Frédéric Abergel

On some control problems in fluid mechanics

The issue of minimizing turbulence in an evolutionary Navier-Stokes flow is addressed from the point of view of optimal control. We derive theoretical results for various physical situations: distributed control, Bénard-type problems with boundary control, and flow in a channel. For each case that we consider, our results include the formulation of the problem as an optimal control problem and proof of the existence of an optimal control (which is not expected to be unique). Finally, we describe a numerical algorithm based on the gradient method for the corresponding cost function. For readers who are not interested in the mathematical details and the mathematical justifications, a nontechnical description of our results is included in Section 5.

Econophysics review: II. Agent-based models

This article is the second part of a review of recent empirical and theoretical developments usually grouped under the heading Econophysics. In the first part, we reviewed the statistical properties of financial time series, the statistics exhibited in order books and discussed some studies of correlations of asset prices and returns. This second part deals with models in Econophysics from the point of view of agent-based modeling. Of the large number of multi-agent-based models, we have identified three representative areas. First, using previous work originally presented in the fields of behavioral finance and market microstructure theory, econophysicists have developed agent-based models of order-driven markets that we discuss extensively here. Second, kinetic theory models designed to explain certain empirical facts concerning wealth distribution are reviewed. Third, we briefly summarize game theory models by reviewing the now classic minority game and related problems.

Modelling Bid and Ask Prices Using Constrained Hawkes Processes: Ergodicity and Scaling Limit ∗

We introduce a multivariate point process describing the dynamics of the Bid and Ask price of a financial asset. The point process is similar to a Hawkes process, with additional constraints on its intensity corresponding to the natural ordering of the best Bid and Ask prices. We study this process in the special case where the fertility function is exponential so that the process is entirely described by an underlying Markov chain including the constraint variable. Natural, explicit conditions on the parameters are established that ensure the ergodicity of the chain. Moreover, scaling limits are derived for the integrated point process.

Market microstructure: confronting many viewpoints

Introduction About the Editors PART I ECONOMIC MICROSTRUCTURE THEORY 1 Algorithmic Trading: Issues and Preliminary Evidence Thierry Foucault 1.1 Introduction 1.2 What is algorithmic trading? 1.2.1 Definition and typology 1.2.2 Scope and profitability 1.3 Market structure and algorithmic trading 1.4 Costs and benefits of algorithmic trading 1.4.1 Algorithmic trading reduces search costs 1.4.2 Algorithmic trading has an ambiguous effect on adverse selection costs 1.4.3 Algorithmic trading and price discovery 1.4.4 Welfare effects 1.4.5 Algorithmic trading as a source of risk 1.5 Empirical evidence 1.5.1 Algorithmic trading and market liquidity 1.5.2 Algorithmic trading and volatility 1.5.3 Algorithmic trading and price discovery 1.5.4 Algorithmic trading and market stability 1.6 Conclusions Appendix Acknowledgment References 2 Order Choice and Information in Limit Order Markets 41 Ioanid Ro u 2.1 Introduction 2.2 Order choice with symmetric information 2.3 Order choice with asymmetric information 2.4 The information content of orders 2.5 Questions for future research References PART II HIGH FREQUENCY DATA MODELING 3 Some Recent Results on High Frequency Correlation Nicolas Huth and Frederic Abergel 3.1 Introduction 3.2 Data description 3.3 Multivariate event time 3.3.1 Univariate case 3.3.2 Multivariate case 3.3.3 Empirical results 3.4 High frequency lead/lag 3.4.1 The Hayashi-Yoshida cross-correlation function 3.4.2 Empirical results 3.5 Intraday seasonality of correlation 3.5.1 Empirical results 3.6 Conclusion Acknowledgment References 4 Statistical Inference for Volatility and Related Limit Theorems Nakahiro Yoshida 4.1 Introduction 4.2 QLA for an ergodic diffusion process 4.3 QLA for volatility in the finite time-horizon 4.4 Nonsynchronous covariance estimation 4.4.1 Consistent estimator 4.4.2 Functional limit theorem 4.4.3 Application of YUIMA 4.4.4 Lead-lag estimation 4.5 YUIMA II for statistical analysis and simulation for stochastic differential equations 4.6 Higher order asymptotics and finance 4.6.1 Martingale expansion 4.6.2 Small sigma expansion Acknowledgments References PART III MARKET IMPACT 5 Models for the Impact of All Order Book Events Zoltan Eisler, Jean-Philippe Bouchaud, and Julien Kockelkoren 5.1 Introduction 5.2 A short summary of market order impact models 5.3 Many-event impact models 5.3.1 Notation and definitions 5.3.2 The transient impact model (TIM) 5.3.3 The history dependent impact model (HDIM) 5.4 Model calibration and empirical tests 5.4.1 Data 5.4.2 The case of large ticks 5.4.3 The case of small ticks 5.5 Conclusion Appendix Acknowledgments References 6 Limit Order Flow, Market Impact, and Optimal Order Sizes: Evidence from NASDAQ TotalView-ITCH Data Nikolaus Hautsch and Ruihong Huang 6.1 Introduction 6.2 Market environment and data 6.3 Major order flow and order book characteristics 6.4 An econometric model for the market impact of limit orders 6.4.1 A cointegrated VAR model for the limit order book 6.4.2 Estimating market impact 6.5 Market impact at NASDAQ 6.6 Optimal order size 6.7 Conclusions Acknowledgment References PART IV OPTIMAL TRADING Introduction: Trading and Market Micro-structure Charles-Albert Lehalle References 7 Collective Portfolio Optimization in Brokerage Data: The Role of Transaction Cost Structure Damien Challet and David Morton de Lachapelle 7.1 Introduction 7.2 Description of the data 7.3 Results 7.4 The influence of transaction costs on trading behaviour from optimal mean-variance portfolios 7.5 Discussion and outlook Acknowledgments References 8 Optimal Execution of Portfolio Transactions with Short-Term Alpha Adriana M. Criscuolo and Henri Waelbroeck 8.1 Introduction 8.2 Short-term alpha decay and hidden order arbitrage theory 8.3 Total cost definition and constraints 8.3.1 Equations without the risk term 8.3.2 Equations including risk without the alpha term 8.4 Total cost optimization 8.4.1 Results for lambda = 0 and the arbitrary alpha term 8.4.2 Risk-adjusted optimization 8.5 Conclusions 8.5.1 Main results in the absence of short-term alpha 8.5.2 Main results with short-term alpha 8.5.3 Institutional trading practices Proviso References Combined References Index

LONG TIME BEHAVIOUR OF A HAWKES PROCESS-BASED LIMIT ORDER BOOK

This article presents a Hawkes process-based limit order book model with a particular interest in the long-time bahaviour. A general approach based on ergodic theory and martingale convergence theorems is used.

The Times Change: Multivariate Subordination, Empirical Facts

The normality of multi-asset returns in event time is shown empirically. A multivariate subordination mechanism is proposed in order to explain this phenomenon.

Econophysics of agent-based models

Jan Tinbergen, the first recipient of the Nobel Memorial Prize in Economics in 1969, obtained his PhD in physics at the University of Leiden under the supervision of Paul Ehrenfest in 1929. Among many achievements as an economist after his training as a physicist, Tinbergen proposed the so-called Gravity Model of international trade. The model predicts that the intensity of trade between two countries is described by a formula similar to Newton’s law of gravitation, where mass is replaced by Gross Domestic Product. Since Tinbergen’s proposal, the Gravity Model has become the standard model of non-zero trade flows in macroeconomics. However, its intrinsic limitation is the prediction of a completely connected network, which fails to explain the observed intricate topology of international trade. Recent network models overcome this limitation by describing the real network as a member of a maximum-entropy statistical ensemble. The resulting expressions are formally analogous to quantum statistics: the international trade network is found to closely follow the Fermi-Dirac statistics in its purely binary topology, and the recently proposed mixed Bose-Fermi statistics in its full (binary plus weighted) structure. This seemingly esoteric result is actually a simple effect of the heterogeneity of world countries, that imposes strong structural constraints on the network. Our discussion highlights similarities and differences between macroeconomics and statistical-physics approaches to economic networks. Tiziano Squartini Instituut-Lorentz for Theoretical Physics, Leiden Institute of Physics, University of Leiden, Niels Bohrweg 2, 2333 CA Leiden, e-mail: squartini@lorentz.leidenuniv.nl Diego Garlaschelli Instituut-Lorentz for Theoretical Physics, Leiden Institute of Physics, University of Leiden, Niels Bohrweg 2, 2333 CA Leiden, e-mail: garlaschelli@lorentz.leidenuniv.nl 1 ar X iv :1 30 4. 32 52 v1 [ qfi n. G N ] 1 1 A pr 2 01 3 2 Tiziano Squartini and Diego GarlaschelliThis article is the second part of a review of recent empirical and theoretical developments usually grouped under the heading Econophysics. In the first part, we reviewed the statistical properties of financial time series, the statistics exhibited in order books and discussed some studies of correlations of asset prices and returns. This second part deals with models in Econophysics from the point of view of agent-based modeling. Of the large number of multiagent- based models, we have identified three representative areas. First, using previous work originally presented in the fields of behavioral finance and market microstructure theory, econophysicists have developed agent-based models of order-driven markets that we discuss extensively here. Second, kinetic theory models designed to explain certain empirical facts concerning wealth distribution are reviewed. Third, we briefly summarize game theory models by reviewing the now classic minority game and related problems.

Econophysics review: I. Empirical facts

This article and the companion paper aim at reviewing recent empirical and theoretical developments usually grouped under the term Econophysics. Since the name was coined in 1995 by merging the words ‘Economics’ and ‘Physics’, this new interdisciplinary field has grown in various directions: theoretical macroeconomics (wealth distribution), microstructure of financial markets (order book modeling), econometrics of financial bubbles and crashes, etc. We discuss the interactions between Physics, Mathematics, Economics and Finance that led to the emergence of Econophysics. We then present empirical studies revealing the statistical properties of financial time series. We begin the presentation with the widely acknowledged ‘stylized facts’, which describe the returns of financial assets—fat tails, volatility clustering, autocorrelation, etc.—and recall that some of these properties are directly linked to the way ‘time’ is taken into account. We continue with the statistical properties observed on order books in financial markets. For the sake of illustrating this review, (nearly) all the stated facts are reproduced using our own high-frequency financial database. Finally, contributions to the study of correlations of assets such as random matrix theory and graph theory are presented. The companion paper will review models in Econophysics from the point of view of agent-based modeling.

Trade-Throughs: Empirical Facts - Application to Lead-Lag Measures

Order splitting is a well-known behavior in trading: traders constantly scan the limit order book and choose to limit the size of their orders to the quantity available at the best limit. Order splitting allows traders not to reveal their intention to the market so as not to move too much the price against them. In this note, we focus on the other trades, called trade-throughs, which are trades that go through the best available price in the order book. We provide various statistics on trade-throughs: their liquidity, their intraday distribution and the spread relaxation that follows them. We also present a new method to get empirical distributions of lead-lag parameters between assets, sectors or even markets. This empirical study is based on tick-by-tick data of major EU and US equity futures from TRTH (Thomson Reuters Tick History) database.

Nonquadratic Local Risk-Minimization for Hedging Contingent Claims in Incomplete Markets

We introduce a new criterion to perform hedging of contingent claims in incomplete markets. Our approach is close to the one proposed by Schweizer [Stochastic Process. Appl., 37 (1991), pp. 339-363] in that it uses the concept of locally risk-minimizing strategies. But we aim at being more general by defining the local risk as a general, nonnecessarily quadratic, convex function of the local cost process. We derive the corresponding optimal strategies and value function in both discrete and continuous time settings. Finally we give an application of our hedging method in the stochastic volatility case as well as in the jump diffusion case. We work with a single traded asset, but our approach may be generalized to deal with claims depending on multiple assets.