ROSENBAUM Mathieu

In this paper we provide an expansion formula for Hawkes processes which involves the addition of jumps at deterministic times to the Hawkes process in the spirit of the wellknown integration by parts formula (or more precisely the Mecke formula) for Poisson functional. Our approach allows us to provide an expansion of the premium of a class of cyber insurance derivatives (such as reinsurance contracts including generalized Stop-Loss contracts) or risk management instruments (like Expected Shortfall) in terms of so-called shifted Hawkes processes. From the actuarial point of view, these processes can be seen as "stressed" scenarios. Our expansion formula for Hawkes processes enables us to provide lower and upper bounds on the premium (or the risk evaluation) of such cyber contracts and to quantify the surplus of premium compared to the standard modeling with a homogenous Poisson process.

This thesis is composed of two independent parts. The first part focuses on the application of the Principal-Agent problem (c.f. Cvitanic & Zhang (2013) and Cvitanic. et al. (2018)) for solving modeling problems in energy markets. The second one deals with the joint laws of a martingale and its current maximum.We first focus on the electricity capacity market, and in particular capacity remuneration mechanisms. Given the increasing share of renewable energies in the electricity production, "classical" power plants (e.g. gas or coal) are less and less used, which makes them unprofitable and not economically viable. However, their closure would expose consumers to the risk of a blackout in the event of a peak in electricity demand, since electricity cannot be stored. Thus, generation capacity must always be maintained above demand, which requires a "capacity payment mechanism" to remunerate seldom used power plants, which can be understood as an insurance to be paid against electricity blackouts.We then address the issue of incentives for decarbonization. The objective is to propose a model of an instrument that can be used by a public agent (the state) to encourage the different sectors to reduce their carbon emissions in a context of moral hazard (where the state does not observe the effort of the actors and therefore cannot know whether a decrease in emissions comes from a decrease in production and consumption or from a management effort. The second part (independent) is motivated by model calibration and arbitrage on a financial market with barrier options. It presents a result on the joint laws of a martingale and its current maximum. We consider a family of probabilities in dimension 2, and we give necessary and sufficient conditions ensuring the existence of a martingale such that its marginal laws coupled with those of its current maximum coincide with the given probabilities.We follow the methodology of Hirsch and Roynette (2012) based on a martingale construction by DHS associated with a well-posed Fokker-Planck PDE verified by the given marginal laws under regularity assumptions, then in a general framework with regularization and boundary crossing.

This thesis is divided into three parts. In the first part, we apply Principal-Agent theory to some market microstructure problems. First, we develop an incentive policy to improve the quality of market liquidity in the context of market-making activity in a bed and a dark pool managed by the same exchange. We then adapt this incentive design to the regulation of market-making activity when several market-makers compete on a platform. We also propose a form of incentive based on the choice of asymmetric tick sizes for buying and selling an asset. We then address the issue of designing a derivatives market, using a quantization method to select the options listed on the platform, and Principal-Agent theory to create incentives for an options market-maker. Finally, we develop an incentive mechanism robust to the model specification to increase investment in green bonds.The second part of this thesis is devoted to high-dimensional options market-making. The second part of this paper is devoted to the market-making of high-dimensional options. Assuming constant Greeks, we first propose a model to deal with long-maturity options. Then we propose an approximation of the value function to handle non-constant Greeks and short maturity options. Finally, we develop a model for the high frequency dynamics of the implied volatility surface. Using multidimensional Hawkes processes, we show how this model can reproduce many stylized facts such as the skew, the smile and the term structure of the surface.The last part of this thesis is devoted to optimal trading problems in high dimension. First, we develop a model for optimal trading of stocks listed on several platforms. For a large number of platforms, we use a deep reinforcement learning method to compute the optimal trader controls. Then, we propose a methodology to solve trading problems in an approximately optimal way without using stochastic control theory. We present a model in which an agent exhibits approximately optimal behavior if it uses the gradient of the macroscopic trajectory as a short-term signal. Finally, we present two new developments on the optimal execution literature. First, we show that we can obtain an analytical solution to the Almgren-Chriss execution problem with geometric Brownian motion and quadratic penalty. Second, we propose an application of the latent order book model to the optimal execution problem of a portfolio of assets, in the context of liquidity stress tests.

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Recent empirical analyses have revealed the existence of the Zumbach effect. This discovery led to the development of the quadratic Hawkes process, adapted to reproduce this effect. Since this model does not relate to the price formation process, we extended it to the order book with a generalized quadratic Hawkes process (GQ-Hawkes). Using market data, we show that there is a Zumbach-like effect that decreases future liquidity. Microfounding the Zumbach effect, it is responsible for a potential destabilization of financial markets. Moreover, the exact calibration of a QM-Hawkes process tells us that markets are at the edge of criticality. This empirical evidence has therefore prompted us to analyze an order book model constructed with a Zumbach-type coupling. We therefore introduced the Santa Fe quadratic model and proved numerically that there is a phase transition between a stable market and an unstable market subject to liquidity crises. Thanks to a finite size analysis we were able to determine the critical exponents of this transition, belonging to a new universality class. Not being analytically solvable, this led us to introduce simpler models to describe liquidity crises. Putting aside the microstructure of the order book, we obtain a class of spread models where we have computed the critical parameters of their transitions. Even if these exponents are not those of the Santa Fe quadratic transition, these models open new horizons to explore the spread dynamics. One of them has a nonlinear coupling that reveals a metastable state. This elegant alternative scenario does not need critical parameters to obtain an unstable market, even if the empirical evidence is not in its favor. Finally, we looked at order book dynamics from another angle: reaction-diffusion. We modeled a liquidity that reveals itself in the order book with a certain frequency. Solving this model in equilibrium reveals that there is a stability condition on the parameters beyond which the order book empties completely, corresponding to a liquidity crisis. By calibrating it on market data, we were able to qualitatively analyze the distance to this unstable region.

The main objective of this thesis is to understand the various aspects of market impact. It consists of four chapters in which market impact is studied in different contexts and at different scales. The first chapter presents an empirical study of the market impact of limit orders in European equity markets. In the second chapter, we have extended the methodology presented for the equity markets to the options markets. This empirical study has shown that our definition of an options meta-order allows us to recover all the results highlighted in the equity markets. The third chapter focuses on market impact in the context of derivatives valuation. This chapter attempts to bring a microstructure component to the valuation of options by proposing a theory of market impact disturbances during the re-hedging process. In the fourth chapter, we explore a fairly simple model for metaorder relaxation. Metaorder relaxation is treated in this section as an informational process that is transmitted to the market. Thus, starting from the point of departure that at the end of the execution of a meta-order the information carried by it is maximal, we propose an interpretation of the relaxation phenomenon as being the result of the degradation of this information at the expense of the external noise of the market.

In this thesis, we are mainly interested in three research topics, relatively independent, but nevertheless related through the thread of interactions and incentives, as highlighted in the introduction constituting the first chapter.In the first part, we present extensions of contract theory, allowing in particular to consider a multitude of players in principal-agent models, with drift and volatility control, in the presence of moral hazard. In particular, Chapter 2 presents a continuous-time optimal incentive problem within a hierarchy, inspired by the one-period model of Sung (2015) and enlightening in two respects: on the one hand, it presents a framework where volatility control occurs in a perfectly natural way, and, on the other hand, it highlights the importance of considering continuous-time models. In this sense, this example motivates the comprehensive and general study of hierarchical models carried out in the third chapter, which goes hand in hand with the recent theory of second-order stochastic differential equations (2EDSR). Finally, in Chapter 4, we propose an extension of the principal-agent model developed by Aïd, Possamaï, and Touzi (2019) to a continuum of agents, whose performances are in particular impacted by a common hazard. In particular, these studies guide us towards a generalization of the so-called revealing contracts, initially proposed by Cvitanić, Possamaï and Touzi (2018) in a single-agent model.In the second part, we present two applications of principal-agent problems to the energy domain. The first one, developed in Chapter 5, uses the formalism and theoretical results introduced in the previous chapter to improve electricity demand response programs, already considered by Aïd, Possamaï and Touzi (2019). Indeed, by taking into account the infinite number of consumers that a producer has to supply with electricity, it is possible to use this additional information to build the optimal incentives, in particular to better manage the residual risk implied by weather hazards. In a second step, chapter 6 proposes, through a principal-agent model with adverse selection, an insurance that could prevent some forms of precariousness, in particular fuel precariousness.Finally, we end this thesis by studying in the last part a second field of application, that of epidemiology, and more precisely the control of the diffusion of a contagious disease within a population. In chapter 7, we first consider the point of view of individuals, through a mean-field game: each individual can choose his rate of interaction with others, reconciling on the one hand his need for social interactions and on the other hand his fear of being contaminated in turn, and of contributing to the wider diffusion of the disease. We prove the existence of a Nash equilibrium between individuals, and exhibit it numerically. In the last chapter, we take the point of view of the government, wishing to incite the population, now represented as a whole, to decrease its interactions in order to contain the epidemic. We show that the implementation of sanctions in case of non-compliance with containment can be effective, but that, for a total control of the epidemic, it is necessary to develop a conscientious screening policy, accompanied by a scrupulous isolation of the individuals tested positive.

The speed of information exchange in the Internet network is measured using latency: a time that measures the time elapsed between the sending of the first bit of information of a request and the reception of the first bit of information of the response. In this thesis realized in collaboration with Citrix, we are interested in the study and modeling of latency data in a context of Internet traffic optimization. Citrix collects data through two different channels, generating latency measures suspected to share common properties. In a first step, we address a distributional fitting problem where the co-variates and the responses are probability measures imaged from each other by a deterministic transport, and the observables are independent samples drawn according to these laws. We propose an estimator of this transport and show its convergence properties. We show that our estimator can be used to match the distributions of the latency measures generated by the two channels.In a second step we propose a modeling strategy to predict the process obtained by computing the moving median of the latency measures on regular partitions of the interval [0, T] with a mesh size D > 0. We show that the conditional mean of this process, which plays a major role in Internet traffic optimization, is correctly described by a Fourier series decomposition and that its conditional variance is organized in clusters that we model using an ARMA Seasonal-GARCH process, i.e., an ARMA-GARCH process with added deterministic seasonal terms. The predictive performance of this model is compared to the reference models used in the industry. A new measure of the amount of residual information not captured by the model based on a certain entropy criterion is introduced.We then address the problem of fault detection in the Internet network. We propose an algorithm for detecting changes in the distribution of a stream of latency data based on the comparison of two sliding windows using a certain weighted Wasserstein distance.Finally, we describe how to select the training data of predictive algorithms in order to reduce their size to limit the computational cost without impacting the accuracy.

This thesis is organized in three parts. The first part examines the relationship between microscopic and macroscopic market dynamics by focusing on the properties of volatility. In the second part, we focus on the stochastic optimal control of point processes. Finally, in the third part, we study two market design problems. We start this thesis by studying the links between the no-arbitrage principle and the volatility irregularity. Using a scaling method, we show that we can effectively connect these two notions by analyzing the market impact of metaorders. More precisely, we model the market order flow using linear Hawkes processes. We then show that the no-arbitrage principle and the existence of a non-trivial market impact imply that volatility is rough and more precisely that it follows a rough Heston model. We then examine a class of microscopic models where the order flow is a quadratic Hawkes process. The objective is to extend the rough Heston model to continuous models allowing to reproduce the Zumbach effect. Finally, we use one of these models, the quadratic rough Heston model, for the joint calibration of the SPX and VIX volatility slicks. Motivated by the intensive use of point processes in the first part, we are interested in the stochastic control of point processes in the second part. Our objective is to provide theoretical results for applications in finance. We start by considering the case of Hawkes process control. We prove the existence of a solution and then propose a method to apply this control in practice. We then examine the scaling limits of stochastic control problems in the context of population dynamics models. More precisely, we consider a sequence of models of discrete population dynamics which converge to a model for a continuous population. For each model we consider a control problem. We prove that the sequence of optimal controls associated to the discrete models converges to the optimal control associated to the continuous model. This result is based on the continuity, with respect to different parameters, of the solution of a backward-looking schostatic differential equation.In the last part we consider two market design problems. First, we examine the question of the organization of a liquid derivatives market. Focusing on an options market, we propose a two-step method that can be easily applied in practice. The first step is to select the options that will be listed on the market. For this purpose, we use a quantization algorithm that allows us to select the options most in demand by investors. We then propose a pricing incentive method to encourage market makers to offer attractive prices. We formalize this problem as a principal-agent problem that we solve explicitly. Finally, we find the optimal duration of an auction for markets organized in sequential auctions, the case of zero duration corresponding to the case of a continuous double auction. We use a model where the market takers are in competition and we consider that the optimal duration is the one corresponding to the most efficient price discovery process. After proving the existence of a Nash equilibrium for the competition between market takers, we apply our results on market data. For most assets, the optimal duration is between 2 and 10 minutes.

The subject of the thesis is parametric and non-parametric estimation in jump process models. The thesis is composed of 3 parts which regroup 4 works. The first part, which is composed of two chapters, deals with the estimation of drift and volatility parameters by contrast methods from discrete observations, with the main objective of minimizing the conditions on the observation step, so that it can for example go arbitrarily slowly towards 0. The second part of the thesis concerns asymptotic developments, and bias correction, for the estimation of the integrated volatility. The third part of the thesis, concerns the adaptive estimation of the stationary measure for jump processes.

In this thesis, we examine several stochastic approximation methods for both the computation of financial risk measures and the pricing of derivatives.Since explicit formulas are rarely available for such quantities, the need for fast, efficient and reliable analytical approximations is of paramount importance to financial institutions.In the first part, we study several multilevel Monte Carlo approximation methods and apply them to two practical problems: the estimation of quantities involving nested expectations (such as initial margin) and the discretization of integrals appearing in rough models for the forward variance for VIX option pricing.In both cases, we analyze the asymptotic optimality properties of the corresponding multilevel estimators and numerically demonstrate their superiority over a classical Monte Carlo method.In the second part, motivated by the numerous examples from credit risk modeling, we propose a general metamodeling framework for large sums of weighted Bernoulli random variables, which are conditionally independent with respect to a common factor X. Our generic approach is based on the polynomial decomposition of the chaos of the common factor and on a Gaussian approximation. L2 error estimates are given when the factor X is associated with classical orthogonal polynomials.Finally, in the last part of this thesis, we focus on the short-time asymptotics of U.S. implied volatility and U.S. option prices in local volatility models. We also propose a law approximation of the VIX index in rough models for forward variance, expressed in terms of lognormal proxies, and derive expansion results for VIX options with explicit coefficients.

We address the mechanism design problem of an exchange setting suitable make-take fees to attract liquidity on its platform. Using a principal-agent approach, we provide the optimal compensation scheme of a market maker in quasi-explicit form. This contract depends essentially on the market maker inventory trajectory and on the volatility of the asset. We also provide the optimal quotes that should be displayed by the market maker. The simplicity of our formulas allows us to analyze in details the effects of optimal contracting with an exchange, compared to a situation without contract. We show in particular that it improves liquidity and reduces trading costs for investors. We extend our study to an oligopoly of symmetric exchanges and we study the impact of such common agency policy on the system.

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This thesis consists of two interrelated parts. In the first part, we empirically study the behavior of high-frequency traders on European financial markets. In the second part, we use the results obtained to build new multi-agent models. The main objective of these models is to provide regulators and trading platforms with innovative tools to implement microstructure relevant rules and to quantify the impact of various participants on market quality.In the first part, we perform two empirical studies on unique data provided by the French regulator. We have access to all orders and trades of CAC 40 assets, at the microsecond scale, with the identities of the actors involved. We begin by comparing the behavior of high-frequency traders to that of other players, particularly during periods of stress, in terms of liquidity provision and trading activity. We then deepen our analysis by focusing on liquidity consuming orders. We study their impact on the price formation process and their information content according to the different categories of flows: high-frequency traders, participants acting as clients and participants acting as principal.In the second part, we propose three multi-agent models. Using a Glosten-Milgrom approach, our first model constructs the entire order book (spread and volume available at each price) from the interactions between three types of agents: an informed agent, an uninformed agent and market makers. This model also allows us to develop a methodology for predicting the spread in case of a change in the price step and to quantify the value of the priority in the queue. In order to focus on an individual scale, we propose a second approach where the specific dynamics of the agents are modeled by nonlinear Hawkes-type processes that depend on the state of the order book. In this framework, we are able to compute several relevant microstructure indicators based on individual flows. In particular, it is possible to classify market makers according to their own contribution to volatility. Finally, we introduce a model where liquidity providers optimize their best bid and offer prices according to the profit they can generate and the inventory risk they face. We then theoretically and empirically highlight an important new relationship between inventory and volatility.

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We address the mechanism design problem of an exchange setting suitable make-take fees to attract liquidity on its platform. Using a principal-agent approach, we provide the optimal compensation scheme of a market maker in quasi-explicit form. This contract depends essentially on the market maker inventory trajectory and on the volatility of the asset. We also provide the optimal quotes that should be displayed by the market maker. The simplicity of our formulas allows us to analyze in details the effects of optimal contracting with an exchange, compared to a situation without contract. We show in particular that it improves liquidity and reduces trading costs for investors. We extend our study to an oligopoly of symmetric exchanges and we study the impact of such common agency policy on the system.

The main objective of this thesis is to understand the interactions between financial agents and the order book. We consider in the first chapter the control problem of an agent trying to take into account the available liquidity in the order book in order to optimize the placement of a unit order. Our strategy reduces the risk of adverse selection. Nevertheless, the added value of this approach is weakened in the presence of latency: predicting future price movements is of little use if agents' reaction time is slow.In the next chapter, we extend our study to a more general execution problem where agents trade non-unitary quantities in order to limit their impact on the price. In the third chapter, we build on the previous approach to solve this time market making problems rather than execution problems. This allows us to propose relevant strategies compatible with the typical actions of market makers. Then, we model the behavior of directional high frequency traders and institutional brokers in order to simulate a market where our three types of agents interact optimally with each other.We propose in the fourth chapter an agent model where the flow dynamics depend not only on the state of the order book but also on the market history. To do so, we use generalizations of nonlinear Hawkes processes. In this framework, we are able to compute several relevant indicators based on individual flows. In particular, it is possible to classify market makers according to their contribution to volatility.To solve the control problems raised in the first part of the thesis, we have developed numerical schemes. Such an approach is possible when the dynamics of the model are known. When the environment is unknown, stochastic iterative algorithms are usually used. In the fifth chapter, we propose a method to accelerate the convergence of such algorithms.The approaches considered in the previous chapters are suitable for liquid markets using the order book mechanism. However, this methodology is not necessarily relevant for markets governed by specific operating rules. To address this issue, we propose, first, to study the behavior of prices in the very specific electricity market.

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This thesis aims at understanding several aspects of the roughness of volatility observed universally on financial assets. This is done in six steps. In the first part, we explain this property from the typical behaviors of agents in the market. More precisely, we build a microscopic price model based on Hawkes processes reproducing the important stylized facts of the market microstructure. By studying the long-run price behavior, we show the emergence of a rough version of the Heston model (called rough Heston model) with leverage. Using this original link between Hawkes processes and Heston models, we compute in the second part of this thesis the characteristic function of the log-price of the rough Heston model. This characteristic function is given in terms of a solution of a Riccati equation in the case of the classical Heston model. We show the validity of a similar formula in the case of the rough Heston model, where the Riccati equation is replaced by its fractional version. This formula allows us to overcome the technical difficulties due to the non-Markovian character of the model in order to value derivatives. In the third part, we address the issue of risk management of derivatives in the rough Heston model. We present hedging strategies using the underlying asset and the forward variance curve as instruments. This is done by specifying the infinite-dimensional Markovian structure of the model. Being able to value and hedge derivatives in the rough Heston model, we confront this model with the reality of financial markets in the fourth part. More precisely, we show that it reproduces the behavior of implied and historical volatility. We also show that it generates the Zumbach effect, which is a time-reversal asymmetry observed empirically on financial data. In the fifth part, we study the limiting behavior of the implied volatility at low maturity in the framework of a general stochastic volatility model (including the rough Bergomi model), by applying a density development of the asset price. While the approximation based on Hawkes processes has addressed several questions related to the rough Heston model, in Part 6 we consider a Markovian approximation applying to a more general class of rough volatility models. Using this approximation in the particular case of the rough Heston model, we obtain a numerical method for solving the fractional Riccati equations. Finally, we conclude this thesis by studying a problem not related to the rough volatility literature. We consider the case of a platform seeking the best make-take fee scheme to attract liquidity. Using the principal-agent framework, we describe the best contract to offer to the market maker as well as the optimal quotes displayed by the latter. We also show that this policy leads to better liquidity and lower transaction costs for investors.

This thesis addresses different aspects of market microstructure modeling and market making problems, with a particular focus on the practitioner's point of view. The order book, at the heart of the financial market, is a complex high-dimensional queueing system. We aim to improve the knowledge of the LOB for the research community, propose new modeling ideas and develop applications for Market Makers. In particular, we thank the Automated Market Making team for providing the high-quality high-frequency database and a powerful computational grid, without which this research would not have been possible. Chapter 1 presents the motivation for this research and summarizes the main results of the different works. Chapter 2 focuses entirely on the LOB and aims at proposing a new model that better reproduces some stylized facts. Through this research, not only do we confirm the influence of historical order flow on the arrival of new ones, but a new model is also provided that replicates much better the dynamics of the LOB, in particular the volatility realized in high and low frequency. In Chapter 3, the objective is to study market making strategies in a more realistic context. This research contributes to two aspects: on the one hand the new proposed model is more realistic but still simple to apply for strategy design, on the other hand the practical Market Making strategy is much improved compared to a naive strategy and is promising for practical application. High frequency prediction with deep learning method is studied in Chapter 4. Numerous results of the 1-step and multi-step prediction found the non-linearity, stationarity and universality of the relationship between the microstructure indicators and the price change, as well as the limitation of this approach in practice.

This thesis contains two parts that study two different topics. Chapters 1-4 are devoted to problems of discretization of processes with stopping times. In Chapter 1 we study the optimal discretization error for stochastic integrals with respect to a continuous multidimensional Brownian semimartingale. In this framework we establish a trajectory lower bound for the renormalized quadratic variation of the error. We provide a sequence of stopping times that gives an asymptotically optimal discretization. This sequence is defined as the output time of random ellipsoids by the semimartingale. Compared to the previous results we allow a rather large class of semimartingales. We prove that the lower bound is exact. In Chapter 2 we study the adaptive version of the model of the optimal discretization of stochastic integrals. In Chapter 1 the construction of the optimal strategy uses the knowledge of the diffusion coefficient of the considered semimartingale. In this work we establish an asymptotically optimal discretization strategy that is adaptive to the model and does not use any information about the model. We prove the optimality for a rather general class of discretization grids based on kernel techniques for adaptive estimation. In Chapter 3 we study the convergence of renormalized discretization error laws of Itô processes for a concrete and rather general class of discretization grids given by stopping times. Previous works on the subject consider only the case of dimension 1. Moreover they concentrate on particular cases of grids, or prove results under abstract assumptions. In our work the boundary distribution is given explicitly in a clear and simple form, the results are shown in the multidimensional case for the process and for the discretization error. In Chapter 4 we study the parametric estimation problem for diffusion processes based on time-lapse observations. Previous works on the subject consider deterministic, strongly predictable or random observation times independent of the process. Under weak assumptions, we construct a suite of consistent estimators for a large class of observation grids given by stopping times. An asymptotic analysis of the estimation error is performed. Furthermore, for the parameter of dimension 1, for any sequence of estimators that verifies an unbiased LCT, we prove a uniform lower bound for the asymptotic variance. We show that this bound is exact. Chapters 5-6 are devoted to the uncertainty quantification problem for stochastic approximation bounds. In Chapter 5 we analyze the uncertainty quantification for stochastic approximation limits (SA). In our framework the limit is defined as a zero of a function given by an expectation. This expectation is taken with respect to a random variable for which the model is supposed to depend on an uncertain parameter. We consider the limit of SA as a function of this parameter. We introduce an algorithm called USA (Uncertainty for SA). It is a procedure in increasing dimension to compute the basic chaos expansion coefficients of this function in a basis of a well chosen Hilbert space. The convergence of USA in this Hilbert space is proved. In Chapter 6 we analyze the convergence rate in L2 of the USA algorithm developed in Chapter 5. The analysis is non-trivial because of the infinite dimension of the procedure. The rate obtained depends on the model and the parameters used in the USA algorithm. Its knowledge allows to optimize the rate of growth of the dimension in USA.

We address the mechanism design problem of an exchange setting suitable make-take fees to attract liquidity on its platform. Using a principal-agent approach, we provide the optimal compensation scheme of a market maker in quasi-explicit form. This contract depends essentially on the market maker inventory trajectory and on the volatility of the asset. We also provide the optimal quotes that should be displayed by the market maker. The simplicity of our formulas allows us to analyze in details the effects of optimal contracting with an exchange, compared to a situation without contract. We show in particular that it improves liquidity and reduces trading costs for investors. We extend our study to an oligopoly of symmetric exchanges and we study the impact of such common agency policy on the system.

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This thesis focuses on valuation and financial strategies for hedging risks in energy markets. These markets present particularities that distinguish them from standard financial markets, notably illiquidity and incompleteness. Illiquidity is reflected in high transaction costs and constraints on volumes traded. Incompleteness is the inability to perfectly replicate derivatives. We focus on different aspects of market incompleteness. The first part deals with valuation in Lévy models. We obtain an approximate formula for the indifference price and we measure the minimum premium to be brought over the Black-Scholes model. The second part concerns the valuation of spread options in the presence of stochastic correlation. Spread options deal with the price difference between two underlying assets -- for example gas and electricity -- and are widely used in the energy markets. We propose an efficient numerical procedure to calculate the price of these options. Finally, the third part deals with the valuation of a product with an exogenous risk for which forecasts exist. We propose an optimal dynamic strategy in the presence of volume risk, and apply it to the valuation of wind farms. In addition, a section is devoted to asymptotic optimal strategies in the presence of transaction costs.

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The subject of this thesis is the statistical inference of multidimensional Ornstein-Uhlenbeck processes. In a first part, we introduce a model of stochastic graphs defined as binary observations of trajectories. We then show that it is possible to deduce the dynamics of the underlying trajectory from the binary observations. For this, we construct statistics from the graph and show new convergence properties in the context of a long time and high frequency observation. We also analyze the properties of stochastic graphs from the point of view of evolving networks. In a second part, we work under the assumption of complete information and continuous time and add a sparsity assumption concerning the textit{drift} parameter of the Ornstein-Uhlenbeck process. We then show sharp oracle properties of the Lasso estimator, prove a lower bound on the estimation error in the minimax sense and show asymptotic optimality properties of the Adaptive Lasso estimator. We then apply these methods to estimate the speed of return at the average of daily returns of US stocks as well as the prices of dividend futures for the EURO STOXX 50 index.

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This thesis is devoted to the study of price formation in financial markets, especially when these markets are composed of a large number of agents. We begin with an empirical study of an emerging market -- bitcoin -- in order to better understand how individual actions affect prices -- the so-called "market impact". We then develop a theoretical model of impact based on the concept of the heterogeneous agent, which manages to replicate empirical observations of a concave impact in a non-manipulable market. The heterogeneous agent framework allows us to revisit the concepts of supply and demand in a dynamic framework, to better understand the impact of the market mechanism on liquidity, and to lay the foundations of a realistic market simulator. Finally, we show, through the empirical study of several bubbles and crashes on the bitcoin market, the crucial role of the micro-structure in the understanding of extreme phenomena.

Several new estimation methods have been recently proposed for the linear regression model with observation error in the design. Different assumptions on the data generating process have motivated different estimators and analysis. In particular, the literature considered (1) observation errors in the design uniformly bounded by some $\bar \delta$, and (2) zero mean independent observation errors. Under the first assumption, the rates of convergence of the proposed estimators depend explicitly on $\bar \delta$, while the second assumption has been applied when an estimator for the second moment of the observational error is available. This work proposes and studies two new estimators which, compared to other procedures for regression models with errors in the design, exploit an additional $l_{\infty}$-norm regularization. The first estimator is applicable when both (1) and (2) hold but does not require an estimator for the second moment of the observational error. The second estimator is applicable under (2) and requires an estimator for the second moment of the observation error. Importantly, we impose no assumption on the accuracy of this pilot estimator, in contrast to the previously known procedures. As the recent proposals, we allow the number of covariates to be much larger than the sample size. We establish the rates of convergence of the estimators and compare them with the bounds obtained for related estimators in the literature. These comparisons show interesting insights on the interplay of the assumptions and the achievable rates of convergence.

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The main objective of this thesis is to provide new theoretical results concerning the performance of investments based on stochastic models. To do so, we consider the optimal investment strategy in the framework of a risky asset model with constant volatility and a hidden Ornstein Uhlenbeck process. In the first chapter, we present the context and the objectives of this study. We present, also, the different methods used, as well as the main results obtained. In the second chapter, we focus on the feasibility of trend calibration. We answer this question with analytical results and numerical simulations. We close this chapter by also quantifing the impact of a calibration error on the trend estimate and exploit the results to detect its sign. In the third chapter, we assume that the agent is able to calibrate the trend well and we study the impact that the non-observability of the trend has on the performance of the optimal strategy. To do so, we consider the case of a logarithmic utility and an observed or unobserved trend. In each of the two cases, we explain the asymptotic limit of the expectation and the variance of the logarithmic return as a function of the signal-to-noise ratio and the speed of reversion to the mean of the trend. We conclude this study by showing that the asymptotic Sharpe ratio of the optimal strategy with partial observations cannot exceed 2/(3^1.5)∗100% of the asymptotic Sharpe ratio of the optimal strategy with complete information. The fourth chapter studies the robustness of the optimal strategy with calibration error and compares its performance to a technical analysis strategy. To do so, we characterize, analytically, the asymptotic expectation of the logarithmic return of each of these two strategies. We show, through our theoretical results and numerical simulations, that a technical analysis strategy is more robust than the poorly calibrated optimal strategy.

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We show that simple explicit formulas can be obtained for several relevant quantities related to the laws of the uniformly sampled Brownian bridge, Brownian meander and three dimensional Bessel process. To prove such results, we use the distribution of a triplet of random variables associated to the pseudo-Brownian bridge together with various relationships between the laws of these four processes.

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We extend the results of [14, 27, 29] about the convergence of nearly unstable AR(p) processes to the infinite order case. To do so, we proceed as in [19, 20] by using limit theorems for some well chosen geometric sums. We prove that when the coefficients sequence has a light tail, nearly unstable AR($\mathrm{\infty }$) processes behave as Ornstein-Uhlenbeck models. However, in the heavy tail case, we show that fractional diffusions arise as limiting laws for such processes.

No summary available.

This thesis explores theoretically and empirically the implications of the joint stock/option dynamics on various issues related to options trading. First, we study the joint dynamics between an option on a stock and an option on the market index. The CAPM model provides an adequate mathematical framework for this study because it allows to model the joint dynamics of a stock and its market index. Moving to option prices, we show that beta and idiosyncratic volatility, parameters of the model, allow us to characterize the relationship between the implied volatility surfaces of the stock and the index. We then turn to the estimation of the beta parameter under the risk-neutral probability using option prices. This measure, called implied beta, represents the information contained in the option prices about the realization of the beta parameter in the future.For this reason, we try to see, if implied beta has any predictive power of the future beta.By conducting an empirical study, we conclude that implied beta does not improve the predictive ability compared to the historical beta which is computed through the linear regression of the stock returns on the index returns. Better yet, we note that the oscillation of the implied beta around the future beta can lead to arbitrage opportunities, and we propose an arbitrage strategy that allows to monetize this gap. On the other hand, we show that the implied beta estimator could be used to hedge options on the stock using index instruments, this hedging concerns notably the volatility risk and also the delta risk. In the second part of our work, we are interested in the problem of market making on options. In this study, we assume that the model of the underlying's dynamics under the risk-neutral probability could be misspecified, which reflects a mismatch between the implied distribution of the underlying and its historical distribution.First, we consider the case of a risk-neutral market maker who aims to maximize the expectation of his future wealth. Using a stochastic optimal control approach, we determine the optimal call and put prices on the option and interpret the effect of price inefficiency on the optimal strategy. In a second step, we consider that the market maker is risk averse and therefore tries to reduce the uncertainty associated with his inventory. By solving an optimization problem based on a mean-variance criterion, we obtain analytical approximations of the optimal buying and selling prices. We also show the effects of inventory and price inefficiency on the optimal strategy. We then turn to the market making of options in a higher dimension. Thus, following the same reasoning, we present a framework for the market making of two options with different underlyings with the constraint of variance reduction related to the inventory risk held by the market maker. In the last part of our work, we study the joint dynamics between the implied volatility at the currency and the underlying, and we try to establish the link between these joint dynamics and the implied skew. We are interested in an indicator called "Skew Stickiness Ratio" which has been introduced in the recent literature. This indicator measures the sensitivity of the implied volatility of the currency to the movements of the underlying. We propose a method that allows us to estimate the value of this indicator under the risk-neutral probability without the need to admit assumptions on the dynamics of the underlying. [.].

No summary available.

This thesis consists of two related parts, the first on the order book and the second on tick value effects. In the first part, we present our backlog modeling framework. The tail-reactive model is first introduced, in which we revise the traditional zero-intelligence approach by adding dependence on the order book state. An empirical study shows that this model is very realistic and reproduces many interesting microscopic features of the underlying asset such as the backlog distribution. We also show that it can be used as an efficient market simulator, allowing the evaluation of complex investment tactics. We then extend the tail-reactive model to a general Markovian framework. Ergodicity conditions are discussed in detail in this setting. In the second part of this thesis, we are interested in studying the role played by the tick value at two microscopic and macroscopic scales. First, an empirical study of the consequences of a change in tick value is performed using data from the Japanese 2014 tick size reduction pilot program. A prediction formula for the effects of a tick value change on transaction costs is derived. Then, a multi-agent model is introduced to explain the relationships between market volume, price dynamics, bid-ask spread, tick value and equilibrium order book state.

In this thesis, we consider two types of applications of feedback effects in finance. These effects come into play when market participants execute sequences of trades or take part in chain reactions, which generate peaks of activity. The first part presents a dynamic optimal execution model in the presence of an exogenous stochastic market order flow. We start from the benchmark model of Obizheva and Wang, which defines an optimal execution framework with a mixed price impact. We add an order flow modeled using Hawkes processes, which are jump processes with a self-excitation property. Using stochastic control theory, we determine the optimal strategy analytically. Then we determine the conditions for the existence of Price Manipulation Strategies, as introduced by Huberman and Stanzl. These strategies can be excluded if the self-excitation of the order flow exactly offsets the price resilience. In a second step, we propose a calibration method for the model, which we apply on high frequency financial data from CAC40 stock prices. On these data, we find that the model explains a non-negligible part of the price variance. An evaluation of the optimal strategy in backtesting shows that it is profitable on average, but that realistic transaction costs are sufficient to prevent price manipulation. Then, in the second part of the thesis, we focus on the modeling of intraday volatility. In the literature, most of the backward-looking volatility models focus on the daily time scale, i.e., on day-to-day price changes. The objective here is to extend this type of approach to shorter time scales. We first present an ARCH-type model with the particularity of taking into account separately the contributions of past intraday and night-time returns. A calibration method for this model is studied, as well as a qualitative interpretation of the results on US and European stock returns. In the next chapter, we further reduce the time scale considered. We study a high-frequency volatility model, the idea of which is to generalize the Hawkes process framework to better reproduce some empirical market characteristics. In particular, by introducing quadratic feedback effects inspired by the QARCH discrete time model we obtain a power law distribution for volatility as well as time skewness.

Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high frequency finance. However, in practice, the statistical estimation results seem to show that very often, only "nearly unstable Hawkes processes" are able to fit the data properly. By nearly unstable, we mean that the L1 norm of their kernel is close to unity. We study in this work such processes for which the stability condition is almost violated. Our main result states that after suitable rescaling, they asymptotically behave like integrated Cox Ingersoll Ross models. Thus, modeling financial order flows as nearly unstable Hawkes processes may be a good way to reproduce both their high and low frequency stylized facts. We then extend this result to the Hawkes based price model introduced by Bacry et al. We show that under a similar criticality condition, this process converges to a Heston model. Again, we recover well known stylized facts of prices, both at the microstructure level and at the macroscopic scale.

We identify the distribution of a natural triplet associated with the pseudo-Brownian bridge. In particular, for $B$ a Brownian motion and $T_1$ its first hitting time of the level one, this remarkable law allows us to understand some properties of the process $(B_{uT_1}/\sqrt{T_1},~u\leq 1)$ under uniform random sampling.

No summary available.

This volume provides a broad insight on current, high level researches in probability theory.

This thesis was realized within the company Invivoo. The main objective was to find investment strategies: to have a high gain and a low risk. The research work was mainly focused on this last point. In this sense, we wanted to generalize a model fidèle to the reality of the financiers' markets, both for low and high frequency data and, at very high frequency, variation by variation.

High Frequency finance has recently evolved from statistical modeling and analysis of financial data – where the initial goal was to reproduce stylized facts and develop appropriate inference tools – toward trading optimization, where an agent seeks to execute an order (or a series of orders) in a stochastic environment that may react to the trading algorithm of the agent (market impact, invoentory). This context poses new scientific challenges addressed by the minisymposium OPSTAHF.

Recent regulatory changes, known as Reg NMS in the United States or MiFID in Europe, together with the effects of the financial crisis (mainly its impact on liquidity), induced major changes on market microstructure in two main aspects: • the fragmentation of the liquidity around several trading venues, with the appearance of newcomers in Europe like Chi-X, BATS Europe, or Turquoise, some of them being not regulated or “dark". • the rise of a new type of agents, the high frequency traders, liable for 40% to 70% of the transactions. These two effects are linked since the high frequency traders, being the main clients of the trading venues, have an implicit impact on the products offered by these venues. Combining a survey of recent academic findings and empirical evidences, this paper presents what we consider to be the key elements to understand the stakes of these changes, and also provides potential clues to mitigate some of them. A first section is dedicated to exposes the recent modifications in market microstructure. The second one explains the role of the price formation process and how, interacting with liquidity supply and demand, high frequency traders can reshape it. The next section discloses the various strategies used by these new market participants and their profitability. A final section discusses recent tools designed in order to assess and control the high frequency trading activity.

Let B be a Brownian motion and T1 its first hitting time of the level 1. For U a uniform random variable independent of B, we study in depth the distribution of B UT1/√T1, that is the rescaled Brownian motion sampled at uniform time. In particular, we show that this variable is centered.

We consider the hedging error of a derivative due to discrete trading in the presence of a drift in the dynamics of the underlying asset. We suppose that the trader wishes to find rebalancing times for the hedging portfolio which enable him to keep the discretization error small while taking advantage of market trends. Assuming that the portfolio is readjusted at high frequency, we introduce an asymptotic framework in order to derive optimal discretization strategies. More precisely, we formulate the optimization problem in terms of an asymptotic expectation-error criterion. In this setting, the optimal rebalancing times are given by the hitting times of two barriers whose values can be obtained by solving a linear-quadratic optimal control problem. In specific contexts such as in the Black-Scholes model, explicit expressions for the optimal rebalancing times can be derived.

We show that simple explicit formulas can be obtained for several relevant quantities related to the laws of the uniformly sampled Brownian bridge, Brownian meander and three dimensional Bessel process. To prove such results, we use the distribution of a triplet of random variables associated to the pseudo-Brownian bridge together with various relationships between the laws of these four processes.

At the ultra high frequency level, the notion of price of an asset is very ambiguous. Indeed, many different prices can be defined (last traded price, best bid price, mid price, etc.). Thus, in practice, market participants face the problem of choosing a price when implementing their strategies. In this work, we propose a notion of efficient price which seems relevant in practice. Furthermore, we provide a statistical methodology enabling to estimate this price from the order flow.

No summary available.

We identify the distribution of a natural triplet associated with the pseudo-Brownian bridge. In particular, for $B$ a Brownian motion and $T_1$ its first hitting time of the level one, this remarkable law allows us to understand some properties of the process $(B_{uT_1}/\sqrt{T_1},~u\leq 1)$ under uniform random sampling.

The objective of my thesis is to study mathematically a volatility breakout indicator widely used by practitioners in the trading room. The Bollinger Bands indicator belongs to the family of so-called technical analysis methods and is therefore based exclusively on the recent history of the price considered and a principle deduced from past market observations, independently of any mathematical model. My work consists in studying the performance of this indicator in a universe governed by stochastic differential equations (Black-Scholes) whose diffusion coefficient changes its value at an unknown and unobservable random time, for a practitioner wishing to maximize an objective function (for example, a certain expected utility of the portfolio value at a certain maturity). In the framework of the model, the Bollinger indicator can be interpreted as an estimator of the time of the next break. In the case of small volatilities, we show that the behavior of the density of the indicator depends on the volatility, which makes it possible to detect, for a large enough volatility ratio, the volatility regime in which the indicator's distribution is located. Also, in the case of high volatilities, we show by an approach via the Laplace transform, that the asymptotic behavior of the indicator's distribution tails depends on the volatility. This makes it possible to detect the change in the large volatilities. Then, we are interested in a comparative study between the Bollinger indicator and the classical estimator of the quadratic variation for the detection of change in volatility. Finally, we study the optimal portfolio management which is described by a non-standard stochastic problem in the sense that the admissible controls are constrained to be functionals of the observed prices. We solve this control problem by drawing on the work of Pham and Jiao to decompose the initial portfolio allocation problem into a post-breakdown management problem and a pre-breakdown problem, and each of these problems is solved by the dynamic programming method. Thus, a verification theorem is proved for this stochastic control problem.

Let B be a Brownian motion and T its first hitting time of the level 1. For U a uniform random variable independent of B, we study in depth the distribution of T^{-1/2}B_{UT}, that is the rescaled Brownian motion sampled at uniform time. In particular, we show that this variable is centered.

The aim of this thesis is to deepen the academic knowledge on the joint variations of high-frequency financial assets by analyzing them from a novel perspective. We take advantage of a tick-by-tick price database to highlight new stylistic facts about high-frequency correlation, and also to test the empirical validity of multivariate models. In Chapter 1, we discuss why high-frequency correlation is of paramount importance to trading. Furthermore, we review the empirical and theoretical literature on correlation at small time scales. Then we describe the main characteristics of the dataset we use. Finally, we state the results obtained in this thesis. In chapter 2, we propose an extension of the subordination model to the multivariate case. It is based on the definition of a global event time that aggregates the financial activity of all the assets considered. We test the ability of our model to capture notable properties of the empirical multivariate distribution of returns and observe convincing similarities. In Chapter 3, we study high-frequency lead/lag relationships using a correlation function estimator fit to tick-by-tick data. We illustrate its superiority over the standard correlation estimator in detecting the lead/lag phenomenon. We draw a parallel between lead/lag and classical liquidity measures and reveal an arbitrage to determine the optimal pairs for lead/lag trading. Finally, we evaluate the performance of a lead/lag based indicator to forecast short-term price movements. In Chapter 4, we focus on the seasonal profile of intraday correlation. We estimate this profile over four stock universes and observe striking similarities. We attempt to incorporate this stylized fact into a tick-by-tick price model based on Hawkes processes. The model thus constructed captures the empirical correlation profile quite well, despite its difficulty to reach the absolute correlation level.

This research work deals with the problem of identifying and predicting the trends of a financial series considered in a multivariate framework. The framework of this problem, inspired by machine learning, is defined in chapter I. The efficient markets hypothesis, which contradicts the objective of trend prediction, is first recalled, while the different schools of thought in market analysis, which to some extent oppose the efficient markets hypothesis, are also exposed. We explain the techniques of fundamental analysis, technical analysis and quantitative analysis, and we are particularly interested in the techniques of statistical learning allowing the calculation of predictions on time series. The difficulties of dealing with time-dependent and/or non-stationary factors are highlighted, as well as the usual pitfalls of overfitting and careless data manipulation. Extensions of the classical statistical learning framework, especially transfer learning, are presented. The main contribution of this chapter is the introduction of a research methodology allowing the development of numerical models for trend prediction. This methodology is based on an experimental protocol, consisting of four modules. The first module, entitled Data Observation and Modeling Choices, is a preliminary module devoted to the expression of modeling choices, hypotheses and very general objectives. The second module, Database Construction, transforms the target variable and explanatory variables into factors and labels in order to train numerical trend prediction models. The third module, Model Building, is aimed at building numerical trend prediction models. The fourth and final module, Backtesting and Numerical Results, evaluates the accuracy of the trend prediction models on a significant test set, using two generic backtesting procedures. The first procedure returns the recognition rates of upward and downward trends. The second procedure constructs trading rules using the predictions computed on the test set. The result (P&L) of each of the trading rules is the accumulated gains and losses during the test period. Moreover, these backtesting procedures are completed by interpretation functions, which facilitate the analysis of the decision mechanism of the numerical models. These functions can be measures of the predictive ability of the factors, or measures of the reliability of the models as well as of the delivered predictions. They contribute decisively to the formulation of hypotheses better adapted to the data, as well as to the improvement of the methods of representation and construction of databases and models. This is explained in chapter IV. The numerical models, specific to each of the model building methods described in Chapter IV, and aimed at predicting the trends of the target variables introduced in Chapter II, are indeed calculated and backtested. The reasons for switching from one model-building method to another are particularly well documented. The influence of the choice of parameters - and this at each stage of the experimental protocol - on the formulation of conclusions is also highlighted. The PPVR procedure, which does not require any additional calculation of parameters, has thus been used to reliably study the efficient markets hypothesis. New research directions for the construction of predictive models are finally proposed.

This thesis deals with several statistical finance problems and consists of four parts. In the first part, we study the question of estimating the persistence of volatility from discrete observations of a diffusion model over an interval [0,T], where T is a fixed objective time. For this purpose, we introduce a fractional Brownian motion of Hurst index H in the volatility dynamics. We construct an estimation procedure of the parameter H from the high frequency data of the diffusion. We show that the precision of our estimator is n^{-1/(4H+2)}, where n is the observation frequency and we prove its optimality in the minimax sense. These theoretical considerations are followed by a numerical study on simulated and financial data. The second part of the thesis deals with the problem of microstructure noise. For this purpose, we consider observations at frequency n$ and with rounding error a_n tending to zero, of a diffusion model on an interval [0,T], where T is a fixed objective time. In this framework, we propose estimators of the integrated volatility of the asset whose precision is shown to be max(a_n, n^{-1/2}). We also obtain central limit theorems in the case of homogeneous diffusions. This theoretical study is also followed by a numerical study on simulated and financial data. In the third part of this thesis, we establish a simple characterization of Besov spaces and we use it to prove new regularity properties for some stochastic processes. This part may seem disconnected from the problems of statistical finance but it has been inspiring for part 4 of the thesis. In the last part of the thesis, a new microstructure noise index is constructed and studied on financial data. This index, whose calculation is based on the p-variations of the considered asset at different time scales, can be interpreted in terms of Besov spaces. Compared to other indices, it seems to have several advantages. In particular, it allows to highlight original phenomena such as a certain form of additional regularity in the finest scales. It is shown that these phenomena can be partially reproduced by additive microstructure noise or diffusion models with rounding error. Nevertheless, a faithful reproduction seems to require either a combination of two forms of error or a sophisticated form of rounding error.

This thesis deals with several statistical finance problems and consists of four parts. In the first part, we study the question of estimating the persistence of volatility from discrete observations of a diffusion model over an interval [0,T], where T is a fixed objective time. For this purpose, we introduce a fractional Brownian motion of Hurst index H in the volatility dynamics. We construct an estimation procedure of the parameter H from the high frequency data of the diffusion. We show that the precision of our estimator is n^{-1/(4H+2)}, where n is the observation frequency and we prove its optimality in the minimax sense. These theoretical considerations are followed by a numerical study on simulated and financial data. The second part of the thesis deals with the problem of microstructure noise. For this, we consider observations at frequency n and with rounding error α_n tending to zero, of a diffusion model over an interval [0,T], where T is a fixed objective time. In this framework, we propose estimators of the integrated volatility of the asset whose precision is shown to be max(α_n, n^{-1/2}). We also obtain central limit theorems in the case of homogeneous diffusions. This theoretical study is also followed by a numerical study on simulated and financial data. In the third part of this thesis, we establish a simple characterization of Besov spaces and we use it to prove new regularity properties for some stochastic processes. This part may seem disconnected from the problems of statistical finance but it has been inspiring for part 4 of the thesis. In the last part of the thesis, a new microstructure noise index is constructed and studied on financial data. This index, whose calculation is based on the p-variations of the considered asset at different time scales, can be interpreted in terms of Besov spaces. Compared to other indices, it seems to have several advantages. In particular, it allows to highlight original phenomena such as a certain form of additional regularity in the finest scales. It is shown that these phenomena can be partially reproduced by additive microstructure noise or diffusion models with rounding error. Nevertheless, a faithful reproduction seems to require either a combination of two forms of error or a sophisticated form of rounding error.