We investigate and prove the mathematical properties of a general class of one-dimensional unimod... more We investigate and prove the mathematical properties of a general class of one-dimensional unimodal smooth maps perturbed with a heteroscedastic noise. Specifically, we investigate the stability of the associated Markov chain, show the weak convergence of the unique stationary measure to the invariant measure of the map, and show that the average Lyapunov exponent depends continuously on the Markov chain parameters. Representing the Markov chain in terms of random transformation enables us to state and prove the Central Limit Theorem, the large deviation principle, and the Berry-Esséen inequality. We perform a multifractal analysis for the invariant and the stationary measures, and we prove Gumbel's law for the Markov chain with an extreme index equal to 1. In addition, we present an example linked to the financial concept of systemic risk and leverage cycle, and we use the model to investigate the finite sample properties of our asymptotic results Keywords Random dynamical systems • Unimodal maps • Lyapunov exponents • Leverage cycles • Systemic risk Communicated by Francesco Zamponi.
We extend some results of Marmi-Moussa-Yoccoz on the cohomological equations and local conjugacy ... more We extend some results of Marmi-Moussa-Yoccoz on the cohomological equations and local conjugacy classes of interval exchange maps of restricted Roth type. In particular, we answer a question of Krikorian about the codimension of the local conjugacy class of self-similar interval exchange maps associated to the Eierlegende Wollmilchsau and the Ornithorynque.
In this article we discuss the distribution of asset price movements by the market potential func... more In this article we discuss the distribution of asset price movements by the market potential function. From the principle of free energy minimization we analyze two different kinds of market potentials. We obtain a U-shaped potential when market reversion (i.e. contrarian investors) is dominant. On the other hand, if there are more trend followers, flat and logarithmic-like potentials appeared. By using the Cyclical Adjusted Price-to-Earning ratio, which is a common valuation tool, we empirically investigate the market data. By studying long term data we observe the historical change of the market potential of the US stock market. Recent US data shows that the market potential looks more likely as the trending followers' potential. Next, we compare the market potentials for 12 different countries. Though some countries have similar market potentials, there are specific examples like Japan which exhibits very flat potential.
We consider the flow in direction θ on a translation surface and we study the asymptotic behavior... more We consider the flow in direction θ on a translation surface and we study the asymptotic behavior for r → 0 of the time needed by orbits to hit the r-neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as hitting time. For flat tori the limsup of hitting time is equal to the diophantine type of the direction θ. In higher genus, we consider a generalized geometric notion of diophantine type of a direction θ and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that the limsup of hitting time is always less or equal to the square of the diophantine type. For any square-tiled surface with the same topology the diophantine type itself is a lower bound, and any value between the two bounds can be realized, moreover this holds also for a larger class of origamis satisfying a specific topological assumption. Finally, for the so-called Eierlegende Wollmilchsau origami, the equality between limsup of hitting time and diophantine type subsists. Our results apply to L-shaped billiards.
We present some works of Corinna Ulcigrai closely related to Diophantine approximations and gener... more We present some works of Corinna Ulcigrai closely related to Diophantine approximations and generalizing classical notions to the context of interval exchange maps, translation surfaces and Teichmüller dynamics.
HAL (Le Centre pour la Communication Scientifique Directe), 2020
We consider the minimal average action (Mather's β function) for area preserving twist maps of th... more We consider the minimal average action (Mather's β function) for area preserving twist maps of the annulus. The regularity properties of this function share interesting relations with the dynamics of the system. We prove that the β-function associated to a standard-like twist map admits a unique C 1 -holomorphic complex extension, which coincides with this function on the set of real diophantine frequencies.
L'accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisq... more L'accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 113 CHAOTIC BEHAVIOUR IN THE SOLAR SYSTEM [following J. LASKAR] by Stefano MARMI Séminaire BOURBAKI 51ème annee, 1998-99, n° 854 Novembre 1998 0. INTRODUCTION I. Newton certainly believed that the Solar System is topologically unstable. In his view the perturbations among the planets were strong enough to destroy the stability of the Solar System. He even made the hypothesis that God controls the instabilities so as to insure the existence of the Solar System: "but it is not to be conceived that mere mechanical causes could give birth to so many regular motions .... This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent powerful Being" [N, p. 544]. The problem of Solar System stability was (and for many aspects still is) a real one: Halley was able to show, by analyzing the Chaldean observations transmitted by Ptolemy, that Saturn was moving away from the Sun while Jupiter was getting closer. A crude
We investigate and prove the mathematical properties of a general class of one-dimensional unimod... more We investigate and prove the mathematical properties of a general class of one-dimensional unimodal smooth maps perturbed with a heteroscedastic noise. Specifically, we investigate the stability of the associated Markov chain, show the weak convergence of the unique stationary measure to the invariant measure of the map, and show that the average Lyapunov exponent depends continuously on the Markov chain parameters. Representing the Markov chain in terms of random transformation enables us to state and prove the Central Limit Theorem, the large deviation principle, and the Berry-Esséen inequality. We perform a multifractal analysis for the invariant and the stationary measures, and we prove Gumbel's law for the Markov chain with an extreme index equal to 1. In addition, we present an example linked to the financial concept of systemic risk and leverage cycle, and we use the model to investigate the finite sample properties of our asymptotic results
We present an analytical model to study the role of expectation feedbacks and overlapping portfol... more We present an analytical model to study the role of expectation feedbacks and overlapping portfolios on systemic stability of financial systems. Building on [Corsi et al., 2016], we model a set of financial institutions having Value at Risk capital requirements and investing in a portfolio of risky assets, whose prices evolve stochastically in time and are endogenously driven by the trading decisions of financial institutions. Assuming that they use adaptive expectations of risk, we show that the evolution of the system is described by a slow-fast random dynamical system, which can be studied analytically in some regimes. The model shows how the risk expectations play a central role in determining the systemic stability of the financial system and how wrong risk expectations may create panic-induced reduction or over-optimistic expansion of balance sheets. Specifically, when investors are myopic in estimating the risk, the fixed point equilibrium of the system breaks into leverage cycles and financial variables display a bifurcation cascade eventually leading to chaos. We discuss the role of financial policy and the effects of some market frictions, as the cost of diversification and financial transaction taxes, in determining the stability of the system in the presence of adaptive expectations of risk.
By exploiting basic common practice accounting and risk management rules, we propose a simple ana... more By exploiting basic common practice accounting and risk management rules, we propose a simple analytical dynamical model to investigate the effects of micro-prudential changes on macro-prudential outcomes. Specifically, we study the consequence of the introduction of a financial innovation that allow reducing the cost of portfolio diversification in a financial system populated by financial institutions having capital requirements in the form of VaR constraint and following standard mark-to-market and risk management rules. We provide a full analytical quantification of the multivariate feedback effects between investment prices and bank behavior induced by portfolio rebalancing in presence of asset illiquidity and show how changes in the constraints of the bank portfolio optimization endogenously drive the dynamics of the balance sheet aggregate of financial institutions and, thereby, the availability of bank liquidity to the economic system and systemic risk. The model shows that when financial innovation reduces the cost of diversification below a given threshold, the strength (due to higher leverage) and coordination (due to similarity of bank portfolios) of feedback effects increase, triggering a transition from a stationary dynamics of price returns to a non stationary one characterized by steep growths (bubbles) and plunges (bursts) of market prices.
Instabilities in the price dynamics of a large number of financial assets are a clear sign of sys... more Instabilities in the price dynamics of a large number of financial assets are a clear sign of systemic events. By investigating a set of 20 high cap stocks traded at the Italian Stock Exchange, we find that there is a large number of multiple cojumps, i.e. minutes in which a sizable number of stocks displays a discontinuity of the price process. We show that the dynamics of these jumps is not described neither by a multivariate Poisson nor by a multivariate Hawkes model, which are unable to capture simultaneously the time clustering of jumps and the high synchronization of jumps across assets. We introduce a one factor model approach where both the factor and the idiosyncratic jump components are described by a Hawkes process. We introduce a robust calibration scheme which is able to distinguish systemic and idiosyncratic jumps and we show that the model reproduces very well the empirical behaviour of the jumps of the Italian stocks.
Communications in Nonlinear Science and Numerical Simulation
We study a random version of the population-market model proposed by Arlot, Marmi and Papini in A... more 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 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.
This paper investigates the degree of efficiency for the Moscow Stock Exchange. A market is calle... more 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 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 stock 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 o...
We investigate and prove the mathematical properties of a general class of one-dimensional unimod... more We investigate and prove the mathematical properties of a general class of one-dimensional unimodal smooth maps perturbed with a heteroscedastic noise. Specifically, we investigate the stability of the associated Markov chain, show the weak convergence of the unique stationary measure to the invariant measure of the map, and show that the average Lyapunov exponent depends continuously on the Markov chain parameters. Representing the Markov chain in terms of random transformation enables us to state and prove the Central Limit Theorem, the large deviation principle, and the Berry-Esséen inequality. We perform a multifractal analysis for the invariant and the stationary measures, and we prove Gumbel's law for the Markov chain with an extreme index equal to 1. In addition, we present an example linked to the financial concept of systemic risk and leverage cycle, and we use the model to investigate the finite sample properties of our asymptotic results Keywords Random dynamical systems • Unimodal maps • Lyapunov exponents • Leverage cycles • Systemic risk Communicated by Francesco Zamponi.
We extend some results of Marmi-Moussa-Yoccoz on the cohomological equations and local conjugacy ... more We extend some results of Marmi-Moussa-Yoccoz on the cohomological equations and local conjugacy classes of interval exchange maps of restricted Roth type. In particular, we answer a question of Krikorian about the codimension of the local conjugacy class of self-similar interval exchange maps associated to the Eierlegende Wollmilchsau and the Ornithorynque.
In this article we discuss the distribution of asset price movements by the market potential func... more In this article we discuss the distribution of asset price movements by the market potential function. From the principle of free energy minimization we analyze two different kinds of market potentials. We obtain a U-shaped potential when market reversion (i.e. contrarian investors) is dominant. On the other hand, if there are more trend followers, flat and logarithmic-like potentials appeared. By using the Cyclical Adjusted Price-to-Earning ratio, which is a common valuation tool, we empirically investigate the market data. By studying long term data we observe the historical change of the market potential of the US stock market. Recent US data shows that the market potential looks more likely as the trending followers' potential. Next, we compare the market potentials for 12 different countries. Though some countries have similar market potentials, there are specific examples like Japan which exhibits very flat potential.
We consider the flow in direction θ on a translation surface and we study the asymptotic behavior... more We consider the flow in direction θ on a translation surface and we study the asymptotic behavior for r → 0 of the time needed by orbits to hit the r-neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as hitting time. For flat tori the limsup of hitting time is equal to the diophantine type of the direction θ. In higher genus, we consider a generalized geometric notion of diophantine type of a direction θ and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that the limsup of hitting time is always less or equal to the square of the diophantine type. For any square-tiled surface with the same topology the diophantine type itself is a lower bound, and any value between the two bounds can be realized, moreover this holds also for a larger class of origamis satisfying a specific topological assumption. Finally, for the so-called Eierlegende Wollmilchsau origami, the equality between limsup of hitting time and diophantine type subsists. Our results apply to L-shaped billiards.
We present some works of Corinna Ulcigrai closely related to Diophantine approximations and gener... more We present some works of Corinna Ulcigrai closely related to Diophantine approximations and generalizing classical notions to the context of interval exchange maps, translation surfaces and Teichmüller dynamics.
HAL (Le Centre pour la Communication Scientifique Directe), 2020
We consider the minimal average action (Mather's β function) for area preserving twist maps of th... more We consider the minimal average action (Mather's β function) for area preserving twist maps of the annulus. The regularity properties of this function share interesting relations with the dynamics of the system. We prove that the β-function associated to a standard-like twist map admits a unique C 1 -holomorphic complex extension, which coincides with this function on the set of real diophantine frequencies.
L'accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisq... more L'accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 113 CHAOTIC BEHAVIOUR IN THE SOLAR SYSTEM [following J. LASKAR] by Stefano MARMI Séminaire BOURBAKI 51ème annee, 1998-99, n° 854 Novembre 1998 0. INTRODUCTION I. Newton certainly believed that the Solar System is topologically unstable. In his view the perturbations among the planets were strong enough to destroy the stability of the Solar System. He even made the hypothesis that God controls the instabilities so as to insure the existence of the Solar System: "but it is not to be conceived that mere mechanical causes could give birth to so many regular motions .... This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent powerful Being" [N, p. 544]. The problem of Solar System stability was (and for many aspects still is) a real one: Halley was able to show, by analyzing the Chaldean observations transmitted by Ptolemy, that Saturn was moving away from the Sun while Jupiter was getting closer. A crude
We investigate and prove the mathematical properties of a general class of one-dimensional unimod... more We investigate and prove the mathematical properties of a general class of one-dimensional unimodal smooth maps perturbed with a heteroscedastic noise. Specifically, we investigate the stability of the associated Markov chain, show the weak convergence of the unique stationary measure to the invariant measure of the map, and show that the average Lyapunov exponent depends continuously on the Markov chain parameters. Representing the Markov chain in terms of random transformation enables us to state and prove the Central Limit Theorem, the large deviation principle, and the Berry-Esséen inequality. We perform a multifractal analysis for the invariant and the stationary measures, and we prove Gumbel's law for the Markov chain with an extreme index equal to 1. In addition, we present an example linked to the financial concept of systemic risk and leverage cycle, and we use the model to investigate the finite sample properties of our asymptotic results
We present an analytical model to study the role of expectation feedbacks and overlapping portfol... more We present an analytical model to study the role of expectation feedbacks and overlapping portfolios on systemic stability of financial systems. Building on [Corsi et al., 2016], we model a set of financial institutions having Value at Risk capital requirements and investing in a portfolio of risky assets, whose prices evolve stochastically in time and are endogenously driven by the trading decisions of financial institutions. Assuming that they use adaptive expectations of risk, we show that the evolution of the system is described by a slow-fast random dynamical system, which can be studied analytically in some regimes. The model shows how the risk expectations play a central role in determining the systemic stability of the financial system and how wrong risk expectations may create panic-induced reduction or over-optimistic expansion of balance sheets. Specifically, when investors are myopic in estimating the risk, the fixed point equilibrium of the system breaks into leverage cycles and financial variables display a bifurcation cascade eventually leading to chaos. We discuss the role of financial policy and the effects of some market frictions, as the cost of diversification and financial transaction taxes, in determining the stability of the system in the presence of adaptive expectations of risk.
By exploiting basic common practice accounting and risk management rules, we propose a simple ana... more By exploiting basic common practice accounting and risk management rules, we propose a simple analytical dynamical model to investigate the effects of micro-prudential changes on macro-prudential outcomes. Specifically, we study the consequence of the introduction of a financial innovation that allow reducing the cost of portfolio diversification in a financial system populated by financial institutions having capital requirements in the form of VaR constraint and following standard mark-to-market and risk management rules. We provide a full analytical quantification of the multivariate feedback effects between investment prices and bank behavior induced by portfolio rebalancing in presence of asset illiquidity and show how changes in the constraints of the bank portfolio optimization endogenously drive the dynamics of the balance sheet aggregate of financial institutions and, thereby, the availability of bank liquidity to the economic system and systemic risk. The model shows that when financial innovation reduces the cost of diversification below a given threshold, the strength (due to higher leverage) and coordination (due to similarity of bank portfolios) of feedback effects increase, triggering a transition from a stationary dynamics of price returns to a non stationary one characterized by steep growths (bubbles) and plunges (bursts) of market prices.
Instabilities in the price dynamics of a large number of financial assets are a clear sign of sys... more Instabilities in the price dynamics of a large number of financial assets are a clear sign of systemic events. By investigating a set of 20 high cap stocks traded at the Italian Stock Exchange, we find that there is a large number of multiple cojumps, i.e. minutes in which a sizable number of stocks displays a discontinuity of the price process. We show that the dynamics of these jumps is not described neither by a multivariate Poisson nor by a multivariate Hawkes model, which are unable to capture simultaneously the time clustering of jumps and the high synchronization of jumps across assets. We introduce a one factor model approach where both the factor and the idiosyncratic jump components are described by a Hawkes process. We introduce a robust calibration scheme which is able to distinguish systemic and idiosyncratic jumps and we show that the model reproduces very well the empirical behaviour of the jumps of the Italian stocks.
Communications in Nonlinear Science and Numerical Simulation
We study a random version of the population-market model proposed by Arlot, Marmi and Papini in A... more 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 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.
This paper investigates the degree of efficiency for the Moscow Stock Exchange. A market is calle... more 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 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 stock 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 o...
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