Applied Mathematics

By studying all the trades and best bids/asks of ultra high frequency snapshots recorded from the order books of a basket of 10 futures assets, we bring qualitative empirical evidence that the impact of a single 715 trade depends on the intertrade time lags. We find that when the trading rate becomes faster, the return variance per trade or the impact, as measured by the price variation in the direction of the trade, strongly increases. We provide evidence that these properties persist at coarser time scales. We also show that the spread value is an increasing function of the activity. This suggests that order books are more likely empty when the trading rate is high.

We define a numerical method that provides a non-parametric estimation of the kernel shape in symmetric multivariate Hawkes processes. This method relies on second order statistical properties of Hawkes processes that relate the covariance matrix of the process to the kernel matrix. The square root of the correlation function is computed using a minimal phase recovering method. We illustrate our method on some examples and provide an empirical study of the estimation errors. Within this framework, we analyze high frequency financial price data modeled as 1D or 2D Hawkes processes. We find slowly
decaying (power-law) kernel shapes suggesting a long memory nature of self-excitation phenomena at the microstructure level of price dynamics.

In this work, we bring to light a quantity, referred to as implicit spread, playing the role of spread for large tick assets, for which the effective spread is almost always equal to one tick. The relevance of this new parameter is shown both theoretically and numerically. This implicit spread allows us to quantify the tick sizes of large tick assets and to define a notion of optimal tick size. Moreover, our results open the possibility of forecasting the behavior of relevant market quantities after a change in the tick value and to give a way to modify it in order to reach an optimal tick size.

Bibliography My greatest gratitude goes to my advisors, Emmanuel Bacry and Mark Hoffmann for their exceptional guidance. They have accepted me as a PhD student without hesitation even though I had been outside the academic world for a few years. Their support, both academic and personal was invaluable for the completion of this work. To Emmanuel, I am grateful for his constant availability, truly day or night, and for his vast knowledge and recommendations in all the aspects of my work, be it mathematics, finance or programming. To Marc, I am grateful for his hands off guidance and for allowing me to pursue interests that were not always to his best liking. "Our middle ground", as he calls it, was clearly biased in my direction. To both of you, thank you for your pleasantness, your constant motivations, understanding, and most of all for your patience at times I would not have been so kind with myself. To Jean-Francois Muzy, I extend my warmest regards and acknowledgements. Calling him my third advisor would not be an exaggeration. I thank you for your time and contribution, and also for your friendship and continuous support and motivations. Robert Almgren, my former boss at Bank of America in NY contributed significantly to my decision to pursue my PhD. I especially thank him for his guidance and advice, during my tenure with him and beyond. During my PhD, I really appreciated our numerous and lengthy conversations, and the invaluable ideas he shared with me. Thank you for being the gold standard of mentors and role models. I cannot thank Mathieu H. Rosembaum enough. One of the best things I am taking from this experience is my friendship with him. Thank you for always being 2 CONTENTS 3 there in my time of need. I am really going to miss our long and sometimes surreal conversations while we were stuck in traffic on the way to Paris. To the Financial Risks Chair of the Risk Foundation: thank you for your support and for part funding my PhD. Also, thanks to the GIS (X-ENSAE-HEC), for part funding my PhD. I am honored to have Fabrizio Lillo and Nour Meddahi as my thesis examiners. Many thanks to Jean-Philippe Bouchaud for being in my jury. Thank you all for your time and effort, and for traveling to Paris to be present at my defense. I also want to mention Charles-Albert Lehalle for introducing me to Emmanuel. Without his help, my career today would have been very different indeed. Thank you also for your numerous contributions and for your passion in market microstructure. Many thanks also for Frederique Abergel, Arthur Breitman and Carl Graham for the interesting discussions we had. These last three years at Polytechnique have been extremely enjoyable largely because working atmosphere at CMAP. I was immediately taken in as a member of the family. To Nassera, Alexandra, Sylvain, Nathalie, Wallis, Sandra: thank you for making even the most exasperating problems bearable, and for always being there when I needed you. I send my deepest recognitions to my fellow office partners, Camille, Clement, Isabelle, Emilie, Florent, Francisco, Sylvie, Soledad, and Zixian. You have made those three years extremely pleasant. Very special regards go to Julio Alexey Cen (aka Zhihao), for, well... being Julio. To Laurent Duvernet, my thesis brother, thank you for your motivations and help, and for painting such a pleasant picture of life in a PhD under Emmanuel and Marc. Your recommendations were certainly a major factor in my pursuit.

We define a numerical method that provides a non-parametric estimation of the kernel shape in symmetric multivariate Hawkes processes. This method relies on second order statistical properties of Hawkes processes that relate the covariance matrix of the process to the kernel matrix. The square root of the correlation function is computed using a minimal phase recovering method. We illustrate our method on some examples and provide an empirical study of the estimation errors. Within this framework, we analyze high frequency financial price data modeled as 1D or 2D Hawkes processes. We find slowly decaying (power-law) kernel shapes suggesting a long memory nature of self-excitation phenomena at the microstructure level of price dynamics.

Bibliography My greatest gratitude goes to my advisors, Emmanuel Bacry and Mark Hoffmann for their exceptional guidance. They have accepted me as a PhD student without hesitation even though I had been outside the academic world for a few years. Their support, both academic and personal was invaluable for the completion of this work. To Emmanuel, I am grateful for his constant availability, truly day or night, and for his vast knowledge and recommendations in all the aspects of my work, be it mathematics, finance or programming. To Marc, I am grateful for his hands off guidance and for allowing me to pursue interests that were not always to his best liking. "Our middle ground", as he calls it, was clearly biased in my direction. To both of you, thank you for your pleasantness, your constant motivations, understanding, and most of all for your patience at times I would not have been so kind with myself. To Jean-Francois Muzy, I extend my warmest regards and acknowledgements. Calling him my third advisor would not be an exaggeration. I thank you for your time and contribution, and also for your friendship and continuous support and motivations. Robert Almgren, my former boss at Bank of America in NY contributed significantly to my decision to pursue my PhD. I especially thank him for his guidance and advice, during my tenure with him and beyond. During my PhD, I really appreciated our numerous and lengthy conversations, and the invaluable ideas he shared with me. Thank you for being the gold standard of mentors and role models. I cannot thank Mathieu H. Rosembaum enough. One of the best things I am taking from this experience is my friendship with him. Thank you for always being 2 CONTENTS 3 there in my time of need. I am really going to miss our long and sometimes surreal conversations while we were stuck in traffic on the way to Paris. To the Financial Risks Chair of the Risk Foundation: thank you for your support and for part funding my PhD. Also, thanks to the GIS (X-ENSAE-HEC), for part funding my PhD. I am honored to have Fabrizio Lillo and Nour Meddahi as my thesis examiners. Many thanks to Jean-Philippe Bouchaud for being in my jury. Thank you all for your time and effort, and for traveling to Paris to be present at my defense. I also want to mention Charles-Albert Lehalle for introducing me to Emmanuel. Without his help, my career today would have been very different indeed. Thank you also for your numerous contributions and for your passion in market microstructure. Many thanks also for Frederique Abergel, Arthur Breitman and Carl Graham for the interesting discussions we had. These last three years at Polytechnique have been extremely enjoyable largely because working atmosphere at CMAP. I was immediately taken in as a member of the family. To Nassera, Alexandra, Sylvain, Nathalie, Wallis, Sandra: thank you for making even the most exasperating problems bearable, and for always being there when I needed you. I send my deepest recognitions to my fellow office partners, Camille, Clement, Isabelle, Emilie, Florent, Francisco, Sylvie, Soledad, and Zixian. You have made those three years extremely pleasant. Very special regards go to Julio Alexey Cen (aka Zhihao), for, well... being Julio. To Laurent Duvernet, my thesis brother, thank you for your motivations and help, and for painting such a pleasant picture of life in a PhD under Emmanuel and Marc. Your recommendations were certainly a major factor in my pursuit.

By studying all the trades and best bids/asks of ultra high frequency snapshots recorded from the order books of a basket of 10 futures assets, we bring qualitative empirical evidence that the impact of a single trade depends on the intertrade time lags. We find that when the trading rate becomes faster, the return variance per trade or the impact, as measured by the price variation in the direction of the trade, strongly increases. We provide evidence that these properties persist at coarser time scales. We also show that the spread value is an increasing function of the activity. This suggests that order books are more likely empty when the trading rate is high.

In this work, we provide a framework linking microstructural properties of an asset to the tick value of the exchange. In particular, we bring to light a quantity, referred to as implicit spread, playing the role of spread for large tick assets, for which the effective spread is almost always equal to one tick. The relevance of this new parameter is shown both empirically and theoretically. This implicit spread allows us to quantify the tick sizes of large tick assets and to define a notion of optimal tick size. Moreover, our results open the possibility of forecasting the behavior of relevant market quantities after a change in the tick value and to give a way to modify it in order to reach an optimal tick size. Thus, we provide a crucial tool for regulators and trading platforms in the context of high frequency trading.

In this work, we provide a framework linking microstructural properties of an asset to the tick value of the exchange. In particular, we bring to light a quantity, referred to as implicit spread, playing the role of spread for large tick assets, for which the effective spread is almost always equal to one tick. The relevance of this new parameter is shown both empirically and theoretically. This implicit spread allows us to quantify the tick sizes of large tick assets and to define a notion of optimal tick size. Moreover, our results open the possibility of forecasting the behavior of relevant market quantities after a change in the tick value and to give a way to modify it in order to reach an optimal tick size. 1 ar

We consider two inverse problems for the multi-channel two-dimensional Schrödinger equation at xed positive energy, i.e. the equation −∆ψ + V (x)ψ = Eψ at xed positive E, where V is a matrixvalued potential. The rst is the Gel'fand inverse problem on a bounded domain D at xed energy and the second is the inverse xed-energy scattering problem on the whole plane R 2 . We present in this paper two algorithms which give ecient approximate solutions to these problems: in particular, in both cases we show that the potential V is reconstructed with Lipschitz stability by these algorithms up to O(E −(m−2)/2 ) in the uniform norm as E → +∞, under the assumptions that V is m-times dierentiable in L 1 , for m ≥ 3, and has sucient boundary decay.

Je tiens à remercier tout d'abord Roman Novikov, qui a dirigé mes recherches pendant ces années de thèse. Je lui suis très reconnaissant de m'avoir aidé et encouragé à aller rapidement au cœur de problèmes de grand intérêt, comme le problème de Calderón. Travailler avec lui a été une expérience extrêmement enrichissante.