$ and
$Z=B^{\mathcal{H}}_{m}-B^{\mathcal{H}}_{0}$. After equations (6) and (3) and
the property of stationarity of the increments of an fBm, we have

$\begin{array}[]{ccl}I^{2}_{m}&=&1-H(J_{m,.+m}|J_{m,.})\\\
&=&1+\mathbb{P}(J_{m,.}=1)\left[f\left(\mathbb{P}(J_{m,.+m}=1|J_{m,.}=1)\right)+f\left(\mathbb{P}(J_{m,.+m}=0|J_{m,.}=1)\right)\right]\\\
&&+\mathbb{P}(J_{m,.}=0)\left[f\left(\mathbb{P}(J_{m,.+m}=1|J_{m,.}=0)\right)+f\left(\mathbb{P}(J_{m,.+m}=0|J_{m,.}=0)\right)\right]\\\
&=&1+\mathbb{P}(Z>0)\left[f\left(\mathbb{P}(Y>0|Z>0)\right)+f\left(\mathbb{P}(Y\leq
0|Z>0)\right)\right]\\\ &&+\mathbb{P}(Z\leq
0)\left[f\left(\mathbb{P}(Y>0|Z\leq 0)\right)+f\left(\mathbb{P}(Y\leq 0|Z\leq
0)\right)\right].\end{array}$

Noting on the one hand that $\mathbb{P}(Z\leq 0)=\mathbb{P}(Z>0)=1/2$ and on
the other hand that the events $Y>0$ and $Y\leq 0$ are complementary, we get

$\begin{array}[]{ccl}I^{2}_{m}&=&1+\frac{1}{2}\left[f\left(\mathbb{P}(Y>0|Z>0)\right)+f\left(1-\mathbb{P}(Y>0|Z>0)\right)\right]\\\
&&+\frac{1}{2}\left[f\left(\mathbb{P}(Y>0|Z\leq
0)\right)+f\left(1-\mathbb{P}(Y>0|Z\leq 0)\right)\right].\end{array}$ (12)

The vector $(Y,Z)$ is Gaussian of mean zero and of covariance $\Sigma_{YZ}$,
such that:

$\Sigma_{YZ}=m^{2\mathcal{H}}\sigma^{2}\left(\begin{array}[]{cc}1&\rho\\\
\rho&1\end{array}\right),$

where $\rho=2^{2\mathcal{H}-1}-1$. Indeed, exploiting equation (8), we get:

$\begin{array}[]{ccl}\text{Cov}(Y,Z)&=&\mathbb{E}(B^{\mathcal{H}}_{2m}B^{\mathcal{H}}_{m})-\mathbb{E}(B^{\mathcal{H}}_{2m}B^{\mathcal{H}}_{0})-\mathbb{E}((B^{\mathcal{H}}_{m})^{2})+\mathbb{E}(B^{\mathcal{H}}_{m}B^{\mathcal{H}}_{0})\\\
&=&\frac{\sigma^{2}}{2}\left((2m)^{2\mathcal{H}}-2m^{2\mathcal{H}}\right)=m^{2\mathcal{H}}\sigma^{2}\rho.\end{array}$

Simple matrix manipulations provide the determinant
$|\Sigma_{YZ}|=m^{4\mathcal{H}}\sigma^{4}(1-\rho^{2})$ and

$\Sigma_{YZ}^{-1}=\frac{1}{m^{2\mathcal{H}}\sigma^{2}(1-\rho^{2})}\left(\begin{array}[]{cc}1&-\rho\\\
-\rho&1\end{array}\right).$

As a consequence, we can calculate the following joint probability, thanks to
the successive substitutions $u=(y-\rho
z)/m^{\mathcal{H}}\sigma\sqrt{1-\rho^{2}}$ and $v=z/m^{\mathcal{H}}\sigma$:

$\begin{array}[]{ccl}\mathbb{P}(Y>0,Z\leq
0)&=&\int_{-\infty}^{0}\int_{0}^{+\infty}\frac{1}{2\pi|\Sigma_{YZ}|^{1/2}}\exp\left(-\frac{[y\
z]\Sigma_{YZ}^{-1}[y\ z]^{T}}{2}\right)dydz\\\ &=&\frac{1}{2\pi
m^{2\mathcal{H}}\sigma^{2}\sqrt{1-\rho^{2}}}\int_{-\infty}^{0}\int_{0}^{+\infty}\exp\left(-\frac{(y-\rho
z)^{2}+(1-\rho^{2})z^{2}}{2m^{2\mathcal{H}}\sigma^{2}(1-\rho^{2})}\right)dydz\\\
&=&\frac{1}{2\pi m^{\mathcal{H}}\sigma}\int_{-\infty}^{0}\int_{-\rho
z/m^{\mathcal{H}}\sigma\sqrt{1-\rho^{2}}}^{+\infty}\exp\left(-\frac{u^{2}}{2}\right)\exp\left(-\frac{z^{2}}{2m^{2\mathcal{H}}\sigma^{2}}\right)dudz\\\
&=&\frac{1}{2\pi}\int_{-\infty}^{0}\left(1-N\left(\frac{-\rho
v}{\sqrt{1-\rho^{2}}}\right)\right)\exp\left(-\frac{v^{2}}{2}\right)dudv\\\
&=&\frac{1}{2\pi}\int_{0}^{\infty}N\left(\frac{-\rho
v}{\sqrt{1-\rho^{2}}}\right)\exp\left(-\frac{v^{2}}{2}\right)dv.\end{array}$

This is equal to
$\frac{1}{4}-\frac{1}{2\pi}\arctan\left(\frac{\rho}{\sqrt{1-\rho^{2}}}\right)$
[32, Lemma 1]. Noting that $\mathbb{P}(Z\leq 0)=1/2$, we finally get:

$\mathbb{P}(Y>0|Z\leq
0)=\frac{1}{2}-\frac{1}{\pi}\arctan\left(\frac{\rho}{\sqrt{1-\rho^{2}}}\right).$

Similarly

$\mathbb{P}(Y>0|Z>0)=\frac{1}{2}+\frac{1}{\pi}\arctan\left(\frac{\rho}{\sqrt{1-\rho^{2}}}\right).$

Going back to equation (12), we thus get

$I_{m}^{2}=1+f\left(\frac{1}{2}-\frac{1}{\pi}\arctan\left(\frac{\rho}{\sqrt{1-\rho^{2}}}\right)\right)+f\left(\frac{1}{2}+\frac{1}{\pi}\arctan\left(\frac{\rho}{\sqrt{1-\rho^{2}}}\right)\right).$

∎

##  Appendix B Proof of Theorem 2

###### Proof.

The process $\mathcal{X}^{\mathcal{H},\theta}_{t}$ is Gaussian with stationary
increments [29]. Noting
$Y=\mathcal{X}^{\mathcal{H},\theta}_{2m}-\mathcal{X}^{\mathcal{H},\theta}_{m}$
and
$Z=\mathcal{X}^{\mathcal{H},\theta}_{m}-\mathcal{X}^{\mathcal{H},\theta}_{0}$,
we have [29]

$\text{Var}(Y)=\text{Var}(Z)=\sigma^{2}\left(2-h(m\theta)\right)$

and

$\text{Cov}(Y,Z)=\frac{\sigma^{2}}{2}\left(-h(2m\theta)+2h(m\theta)-2\right),$

so that

$\text{corr}(Y,Z)=-1+\frac{2-h(2m\theta)}{4-2h(m\theta)}.$

One sees immediately in the proof of Theorem 1 that equation (9), written for
the sole fBm, is true for any Gaussian process with stationary increments as
soon as one replaces the parameter $\rho$ by the correlation between the
increments $Y$ and $Z$. This leads to Theorem 2. ∎

##  Appendix C Proof of Theorem 3

###### Proof.

We know by equation (4) that

$H^{L+1}_{m}=H^{L}_{m}+H(J_{m,.+Lm}|J_{m,.},...,J_{m,.+(L-1)m}).$

Moreover, the chain rule also leads to [21, Th.2.2.1 and Th.2.5.1]

$H^{L+2}_{m}=H^{L}_{m}+H(J_{m,.+(L+1)m}|J_{m,.},...,J_{m,.+Lm})+H(J_{m,.+Lm}|J_{m,.},...,J_{m,.+(L-1)m}).$

Therefore,

$H^{L+2}_{m}-H^{L+1}_{m}=H(J_{m,.+(L+1)m}|J_{m,.},...,J_{m,.+Lm})\geq 0.$

So the considered mapping is increasing. Then, the second difference of the
function $L\mapsto H^{L}_{m}$ is

$\begin{array}[]{ccl}H^{L+2}_{m}-2H^{L+1}_{m}+H^{L}_{m}&=&H(J_{m,.+(L+1)m}|J_{m,.},...,J_{m,.+Lm})-H(J_{m,.+Lm}|J_{m,.},...,J_{m,.+(L-1)m})\\\
&=&H(J_{m,.+(L+1)m}|J_{m,.},...,J_{m,.+Lm})-H(J_{m,.+(L+1)m}|J_{m,.+m},...,J_{m,.+Lm})\\\
&\leq&0\end{array}$

where we justify the penultimate line by the stationarity of the series and
the last one with the argument that conditioning reduces entropy [21,
Th.2.6.5]. The second difference of $L\mapsto H^{L}_{m}$ being negative, it is
a concave function. ∎