The Reflected Quasipotential V(x) is the solution to a variational problem that arises in the study of reflective Brownian motion. Specifically, the stationary distributions of reflected Brownian motion satisfy a large deviation principle (with respect to a spatial scaling parameter) with V(x) as the rate function. The Skorokhod Problem is an essential device in the construction and analysis of reflected Brownian motion and our value function V(x). Here we characterize V(x) as a solution to a partial differential equation H(DV(x))=0 in the positive n-dimensional orthant with appropriate boundary conditions.  H(p) is the Hamiltonian and DV(x) is the gradient of V(x). V(x) is continuous but not differentiable in general. The characterization  will need to be in terms of viscosity solutions. Solutions are not unique, thus additional qualifications will be needed for uniqueness. In order to prove our uniqueness result we consider a discounted version of V(x) in a truncated region and pass to the limit. In addition to this characterization of V(x) we explore the possibility of cyclic optimal paths in 3 dimensions. / Ph. D.

Cette thèse présente trois principaux sujets de recherche, les deux premiers étant indépendants et le dernier indiquant la relation des deux premières problématiques dans un cas concret.Dans la première partie nous nous intéressons au problème de transport optimal martingale dans l’espace de Skorokhod, dont le premier but est d’étudier systématiquement la tension des plans de transport martingale. On s’intéresse tout d’abord à la semicontinuité supérieure du problème primal par rapport aux distributions marginales. En utilisant la S-topologie introduite par Jakubowski, on dérive la semicontinuité supérieure et on montre la première dualité. Nous donnons en outre deux problèmes duaux concernant la surcouverture robuste d’une option exotique, et nous établissons les dualités correspondantes, en adaptant le principe de la programmation dynamique et l’argument de discrétisation initie par Dolinsky et Soner.La deuxième partie de cette thèse traite le problème du plongement de Skorokhod optimal. On formule tout d’abord ce problème d’optimisation en termes de mesures de probabilité sur un espace élargi et ses problèmes duaux. En utilisant l’approche classique de la dualité; convexe et la théorie d’arrêt optimal, nous obtenons les résultats de dualité. Nous rapportons aussi ces résultats au transport optimal martingale dans l’espace des fonctions continues, d’où les dualités correspondantes sont dérivées pour une classe particulière de fonctions de paiement. Ensuite, on fournit une preuve alternative du principe de monotonie établi par Beiglbock, Cox et Huesmann, qui permet de caractériser les optimiseurs par leur support géométrique. Nous montrons à la fin un résultat de stabilité qui contient deux parties: la stabilité du problème d’optimisation par rapport aux marginales cibles et le lien avec un autre problème du plongement optimal.La dernière partie concerne l’application de contrôle stochastique au transport optimal martingale avec la fonction de paiement dépendant du temps local, et au plongement de Skorokhod. Pour le cas d’une marginale, nous retrouvons les optimiseurs pour les problèmes primaux et duaux via les solutions de Vallois, et montrons en conséquence l’optimalité des solutions de Vallois, ce qui regroupe le transport optimal martingale et le plongement de Skorokhod optimal. Quand au cas de deux marginales, on obtient une généralisation de la solution de Vallois. Enfin, un cas spécial de plusieurs marginales est étudié, où les temps d’arrêt donnés par Vallois sont bien ordonnés. / This PhD dissertation presents three research topics, the first two being independent and the last one relating the first two issues in a concrete case.In the first part we focus on the martingale optimal transport problem on the Skorokhod space, which aims at studying systematically the tightness of martingale transport plans. Using the S-topology introduced by Jakubowski, we obtain the desired tightness which yields the upper semicontinuity of the primal problem with respect to the marginal distributions, and further the first duality. Then, we provide also two dual formulations that are related to the robust superhedging in financial mathematics, and we establish the corresponding dualities by adapting the dynamic programming principle and the discretization argument initiated by Dolinsky and Soner.The second part of this dissertation addresses the optimal Skorokhod embedding problem under finitely-many marginal constraints. We formulate first this optimization problem by means of probability measures on an enlarged space as well as its dual problems. Using the classical convex duality approach together with the optimal stopping theory, we obtain the duality results. We also relate these results to the martingale optimal transport on the space of continuous functions, where the corresponding dualities are derived for a special class of reward functions. Next, We provide an alternative proof of the monotonicity principle established in Beiglbock, Cox and Huesmann, which characterizes the optimizers by their geometric support. Finally, we show a stability result that is twofold: the stability of the optimization problem with respect to target marginals and the relation with another optimal embedding problem.The last part concerns the application of stochastic control to the martingale optimal transport with a payoff depending on the local time, and the Skorokhod embedding problem. For the one-marginal case, we recover the optimizers for both primal and dual problems through Vallois' solutions, and show further the optimality of Vallois' solutions, which relates the martingale optimal transport and the optimal Skorokhod embedding. As for the two-marginal case, we obtain a generalization of Vallois' solution. Finally, a special multi-marginal case is studied, where the stopping times given by Vallois are well ordered.

The robust pricing and hedging approach in Mathematical Finance, pioneered by Hobson (1998), makes statements about non-traded derivative contracts by imposing very little assumptions about the underlying financial model but directly using information contained in traded options, typically call or put option prices. These prices are informative about marginal distributions of the asset. Mathematically, the theory of Skorokhod embeddings provides one possibility to approach robust problems. In this thesis we consider mostly robust pricing and hedging problems of Lookback options (options written on the terminal maximum of an asset) and Convex Vanilla Options (options written on the terminal value of an asset) and extend the analysis which is predominately found in the literature on robust problems by two features: Firstly, options with multiple maturities are available for trading (mathematically this corresponds to multiple marginal constraints) and secondly, restrictions on the total realized variance of asset trajectories are imposed. Probabilistically, in both cases, we develop new optimal solutions to the Skorokhod embedding problem. More precisely, in Part I we start by constructing an iterated Azema-Yor type embedding (a solution to the n-marginal Skorokhod embedding problem, see Chapter 2). Subsequently, its implications are presented in Chapter 3. From a Mathematical Finance perspective we obtain explicitly the optimal superhedging strategy for Barrier/Lookback options. From a probability theory perspective, we find the maximum maximum of a martingale which is constrained by finitely many intermediate marginal laws. Further, as a by-product, we discover a new class of martingale inequalities for the terminal maximum of a cadlag submartingale, see Chapter 4. These inequalities enable us to re-derive the sharp versions of Doob's inequalities. In Chapter 5 a different problem is solved. Motivated by the fact that in some markets both Vanilla and Barrier options with multiple maturities are traded, we characterize the set of market models in this case. In Part II we incorporate the restriction that the total realized variance of every asset trajectory is bounded by a constant. This has been previously suggested by Mykland (2000). We further assume that finitely many put options with one fixed maturity are traded. After introducing the general framework in Chapter 6, we analyse the associated robust pricing and hedging problem for convex Vanilla and Lookback options in Chapters 7 and 8. Robust pricing is achieved through construction of appropriate Root solutions to the Skorokhod embedding problem. Robust hedging and pathwise duality are obtained by a careful development of dynamic pathwise superhedging strategies. Further, we characterize existence of market models with a suitable notion of arbitrage.

Cette thèse porte sur les processus croissants pour l'ordre convexe que nous désignons sous le nom de peacocks. Un résultat remarquable dû à Kellerer stipule qu'un processus stochastique à valeurs réelles est un peacock si et seulement s'il possède les mêmes marginales unidimensionnelles qu'une martingale. Une telle martingale est dite associée à ce processus. Mais dans son article, Kellerer ne donne ni d'exemple de peacock, ni d'idée précise sur la construction d'une martingale associée pour un peacock donné. Ainsi, comme d'autres travaux sur les peacocks, notre étude vise deux objectifs. Il s'agit d'exhiber de nouvelles familles de peacocks et de construire des martingales associées pour certains peacocks. Dans les trois premiers chapitres, nous exhibons diverses classes de peacocks en utilisant successivement les notions de monotonie conditionnelle, de peacock très fort et de positivité totale d'ordre 2. En particulier, nous fournissons plusieurs extensions du résultat de Carr-Ewald-Xiao selon lequel la moyenne arithmétique du mouvement brownien géométrique, encore appelée "option asiatique" est un peacock. L'objet du dernier chapitre est de construire des martingales associées pour une classe de peacocks. Pour cela, nous utilisons les plongements d'Azéma-Yor et de Bertoin-Le Jan. L'originalité de ce chapitre est l'utilisation de la positivité totale d'ordre 2 dans l'étude du plongement d'Azéma-Yor / This thesis deals with real valued stochastic processes which increase in the convex order. We call them peacocks. A remarkable result due to Kellerer states that a real valued process is a peacock if and only if it has the same one-dimensional marginals as a martingale. Such a martingale is said to be associated to this process. But in his article, Kellerer provides neither an example of peacock nor a concrete idea to construct an associated martingale to a given peacock. Hence, as other investigations on peacocks, our study has two purposes. We first exhibit new families of peacocks and then, we contruct associated martingales to certain of them. In the first three chapters, we exhibit several classes of peacocks using successively the notions of conditional monotonicity, very strong peacock and total positivity of order 2. In particular, we provide many extensions of Carr-Ewald-Xiao result which states that the arithmetic mean of geometric Brownian motion, also called "Asian option" is a peacock. The purpose of the last chapter is to construct associated martingales to certain peacocks. To this end, we use Azéma-Yor and Bertoin-Le Jan embedding algorithms. The originality of this chapter is the use of total positivity of order 2 in the study of Azéma-Yor embedding algorithm

The standard approach to pricing financial derivatives is to determine the discounted, risk-neutral expected payoff under a model. This model-based approach leaves us prone to model risk, as no model can fully capture the complex behaviour of asset prices in the real world. Alternatively, we could use the prices of some liquidly traded options to deduce no-arbitrage conditions on the contingent claim in question. Since the reference prices are taken from the market, we are not required to postulate a model and thus the conditions found have to hold under any model. In this thesis we are interested in the pricing of American put options using the latter approach. To this end, we will assume that European options on the same underlying and with the same maturity are liquidly traded in the market. We can then use the market information incorporated into these prices to derive a set of no-arbitrage conditions that are valid under any model. Furthermore, we will show that in a market trading only finitely many American and co-terminal European options it is always possible to decide whether the prices are consistent with a model or there has to exist arbitrage in the market.

We introduce the notion of coupling distances on the space of Lévy measures in order to quantify rates of convergence towards a limiting Lévy jump diffusion in terms of its characteristic triplet, in particular in terms of the tail of the Lévy measure. The main result yields an estimate of the Wasserstein-Kantorovich-Rubinstein distance on path space between two Lévy diffusions in terms of the couping distances. We want to apply this to obtain precise rates of convergence for Markov chain approximations and a statistical goodness-of-fit test for low-dimensional conceptual climate models with paleoclimatic data.

The purpose of this thesis is to develop techniques for analysing interacting particle systems on the half-line. When the number of particles becomes large, stochastic partial differential equations (SPDEs) with Dirichlet boundary conditions will be the natural objects for describing the dynamics of the population's empirical measure. As a source of motivation, we consider systems that arise naturally as models for the pricing of portfolio credit derivatives, although similar applications are found in mathematical neuroscience, stochastic filtering and mean-field games. We will focus on a stochastic McKean--Vlasov system in which a collection of Brownian motions interact through a correlation which is a function of the proportion of particles that have been absorbed at level zero. We prove a law of large numbers where the limiting object is the unique solution to (the weak formulation of) the loss-dependent SPDE: dV_t(x) = 1/2 &part;_xxV_t(x)dt - p(L_t)&part;_xV_t(x)dW_t, V_t(0)=0, where L_t = 1-&lmoust;^&infin;<sub style='position: relative; left: -.8em;'>t</sub></sup>V_t(x)dx, V is a density process on the half-line and W is a Brownian motion. The correlation function is assumed to be piecewise Lipschitz, which encompasses a natural class of credit models. The first of our theoretical developments is to introduce the kernel smoothing method in the dual of the first Sobolev space, H^-1, with the aim of proving uniqueness results for SPDEs. A benefit of this approach is that only first order moment estimates of solutions are required, and in the particle setting this translates into studying the particles at an individual level rather than as a correlated collection. The second idea is to extend Skorokhod's M₁ topology to the space of processes that take values in the tempered distributions. The benefit we gain is that monotone functions have zero modulus of continuity under this topology, so the loss process, L, is easy to control. As a final example, we consider the fluctuations in the convergence of a basic particle system with constant correlation. This gives rise to a central limit theorem, for which the limiting object is a solution to an SPDE with random transport and an additive idiosyncratic driver acting on the first derivative terms. Conditional on the systemic random variables, this driver is a space-time white noise with intensity controlled by the empirical measure of the underlying system. The SPDE has insufficient regularity for us to work in any Sobolev space higher than H^-1, hence we have an example of where our extension to the kernel smoothing method is necessary.

The study of stochastic queueing networks is quite important due to the many applications including transportation, telecommunication, and manufacturing industries. Since there is often no explicit solution to these types of control problems, numerical methods are needed. Following the method of Boué-Dupuis, we use a Dynamic Programming approach of optimization on a controlled Markov Chain that simulates the behavior of a fluid limit of the original process. The search for an optimal control in this case involves a Skorokhod problem to describe the dynamics on the boundary of closed, convex domain. Using relaxed stochastic controls we show that the approximating numerical solution converges to the actual solution as the size of the mesh in the discretized state space goes to zero, and illustrate with an example. / Ph. D.

Diese Dissertation präsentiert neue Techniken der Integration für verschiedene Probleme der Finanzmathematik und einige Anwendungen in der Wahrscheinlichkeitstheorie. Zu Beginn entwickeln wir zwei Zugänge zur robusten stochastischen Integration. Der erste, ähnlich der Ito’schen Integration, basiert auf einer Topologie, erzeugt durch ein äußeres Maß, gegeben durch einen minimalen Superreplikationspreis. Der zweite gründet auf der Integrationtheorie für rauhe Pfade. Wir zeigen, dass das entsprechende Integral als Grenzwert von nicht antizipierenden Riemannsummen existiert und dass sich jedem "typischen Preispfad" ein rauher Pfad im Ito’schen Sinne zuordnen lässt. Für eindimensionale "typische Preispfade" wird sogar gezeigt, dass sie Hölder-stetige Lokalzeiten besitzen. Zudem erhalten wir Verallgemeinerungen von Föllmer’s pfadweiser Ito-Formel. Die Integrationstheorie für rauhe Pfade kann mit dem Konzept der kontrollierten Pfade und einer Topologie, welche die Information der Levy-Fläche enthält, entwickelt werden. Deshalb untersuchen wir hinreichende Bedingungen an die Kontrollstruktur für die Existenz der Levy-Fläche. Dies führt uns zur Untersuchung von Föllmer’s Ito-Formel aus der Sicht kontrollierter Pfade. Para-kontrollierte Distributionen, kürzlich von Gubinelli, Imkeller und Perkowski eingeführt, erweitern die Theorie rauher Pfade auf den Bereich von mehr-dimensionale Parameter. Wir verallgemeinern diesen Ansatz von Hölder’schen auf Besov-Räume, um rauhe Differentialgleichungen zu lösen, und wenden die Ergebnisse auf stochastische Differentialgleichungen an. Zum Schluß betrachten wir stark gekoppelte Systeme von stochastischen Vorwärts-Rückwärts-Differentialgleichungen (FBSDEs) und erweitern die Theorie der Existenz, Eindeutigkeit und Regularität der sogenannten Entkopplungsfelder auf Markovsche FBSDEs mit lokal Lipschitz-stetigen Koeffizienten. Als Anwendung wird das Skorokhodsche Einbettungsproblem für Gaußsche Prozesse mit nichtlinearem Drift gelöst. / In this thesis new robust integration techniques, which are suitable for various problems from stochastic analysis and mathematical finance, as well as some applications are presented. We begin with two different approaches to stochastic integration in robust financial mathematics. The first one is inspired by Ito’s integration and based on a certain topology induced by an outer measure corresponding to a minimal superhedging price. The second approach relies on the controlled rough path integral. We prove that this integral is the limit of non-anticipating Riemann sums and that every "typical price path" has an associated Ito rough path. For one-dimensional "typical price paths" it is further shown that they possess Hölder continuous local times. Additionally, we provide various generalizations of Föllmer’s pathwise Ito formula. Recalling that rough path theory can be developed using the concept of controlled paths and with a topology including the information of Levy’s area, sufficient conditions for the pathwise existence of Levy’s area are provided in terms of being controlled. This leads us to study Föllmer’s pathwise Ito formulas from the perspective of controlled paths. A multi-parameter extension to rough path theory is the paracontrolled distribution approach, recently introduced by Gubinelli, Imkeller and Perkowski. We generalize their approach from Hölder spaces to Besov spaces to solve rough differential equations. As an application we deal with stochastic differential equations driven by random functions. Finally, considering strongly coupled systems of forward and backward stochastic differential equations (FBSDEs), we extend the existence, uniqueness and regularity theory of so-called decoupling fields to Markovian FBSDEs with locally Lipschitz continuous coefficients. These results allow to solve the Skorokhod embedding problem for a class of Gaussian processes with non-linear drift.

Diese Arbeit untersucht eine Reihe multiplikativer Preiseinflussmodelle für das Handeln in einer riskanten Anlage. Unser risikoneutraler Investor versucht seine zu erwartenden Handelserlöse zu maximieren. Zunächst modellieren wir den vorübergehende Preiseinfluss als deterministisches Funktional der Handelsstrategie. Wir stellen den Zusammenhang mit Limit-Orderbüchern her und besprechen die optimale Strategie zum Auf- bzw. Abbau einer Anlageposition bei a priori unbeschränkem Anlagehorizont. Anschließend lösen wir das Optimierungsproblem mit festem Anlagehorizon in zwei Schritten. Mittels Variationsrechnung lässt sich die freie Grenzefläche, die Kauf- und Verkaufsregionen trennt, als lokales Optimum identifizieren, was entscheidend für die Verifikation globaler Optimalität ist. Im zweiten Teil der Arbeit erweitern wir den zwischengeschalteten Markteinflussprozess um eine stochastische Komponente, wodurch optimale Strategien dynamisch an zufällige Liquiditätsschwankungen adaptieren. Wir bestimmen die optimale Liquidierungsstrategie im zeitunbeschränkten Fall als die reflektierende Lokalzeit, die den Markteinfluss unterhalb eines explizit beschriebenen nicht-konstanten Grenzlevels hält. Auch dieser Beweis kombiniert Variationsrechnung und direkten Methoden. Um nun eine Zeitbeschränkung zu ermöglichen, müssen wir Semimartingalstrategien zulassen. Skorochods M1-Toplogie ist der Schlüssel, um die Klasse der möglichen Strategien in einer umfangreichen Familie von Preiseinflussmodellen, welche sowohl additiven, als auch multiplikativen Preiseinfluss umfasst, mit deterministischer oder stochastischer Liquidität, eindeutig von endlichen Variations- auf allgemeine càdlàg Strategien zu erweitern. Nach Einführung proportionaler Transaktionskosten lösen wir das entsprechende eindimensionale freie Grenzproblem des optimalen unbeschränkten Handels und beleuchten mögliche Lösungsansätze für das Liquidierungsproblem, das mit dem Verkauf der letzten Anleihe endet. / In this thesis, we study a class of multiplicative price impact models for trading a single risky asset. We model price impact to be multiplicative so that prices are guaranteed to stay non-negative. Our risk-neutral large investor seeks to maximize expected gains from trading. We first introduce a basic variant of our model, wherein the transient impact is a deterministic functional of the trading strategy. We draw the connection to limit order books and give the optimal strategy to liquidate or acquire an asset position infinite time horizon. We then solve the optimization problem for finite time horizon two steps. Calculus of variations allows to identify the free boundary surface that separates buy and sell regions and moreover show its local optimality, which is a crucial ingredient for the verification giving (global) optimality. In the second part of the thesis, we add stochasticity to the auxiliary impact process. This causes optimal strategies to dynamically adapt to random changes in liquidity. We identify the optimal liquidation strategy in infinite horizon as the reflection local time which keeps the market impact process below an explicitly described non-constant free boundary level. Again the proof technique combines classical calculus of variations and direct methods. To now impose a time constraint, we need to admit semimartingale strategies. Skorokhod's M1 topology is key to uniquely extend the class of admissible controls from finite variation to general càdlàg strategies in a broad class of market models including multiplicative and additive price impact, with deterministic or stochastic liquidity. After introducing proportional transaction costs in our model, we solve the related one-dimensional free boundary problem of unconstrained optimal trading and highlight possible solution methods for the corresponding liquidation problem where trading stops as soon as all assets are sold.