arXiv subjects

Stefano Marmi

Publications and source records attributed to Stefano Marmi.

At least 19 recordsLinked to original sources

arXiv2026

On the minimum of $\sigma$-Brjuno functions

$\sigma$-Brjuno functions were introduced in \cite{MaMoYo_06} as an interesting variant of the classical Brjuno function, where one substitutes the $\log$ singularity at $x=0$ with the power law divergence $x^{-1/\sigma},$ $(\sigma>0).$ As in the classical case, $B_{\sigma}$ is a locally unbounded, highly irregular lower semi continuous function; from semi continuity property it easily follows that $B_{\sigma}$ admits a global minimum but to locate it is quite a challenging problem. We prove that for $\sigma=n \in \mathbb{N}$, the unique global minimum of $B_n$ is achieved at the fixed point $ [0; \overline{n+1}]$. Furthermore, we prove that these minimizers are locally stable, showing that the point of minimum remains constant for $\sigma$ in a neighborhood of $n$. Finally, we discuss the scaling behavior near these minima and we formulate a conjecture about the phase transitions for the location of the minimizer as $\sigma$ varies.

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arXiv2026

Brjuno-Like Functions for nonlinear expanding maps: Fractional Derivatives and Regularity Dichotomies

Cohomological equations appear frequently in dynamical systems. One of the most classical examples is the Liv\v{s}ic equation $$ v(x) = \alpha \circ F(x) - \alpha(x).$$ The existence and regularity of its solutions $\alpha$ is well understood when $F$ is a hyperbolic dynamical system (for instance an expanding map of the circle) and $v$ is a H\"older function. The $\textbf{twisted cohomological equation}$ $$ v(x) = \alpha \circ F(x) - (DF(x))^\beta \, \alpha(x) $$ is much less well understood. Functions similar to the famous Brjuno, Weierstrass, and Takagi functions appear as solutions of this equation. This functional equation also appears in the work of M. Lyubich, and of Avila, Lyubich, and de Melo in their study of deformations of quadratic-like and real-analytic maps. Nevertheless, there are some striking results concerning the (lack of) regularity of solutions $\alpha$ when $F$ is a linear endomorphism of the circle and $v$ is very regular. Notable contributions include works by Berry and Lewis; Ledrappier; Przytycki and Urba\'nski, and more recently by Bara\'nski, B\'ar\'any and Romanowska, as well as by Shen, and by Ren and Shen, on Takagi and Weierstrass (and Weierstrass-like) functions. We study the regularity of solutions $\alpha$ when $F$ is a $\textbf{nonlinear}$ expanding map of the circle and $v$ is not differentiable or even continuous, a setting in which previously used transversality techniques do not appear to be applicable. The new approach uses fractional derivatives to reduce the study of the twisted cohomological equation to that of a corresponding Liv\v{s}ic cohomological equation, and to show that the resulting distributional solutions (in the sense of Schwartz) satisfy certain Central Limit Theorem.

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arXiv2026

Chaos and Synchronization in Financial Leverages Dynamics: Modeling Systemic Risk with Coupled Unimodal Maps

Systemic financial risk refers to the simultaneous failure or destabilization of multiple financial institutions, often triggered by contagion mechanisms or common exposures to shocks. In this paper, we present a dynamical model of bank leverage (the ratio of asset holdings to equity) a quantity that both reflects and drives risk dynamics. We model how banks, constrained by Value-at-Risk (VaR) regulations, adjust their leverage in response to changes in the price of a single asset, assumed to be held in fixed proportion across banks. This leverage-targeting behavior introduces a procyclical feedback loop between asset prices and leverage. In the dynamics, this can manifest as logistic-like behavior with a rich bifurcation structure across model parameters. By analyzing these coupled dynamics in both isolated and interconnected bank models, we outline a framework for understanding how systemic risk can emerge from seemingly rational micro-level behavior.

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arXiv2025

Emergence of Randomness in Temporally Aggregated Financial Tick Sequences

Markets efficiency implies that the stock returns are intrinsically unpredictable, a property that makes markets comparable to random number generators. We present a novel methodology to investigate ultra-high frequency financial data and to evaluate the extent to which tick by tick returns resemble random sequences. We extend the analysis of ultra high-frequency stock market data by applying comprehensive sets of randomness tests, beyond the usual reliance on serial correlation or entropy measures. Our purpose is to extensively analyze the randomness of these data using statistical tests from standard batteries that evaluate different aspects of randomness. We illustrate the effect of time aggregation in transforming highly correlated high-frequency trade data to random streams. More specifically, we use many of the tests in the NIST Statistical Test Suite and in the TestU01 battery (in particular the Rabbit and Alphabit sub-batteries), to prove that the degree of randomness of financial tick data increases together with the increase of the aggregation level in transaction time. Additionally, the comprehensive nature of our tests also uncovers novel patterns, such as non-monotonic behaviors in predictability for certain assets. This study demonstrates a model-free approach for both assessing randomness in financial time series and generating pseudo-random sequences from them, with potential relevance in several applications.

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arXiv2025

The Brjuno and Wilton Functions

The Brjuno and Wilton functions bear a striking resemblance, despite their very different origins; while the Brjuno function $B(x)$ is a fundamental tool in one-dimensional holomorphic dynamics, the Wilton function $W(x)$ stems from the study of divisor sums and self-correlation functions in analytic number theory. We show that these perspectives are unified by the semi-Brjuno function $B_0(x)$. Namely, $B(x)$ and $W(x)$ can be expressed in terms of the even and odd parts of $B_0(x)$, respectively, up to a bounded defect. Based on numerical observations, we further analyze the arising functions $\Delta^+(x) = B^+(x) - 2B_0^+(x)$ and $\Delta^-(x) = W^-(x) - 2B_0^-(x)$, the first of which is H\"older continuous whereas the second exhibits discontinuities at rationals, behaving similarly to the classical popcorn function.

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arXiv2024

Filtering and Statistical Properties of Unimodal Maps Perturbed by Heteroscedastic Noises

We propose a theory of unimodal maps perturbed by an heteroscedastic Markov chain noise and experiencing another heteroscedastic noise due to uncertain observation. We address and treat the filtering problem showing that by collecting more and more observations, one would predict the same distribution for the state of the underlying Markov chain no matter one's initial guess. Moreover we give other limit theorems, emphasizing in particular concentration inequalities and extreme value and Poisson distributions. Our results apply to a family of maps arising from a model of systemic risk in finance.

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arXiv2024

Optimized QUBO formulation methods for quantum computing

Several combinatorial optimization problems can be solved with NISQ devices once that a corresponding quadratic unconstrained binary optimization (QUBO) form is derived. The aim of this work is to drastically reduce the variables needed for these QUBO reformulations in order to unlock the possibility to efficiently obtain optimal solutions for a class of optimization problems with NISQ devices. This is achieved by introducing novel tools that allow an efficient use of slack variables, even for problems with non-linear constraints, without the need to approximate the starting problem. We divide our new techniques in two independent parts, called the iterative quadratic polynomial and the master-satellite methods. Hence, we show how to apply our techniques in case of an NP-hard optimization problem inspired by a real-world financial scenario called Max-Profit Balance Settlement. We follow by submitting several instances of this problem to two D-wave quantum annealers, comparing the performances of our novel approach with the standard methods used in these scenarios. Moreover, this study allows to appreciate several performance differences between the D-wave Advantage and Advantage2 quantum annealers.

quant-ph↗

arXiv2024

Price predictability at ultra-high frequency: Entropy-based randomness test

We use the statistical properties of Shannon entropy estimator and Kullback-Leibler divergence to study the predictability of ultra-high frequency financial data. We develop a statistical test for the predictability of a sequence based on empirical frequencies. We show that the degree of randomness grows with the increase of aggregation level in transaction time. We also find that predictable days are usually characterized by high trading activity, i.e., days with unusually high trading volumes and the number of price changes. We find a group of stocks for which predictability is caused by a frequent change of price direction. We study stylized facts that cause price predictability such as persistence of order signs, autocorrelation of returns, and volatility clustering. We perform multiple testing for sub-intervals of days to identify whether there is predictability at a specific time period during the day.

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arXiv2024

Global and local minima of $\alpha$-Brjuno functions

The main goal of this article is to analyze some peculiar features of the global (and local) minima of $\alpha$-Brjuno functions $B_\alpha$ where $\alpha\in(0,1].$ Our starting point is the result by Balazard--Martin (2020), who showed that the minimum of $B_1$ is attained at $g:=\frac{\sqrt 5 -1}{2}$; analyzing the scaling properties of $B_1$ near $g$ we shall deduce that all preimages of $g$ under the Gauss map are also local minima for $B_1$. Next we consider the problem of characterizing global and local minima of $B_\alpha$ for other values of $\alpha$: we show that for $\alpha\in (g,1)$ the global minimum is again attained at $g$, while for $\alpha$ in a neighbourhood of $1/2$ the function $B_{\alpha}$ attains its minimum at $\gamma:=\sqrt{2}-1$. The fact that the minimum of $B_\alpha$ is attained when $\alpha$ ranges a whole interval of parameters is non trivial. Indeed, we prove that $B_{\alpha}$ is lower semicontinuous for all rational $\alpha,$ but we also exhibit an irrational $\alpha$ for which $B_{\alpha}$ is not lower semicontinuous. %We also prove that if $\alpha$ is rational then $B_{\alpha}$ is lower semicontinuous. This property does not hold in general, in fact we show that $B_{\alpha}$ is not lower semicontinuous for a suitable irrational $\alpha.$

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arXiv2022

The Brjuno functions of the by-excess, odd, even and odd-odd continued fractions and their regularity properties

The Brjuno function was introduced by Yoccoz to study the linearizability of holomorphic germs and other one-dimensional small divisor problems. The Brjuno functions associated with various continued fractions including the by-excess continued fraction were subsequently investigated: it was conjectured that the difference between the classical Brjuno function and the even part of the Brjuno function associated with the by-excess continued fraction extends to a Hölder continuous function of the whole real line. In this paper, we prove this conjecture and we extend its validity to the more general case of Brjuno functions with positive exponents. Moreover, we study the Brjuno functions associated to the odd and even continued fractions introduced by Schweiger. We show that they belong to all $L^p$ spaces, $p\ge1$. We prove that the Brjuno function associated to the odd continued fraction differs from the classical Brjuno function by a Hölder continuous function. On the other hand, the Brjuno function associated to the even continued fraction differs from the classical Brjuno function by a sum of a Hölder continuous function and a Brjuno-type function associated to the odd-odd continued fraction, introduced in the study of the best approximations of the form odd/odd.

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arXiv2022

Volatility of volatility estimation: central limit theorems for the Fourier transform estimator and empirical study of the daily time series stylized facts

We study the asymptotic normality of two feasible estimators of the integrated volatility of volatility based on the Fourier methodology, which does not require the pre-estimation of the spot volatility. We show that the bias-corrected estimator reaches the optimal rate $n^{1/4}$, while the estimator without bias-correction has a slower convergence rate and a smaller asymptotic variance. Additionally, we provide simulation results that support the theoretical asymptotic distribution of the rate-efficient estimator and show the accuracy of the latter in comparison with a rate-optimal estimator based on the pre-estimation of the spot volatility. Finally, using the rate-optimal Fourier estimator, we reconstruct the time series of the daily volatility of volatility of the S\&P500 and EUROSTOXX50 indices over long samples and provide novel insight into the existence of stylized facts about the volatility of volatility dynamics.

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arXiv2022

Efficiency of the Moscow Stock Exchange before 2022

This paper investigates the degree of efficiency for the Moscow Stock Exchange. A market is called efficient if prices of its assets fully reflect all available information. We show that the degree of market efficiency is significantly low for most of the months from 2012 to 2021. We calculate the degree of market efficiency by (i) filtering out regularities in financial data and (ii) computing the Shannon entropy of the filtered return time series. We have developed a simple method for estimating volatility and price staleness in empirical data, in order to filter out such regularity patterns from return time series. The resulting financial time series of stocks' returns are then clustered into different groups according to some entropy measures. In particular, we use the Kullback-Leibler distance and a novel entropy metric capturing the co-movements between pairs of stocks. By using Monte Carlo simulations, we are then able to identify the time periods of market inefficiency for a group of 18 stocks. The inefficiency of the Moscow Stock Exchange that we have detected is a signal of the possibility of devising profitable strategies, net of transaction costs. The deviation from the efficient behavior for a stock strongly depends on the industrial sector it belongs.

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arXiv2022

The Yoccoz-Birkeland livestock population model coupled with random price dynamics

We study a random version of the population-market model proposed by Arlot, Marmi and Papini in Arlot et al. (2019). The latter model is based on the Yoccoz-Birkeland integral equation and describes a time evolution of livestock commodities prices which exhibits endogenous deterministic stochastic behaviour. We introduce a stochastic component inspired from the Black-Scholes market model into the price equation and we prove the existence of a random attractor and of a random invariant measure. We compute numerically the fractal dimension and the entropy of the random attractor and we show its convergence to the deterministic one as the volatility in the market equation tends to zero. We also investigate in detail the dependence of the attractor on the choice of the time-discretization parameter. We implement several statistical distances to quantify the similarity between the attractors of the discretized systems and the original one. In particular, following a work by Cuturi (2013), we use the Sinkhorn distance. This is a discrete and penalized version of the Optimal Transport Distance between two measures, given a transport cost matrix.

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arXiv2021

A convergence criterion for the unstable manifolds of the MacKay approximate renormalisation

We give an explicit arithmetical condition which guarantees the existence of the unstable manifold of the MacKay approximate renormalisation scheme for the breakup of invariant tori in one and a half degrees of freedom Hamiltonian systems, correcting earlier results. Furthermore, when our condition is violated, we give an example of points on which the unstable manifold does not converge.

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arXiv2021

Kelly Betting with Quantum Payoff: a continuous variable approach

The main purpose of this study is to introduce a semi-classical model describing betting scenarios in which, at variance with conventional approaches, the payoff of the gambler is encoded into the internal degrees of freedom of a quantum memory element. In our scheme, we assume that the invested capital is explicitly associated with the quantum analog of the free-energy (i.e. ergotropy functional by Allahverdyan, Balian, and Nieuwenhuizen) of a single mode of the electromagnetic radiation which, depending on the outcome of the betting, experiences attenuation or amplification processes which model losses and winning events. The resulting stochastic evolution of the quantum memory resembles the dynamics of random lasing which we characterize within the theoretical setting of Bosonic Gaussian channels. As in the classical Kelly Criterion for optimal betting, we define the asymptotic doubling rate of the model and identify the optimal gambling strategy for fixed odds and probabilities of winning. The performance of the model are hence studied as a function of the input capital state under the assumption that the latter belongs to the set of Gaussian density matrices (i.e. displaced, squeezed thermal Gibbs states) revealing that the best option for the gambler is to devote all her/his initial resources into coherent state amplitude.

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arXiv2021

Regularity properties of $k$-Brjuno and Wilton functions

We study functions related to the classical Brjuno function, namely $k$-Brjuno functions and the Wilton function. Both appear in the study of boundary regularity properties of (quasi) modular forms and their integrals. We consider various possible versions of them, based on the $α$-continued fraction developments. We study their BMO regularity properties and their behaviour near rational numbers of their finite truncations.

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arXiv2021

Analytic linearization of a generalization of the semi-standard map: radius of convergence and Brjuno sum

One considers a system on $\mathbb{C}^2$ close to an invariant curve which can be viewed as a generalization of the semi-standard map to a trigonometric polynomial with many Fourier modes. The radius of convergence of an analytic linearization of the system around the invariant curve is bounded from below by $\exp(-\frac{2}{d}B(dα)-C)$, where $C\geq 0$ does not depend on $α$, $d\in \mathbb{N}^*$ and $α$ is the frequency of the linear part. For a class of trigonometric polynomials, it is also bounded from above by a similar function. The error function is non decreasing with respect to the smallest coefficient of the trigonometric polynomial.

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arXiv2021

Random-like properties of chaotic forcing

We prove that skew systems with a sufficiently expanding base have approximate exponential decay of correlations, meaning that the exponential rate is observed modulo an error. The fiber maps are only assumed to be Lipschitz regular and to depend on the base in a way that guarantees diffusive behaviour on the vertical component. The assumptions do not imply an hyperbolic picture and one cannot rely on the spectral properties of the transfer operators involved. The approximate nature of the result is the inevitable price one pays for having so mild assumptions on the dynamics on the vertical component. However, the error in the approximation goes to zero when the expansion of the base tends to infinity. The result can be applied beyond the original setup when combined with acceleration or conjugation arguments, as our examples show.

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