The main contribution of the present thesis is in two aspects. The first one, which is the heart of the thesis, is to explore the fundamental relation between rough paths and their signatures. Our main goal is to give a geometric characterization of the kernel of the signature map in different situations. In Chapter Two, we start by establishing a general fact that a continuous Jordan curve on a Riemannian manifold can be arbitrarily well approximated by piecewise minimizing geodesic interpolations which are again Jordan. This result enables us to prove a generalized version of Green’s theorem for planar Jordan curves with finite p-variation 1 &le; p < 2, and to prove that two such Jordan curves have the same signature if and only if they are equal up to reparametrization. In Chapter Three, we investigate the problem for general weakly geometric rough paths. In particular, we show that a weakly geometric rough path has trivial signature if and only if it is tree-like in the sense we will define later on. In Chapter Four, we study the problem in the probabilistic setting. In particular, we show that for a class of stochastic processes, with probability one the sample paths are determined by their signatures up to reparametrization. A fundamental example is Gaussian processes including fractional Brownian motion with Hurst parameter H > 1/4, the Ornstein-Uhlenbeck process and the Brownian bridge. The second one is an application of rough path theory to the study of nonlinear diffusions on manifolds under the framework of nonlinear expectations. In Chapter Five, we begin by studying the geometric rough path nature of G-Brownian motion. This enables us to introduce rough differential equations driven by G-Brownian motion from a pathwise point of view. Next we establish the fundamental relation between rough (pathwise theory) and stochastic (L²-theory) differential equations driven by G-Brownian motion. This is a crucial point of understanding nonlinear diffusions and their generating heat flows on manifolds from an intrinsic point of view. Finally, from the pathwise point of view we construct G-Brownian motion on a compact Riemannian manifold and establish its generating heat flow for a class of G-functions under orthogonal invariance. As an independent interest, we also develop the Euler-Maruyama scheme for stochastic differential equations driven by G-Brownian motion.

Tesis para optar al grado de Doctor en Ciencias de la Ingeniería, Mención Modelación Matemática / Las Ecuaciones Estocásticas en Derivadas Parciales (SPDEs por su sigla en inglés) son una herramienta esencial para el análisis de los límites de escalamiento de diversos modelos microscópicos provenientes de otras áreas de las ciencias tales como la física y la química. Este tipo de ecuaciones corresponde a una ecuación en derivadas parciales clásica a la cual se le ha agregado un término de forzamiento externo aleatorio el que suele ser muy irregular; el ejemplo más sencillo es tal vez la Ecuación del Calor Estocástica, de la cual una de sus versiones es estudiada en la presente tesis. En cualquier caso, la irregularidad de este potencial hace que el análisis de las soluciones de estos problemas sea mucho más complicado. En efecto, hay casos en que dichas soluciones sólo pueden ser entendidas en el sentido de las distribuciones. Hay casos más críticos como la ecuación de Kardar--Parisi--Zhang (KPZ) en en una dimensión espacial donde, si bien se puede probar que posee soluciones Hölder, estas no son lo suficientemente regulares para permitir definir uno de los términos no lineales que aparecen en ella. Durante los últimos 20 años se han desarrollado varias técnicas para el análsis de este tipo de ecuaciones, entre las que destacan la teoría de rough paths geométricos de T. Lyons (1998), los rough paths ramificadosde M. Gubinelli (2010), y la más reciente teoría de estructuras de regularidad de M. Hairer (2014) por la que este último obtuvo la medalla Fields en 2014. Aunque diferentes, todas estas técnicas tienen como idea central el concepto de renormalización. En particular, la renormalización de Wick juega un rol esencial en la renormalización en el marco de las estructuras de regularidad. En este trabajo se desarrollan los productos y polinomios de Wick desde un punto de vista algebraico inspirado en el cálculo umbral de G.-C. Rota. También se explora la teoría general de losrough paths en general y su versión ramificada en particular, probándose nuevos resultados en la dirección de incorporar un análogo de la renormalización de Wick existente en las estructuras de regularidad. Por último, se estudia el modelo de polímero semidiscreto multicapas introducido por I. Corwin and A. Hammond (2014) para el cual se prueba la convergencia de su función de partición hacia la "solución" de la Ecuación del Calor Estocástica multicapas definida por N. O'Connell y J. Warren (2011) algunos años antes. Cabe destacar que al momento de redacción de esta tesis no existen resultados que permitan interpretar este proceso en el continuo como la solución de una SPDE singular como en el caso de la ecuación de KPZ, lo que ha sido una de las principales fuentes de inspiración para este trabajo. / CONICYT/Doctorado Nacional/2013-21130733 CMM - Conicyt PIA AFB170001

We construct a mathematical model of an order driven market where traders can submit limit orders and market orders to buy and sell securities. We adapt the notion of no free lunch of Harrison and Kreps and Jouini and Kallal to our setting and we prove a no-arbitrage theorem for the model of the order driven market. Furthermore, we compute signatures of order books of different financial markets. Signatures, i.e. the full sequence of definite iterated integrals of a path, are one of the fundamental elements of the theory of rough paths. The theory of rough paths provides a framework to describe the evolution of dynamical systems that are driven by rough signals, including rough paths based on Brownian motion and fractional Brownian motion (see the work of Lyons). We show how we can obtain the solution of a polynomial differential equation and its (truncated) signature from the signature of the driving signal and the initial value. We also present and analyse an ODE method for the numerical solution of rough differential equations. We derive error estimates and we prove that it achieves the same rate of convergence as the corresponding higher order Euler schemes studied by Davie and Friz and Victoir. At the same time, it enhances stability. The method has been implemented for the case of polynomial vector fields as part of the CoRoPa software package which is available at http://coropa.sourceforge.net. We describe both the algorithm and the implementation and we show by giving examples how it can be used to compute the pathwise solution of stochastic rough differential equations driven by Brownian rough paths and fractional Brownian rough paths.

Cette thèse est composée de trois chapitres indépendants ayant pour thématique commune la théorie des trajectoires rugueuses. Introduite en 1998 par Terry Lyons, cette approche trajectorielle des équations différentielles stochastiques (EDS) permet l'étude d'EDS dirigées par des processus n'ayant pas la propriété de semi-martingale nécessaire à l'application du cadre de l'intégration d'Itô. C'est par exemple le cas du mouvement brownien fractionnaire pour un indice de Hurst différent d'un demi. Le premier chapitre porte sur les liens entre la théorie des trajectoires rugueuses et celle des structures de régularité qui a été récemment introduite par Martin Hairer pour résoudre une large classe d'équations aux dérivées partielles stochastiques. Nous exposons, avec les outils de cette nouvelle théorie, la définition de l'intégrale rugueuse et de la signature d'une trajectoire irrégulière, ce qui nous mène à la résolution d'équations différentielles rugueuses (EDR). Dans le second chapitre, nous nous intéressons à la construction de flots d'EDR à partir de leurs approximations en temps petit, appelées presque flots. Nous montrons que sous des conditions faibles de régularité du presque flot, bien que l'unicité des solutions de l'EDR associée ne soit plus assurée, il est possible de sélectionner un flot mesurable. Notre cadre général unifie les précédentes approches par flot dues à I. Bailleul, A. M. Davie, P. Friz et N. Victoir. Le dernier chapitre s'attache à l'étude d'une inclusion différentielle perturbée par une trajectoire rugueuse, c'est-à-dire d'une EDR dont la dérive est une fonction multivaluée. Nous démontrons, sans hypothèse de convexité et avec différentes conditions de régularité sur la dérive, l'existence de solution. / This thesis consists of three independent chapters in the theme of rough path theory. Introduced in 1998 by Terry Lyons, this pathwise approach to stochastic differential equations (SDE) allows one to study SDE driven by processes that do not have the semi-martingale property which is required to apply the framework of the Itô integral. This is for example the case of the fractional Brownian motion for a Hurst index different from one-half. The first chapter deals with the links between rough path and regularity structure theories. The latter was recently introduced by Martin Hairer to solve a large class of stochastic partial differential equations. With the tools of this new theory, we show how to build the rough integral and the signature of an irregular path, which leads to solve a rough differential equation (RDE). In the second chapter, we focus on building RDE flows from their approximations at small scale, called almost flows. We show that under weak conditions on regularity of almost flows, although the uniqueness of the associated RDE solutions does not hold, we are able to select a measurable flow. Our general framework unifies the previous approaches by flow due to I. Bailleul, A. M. Davie, P. Friz and N. Victoir. In the last chapter, we study of a differential inclusion perturbed by a rough path, i.e. a RDE whose drift is a multivalued function. We prove, without convexity hypothesis and several conditions on the regularity of the drift, the existence of a solution.

Le présent travail se veut une contribution à l’extension du domaine des applications de la théorie des chemins rugueux à travers l’étude de la convergence des processus discrets, qui permet un nouveau regard sur plusieurs problèmes qui se posent dans le cadre du calcul stochastique classique. Nous étudions la convergence en topologie rugueuse, d’abord des chaînes de Markov sur graphes périodiques, ensuite des marches de Markov cachées, et ce changement de cadre permet d’apporter des informations supplémentaires sur la limite grâce à l’anomalie d’aire, invisible en topologie uniforme. Nous voulons montrer que l’utilité de cet objet dépasse le cadre des équations différentielles. Nous montrons également comment le cadre des chemins rugueux permet d’en- coder la manière dont on plonge un modèle discret dans l’espace des fonctions continues, et que les limites des différents plongements peuvent être différenciées précisément grâce à l’anomalie d’aire. Nous définissons ensuite les temps d’occupation itérés pour une chaîne de Markov et montrons, en utilisant les sommes itérées, qu’ils donnent une structure combinatoire aux marches de Markov cachées. Nous proposons une construction des chemins rugueux en passant par les sommes itérées et la comparons à la construction classique, faite par les intégrales itérées, pour trouver à la limite deux types de chemins rugueux différents, non-géométrique et géométrique respectivement. Pour finir, nous illustrons le calcul et la construction de l’anomalie d’aire et nous donnons quelques résultats supplémentaires sur la convergence des sommes et temps d’occupation itérés. / Through the present work, we hope to contribute to extending the domain of applications of rough paths theory by studying the convergence of discrete processes and thus allowing for a new point of view on several issues appearing in the setting of classical stochastic calculus. We study the convergence, first of Markov chains on periodic graphs, then of hidden Markov walks, in rough path topology, and we show that this change of setting allows to bring forward extra information on the limit using the area anomaly, which is invisible in the uniform topology. We want to show that the utility of this object goes beyond the setting of dierential equations. We also show how rough paths can be used to encode the way we embed a discrete process in the space of continuous functions, and that the limits of these embeddings dier precisely by the area anomaly term. We then define the iterated occupation times for a Markov chain and show using iterated sums that they form an underlying combinatorial structure for hidden Markov walks. We then construct rough paths using iterated sums and compare them to the classical construction, which uses iterated integrals, to get two dierent types of rough paths at the limit: the non-geometric and the geometric one respectively. Finally, we illustrate the computation and construction of the area anomaly and we give some extra results on the convergence of iterated sums and occupation times.

The main motivation behind writing this thesis was to construct numerical methods to approximate solutions to differential equations driven by rough paths, where the solution is considered in the rough path-sense. Rough paths of inhomogeneous degree of smoothness as driving noise are considered. We also aimed to find applications of these numerical methods to stochastic differential equations. After sketching the core ideas of the Rough Paths Theory in Chapter 1, the versions of the core theorems corresponding to the inhomogeneous degree of smoothness case are stated and proved in Chapter 2 along with some auxiliary claims on the continuity of the solution in a certain sense, including an RDE-version of Gronwall's lemma. In Chapter 3, numerical schemes for approximating solutions to differential equations driven by rough paths of inhomogeneous degree of smoothness are constructed. We start with setting up some principles of approximations. Then a general class of local approximations is introduced. This class is used to construct global approximations by pasting together the local ones. A general sufficient condition on the local approximations implying global convergence is given and proved. The next step is to construct particular local approximations in finite dimensions based on solutions to ordinary differential equations derived locally and satisfying the sufficient condition for global convergence. These local approximations require strong conditions on the one-form defining the rough differential equation. Finally, we show that when the local ODE-based schemes are applied in combination with rough polynomial approximations, the conditions on the one-form can be weakened. In Chapter 4, the results of Gyurko & Lyons (2010) on path-wise approximation of solutions to stochastic differential equations are recalled and extended to the truncated signature level of the solution. Furthermore, some practical considerations related to the implementation of high order schemes are described. The effectiveness of the derived schemes is demonstrated on numerical examples. In Chapter 5, the background theory of the Kusuoka-Lyons-Victoir (KLV) family of weak approximations is recalled and linked to the results of Chapter 4. We highlight how the different versions of the KLV family are related. Finally, a numerical evaluation of the autonomous ODE-based versions of the family is carried out, focusing on SDEs in dimensions up to 4, using cubature formulas of different degrees and several high order numerical ODE solvers. We demonstrate the effectiveness and the occasional non-effectiveness of the numerical approximations in cases when the KLV family is used in its original version and also when used in combination with partial sampling methods (Monte-Carlo, TBBA) and Romberg extrapolation.