Les déplacements cellulaires, collectifs ou individuels, sont essentiels pour assurer des fonctions fondamentales de l'organisme (réponse immunitaire, morphogenèse), mais jouent également un rôle crucial dans le développement de certaines pathologies (invasion métastatique).Les processus cellulaires à l'origine du déplacement forment une activité complexe, auto-organisée et fortement multi-échelle en temps mais aussi en espace. Mettre en évidence des principes généraux de la migration est donc un enjeu majeur. Dans cette thèse, nous nous intéressons à la construction de modèles de migration individuelle qui prennent en compte ce caractère multi-échelle de manière minimale.Dans une première partie, nous nous intéressons à des modèles particulaires. Nous décrivons des processus intracellulaires clés de la migration de manière discrète au moyen de processus de population. Puis, par une renormalisation en grand nombre d'individus, taille infinitésimale et dynamique accélérée, nous obtenons des équations de dynamique continue et stochastique, permettant de faire le lien entre la dynamique intracellulaire et le déplacement macroscopique.Nous nous confrontons d'abord à la situation d'un leucocyte se déplaçant dans une artère, et développant des liaisons de différentes natures avec les molécules de la paroi, jusqu'à éventuellement s'arrêter. La dynamique de formation de liaisons est décrite par un processus stochastique de type Naissance et Mort avec Immigration. Ces liaisons correspondent à des forces de résistance au mouvement. Nous obtenons explicitement le temps d'arrêt moyen de la cellule.Puis, nous nous intéressons à la reptation cellulaire, qui se produit grâce à la formation d'excroissances au bord de la cellule, appelées protrusions, qui avancent sur le substrat et exercent des forces de traction. Nous modélisons cette dynamique au moyen d'un processus de population structurée par l'orientation de la protrusion. Le modèle continu limite obtenu peut être étudié pour la migration 1D, et donne lieu à une équation de Fokker-Planck sur la distribution de probabilité de la population de protrusion. L'étude d'une configuration stationnaire permet de mettre en avant une dichotomie entre un état non motile et un état de déplacement directionnel.Dans une seconde partie, nous construisons un modèle déterministe minimal de migration dans un domaine discoïdal non déformable. Nous nous basons sur l'idée selon laquelle les structures responsables de la migration renforcent la polarisation de la cellule, ce qui favorise en retour un déplacement directionnel. Cette boucle positive passe par le transport d'un marqueur moléculaire dont la répartition inhomogène caractérise un état polarisé.Le modèle comporte un problème de convection-diffusion sur la concentration en marqueur, où le champs d'advection correspond à la vitesse d'un fluide de Darcy modélisant le cytosquelette. Son caractère actif est porté par des termes de bord, ce qui fait l'originalité du modèle.Du point de vue analytique, le modèle 1D présente une dichotomie face à une masse critique. Dans les cas sous-critique et critique, il est possible de montrer l'existence globale de solutions faibles, ainsi que la convergence à taux explicite vers l'unique état stationnaire correspondant à un état non polarisé. Au delà de la masse critique et pour des masses intermédiaires, nous mettons en évidence deux états stationnaires supplémentaires correspondant à des profils polarisés. De plus, pour des conditions initiales assez asymétrique, nous démontrons l'apparition d'un blow-up en temps fini.Du point de vue numérique, des tests numériques en 2D sont effectués en volumes finis (Matlab) et éléments finis (FreeFem++). Ils permettent de mettre en évidence à nouveau des états motiles et non motiles. L'effet de perturbations stochastiques est étudié, permettant d'aborder des cas de réponse à des signaux extérieurs chimique (chimiotactisme) ou mécanique (obstacle). / Collective or individual cell displacements are essential in fundamental physiological processes (immune response, embryogenesis) as well as in pathological developments (tumor metastasis). The intracellular processes responsible for cell motion have a complex self-organized activity spanning different time and space scales. Highlighting general principles of migration is therefore a challenging task.In a first part, we build stochastic particular models of migration. To do so, we describe key intracellular processes as discrete in space by using stochastic population models. Then, by a renormalization in large population, infinitesimal size and accelerated dynamics, we obtain continuous stochastic equations for the dynamics of interest, allowing a relation between the intracellular dynamics and the macroscopic displacement.First, we study the case of a leukocyte carried by the blood flow and developing adhesive bonds with the artery wall, until an eventual stop. The binding dynamics is described by a stochastic Birth and Death with Immigration process. These bonds correspond to resistive forces to the motion. We obtain explicitly the mean stopping time of the cell.Then, we study the case of cell crawling, that happens by the formation of protrusions on the cell edge, that grow on the substrate and exert traction forces. We describe this dynamics by a structured population process, where the structure comes from the protrusions' orientations. The limiting continuous model can be analytically studied in the 1D migration case, and gives rise to a Fokker-Planck equation on the probability distribution for the protrusion density. For a stationary profile, we can show the existence of a dichotomy between a non motile state and a directional displacement state.In a second part, we build a deterministic minimal migration model in a discoïdal cell domain. We base our work on the idea such that the structures responsible for migration also reinforce cell polarisation, which favors in return a directional displacement. This positive feedback loop involves the convection of a molecular marker, whose inhomogeneous spatial repartition is characteristic of a polarised state.The model writes as a convection-diffusion problem for the marker's concentration, where the advection field is the velocity field of the Darcy fluid that describes the cytoskeleton. Its active character is carried by boundary terms, which makes the originality of the model.From the analytical point of vue, the 1D model shows a dichotomy depending on a critical mass for the marker. In the subcritical and critical cases, it is possible to show global existence of weak solutions, as well as a rate-explicit convergence of the solution towards the unique stationary profile, corresponding to a non-motile state. Above the critical mass, for intermediate values, we show the existence of two additional stationary solutions corresponding to polarised motile profiles. Moreover, for asymmetric enough initial profiles, we show the finite time apparition of a blowup.Studying a more complex model involving activation of the marker at the cell membrane permits to get rid of this singularity.From the numerical point of vue, numerical experiments are led in 2D either in finite volumes (Matlab) or finite elements (FreeFem++) discretizations. They allow to show both motile and non motile profiles. The effect of stochastic fluctuations in time and space are studied, leading to numerical simulations of cases of responses to an external signal, either chemical (chemotaxis) or mechanical (obstacles).

This thesis consists of three self-contained parts, each with its own abstract, body, references and page numbering. Part I, 'Potential theory, path integrals and the Laplacian of the indicator', finds the transition density of absorbed or reflected Brownian motion in a d-dimensional domain as a Feynman-Kac functional involving the Laplacian of the indicator, thereby relating the hitherto unrelated fields of classical potential theory and path integrals. Part II, 'The problem of alternatives', considers parallel investment in alternative technologies or drugs developed over time, where there can be only one winner. Parallel investment accelerates the search for the winner, and increases the winner's expected performance, but is also costly. To determine which candidates show sufficient performance and/or promise, we find an integral equation for the boundary of the optimal continuation region. Part III, 'Optimal support for renewable deployment', considers the role of government subsidies for renewable technologies. Rapidly diminishing subsidies are cheaper for taxpayers, but could prematurely kill otherwise successful technologies. By contrast, high subsidies are not only expensive but can also prop up uneconomical technologies. To analyse this trade-off we present a new model for technology learning that makes capacity expansion endogenous. There are two reasons for this standalone structure. First, the target readership is divergent. Part I concerns mathematical physics, Part II operations research, and Part III policy. Readers interested in specific parts can thus read these in isolation. Those interested in the thesis as a whole may prefer to read the three introductions first. Second, the separate parts are only partially interconnected. Each uses some theory from the preceding part, but not all of it; e.g. Part II uses only a subset of the theory from Part I. The quickest route to Part III is therefore not through the entirety of the preceding parts. Furthermore, those instances where results from previous parts are used are clearly indicated.

Ausgehend von einem thermodynamisch konsistenten Modell von Dreyer und Duderstadt für Tropfenbildung in Galliumarsenid-Kristallen, das Oberflächenspannung und Spannungen im Kristall berücksichtigt, stellen wir zwei mathematische Modelle zur Evolution der Größe flüssiger Tropfen in Kristallen auf. Das erste Modell behandelt das Regime diffusionskontrollierter Interface-Bewegung, während das zweite Modell das Regime Interface-kontrollierter Bewegung des Interface behandelt. Unsere Modellierung berücksichtigt die Erhaltung von Masse und Substanz. Diese Modelle verallgemeinern das wohlbekannte Mullins-Sekerka-Modell für die Ostwald-Reifung. Wir konzentrieren uns auf arsenreiche kugelförmige Tropfen in einem Galliumarsenid-Kristall. Tropfen können mit der Zeit schrumpfen bzw. wachsen, die Tropfenmittelpunkte sind jedoch fixiert. Die Flüssigkeit wird als homogen im Raum angenommen. Aufgrund verschiedener Skalen für typische Distanzen zwischen Tropfen und typischen Radien der flüssigen Tropfen können wir formal so genannte Mean-Field-Modelle herleiten. Für ein Modell im diffusionskontrollierten Regime beweisen wir den Grenzübergang mit Homogenisierungstechniken unter plausiblen Annahmen. Diese Mean-Field-Modelle verallgemeinern das Lifshitz-Slyozov-Wagner-Modell, welches rigoros aus dem Mullins-Sekerka-Modell hergeleitet werden kann, siehe Niethammer et al., und gut verstanden ist. Mean-Field-Modelle beschreiben die wichtigsten Eigenschaften unseres Systems und sind gut für Numerik und für weitere Analysis geeignet. Wir bestimmen mögliche Gleichgewichte und diskutieren deren Stabilität. Numerische Resultate legen nahe, wann welches der beiden Regimes gut zur experimentellen Situation passen könnte. / Based on a thermodynamically consistent model for precipitation in gallium arsenide crystals including surface tension and bulk stresses by Dreyer and Duderstadt, we propose two different mathematical models to describe the size evolution of liquid droplets in a crystalline solid. The first model treats the diffusion-controlled regime of interface motion, while the second model is concerned with the interface-controlled regime of interface motion. Our models take care of conservation of mass and substance. These models generalise the well-known Mullins-Sekerka model for Ostwald ripening. We concentrate on arsenic-rich liquid spherical droplets in a gallium arsenide crystal. Droplets can shrink or grow with time but the centres of droplets remain fixed. The liquid is assumed to be homogeneous in space. Due to different scales for typical distances between droplets and typical radii of liquid droplets we can derive formally so-called mean field models. For a model in the diffusion-controlled regime we prove this limit by homogenisation techniques under plausible assumptions. These mean field models generalise the Lifshitz-Slyozov-Wagner model, which can be derived from the Mullins-Sekerka model rigorously, see Niethammer et al., and is well-understood. Mean field models capture the main properties of our system and are well adapted for numerics and further analysis. We determine possible equilibria and discuss their stability. Numerical evidence suggests in which case which one of the two regimes might be appropriate to the experimental situation.

L’objectif de cette étude est de simuler l’érosion d’un sol cohésif sous l’effet d’un écoulement incompressible. Le modèle élaboré décrit une vitesse d’érosion interfaciale qui dépend de la contrainte de cisaillement de l’écoulement. La modélisation numérique proposée est une approche eulérienne, où une méthode de pénalisation de domaines est utilisée pour résoudre les équations de Navier-Stokes autour d’un obstacle. L’interface eau/sol est décrite par une fonction Level Set couplée à une loi d’érosion à seuil.L’approximation numérique est basée sur un schéma DDFV (Discrete Duality Finite Volume) autorisant des raffinements locaux sur maillages non-conformes et non-structurés. L’approche par pénalisation a mis en évidence une couche limite d'inconsistance à l'interface fluide/solide lors du calcul de la contrainte de cisaillement. Deux approches sont proposées pour estimer précisément la contrainte de ce problème à frontière libre. La pertinence du modèle à prédire l’érosion interfaciale du sol est confirmée par la présentation de plusieurs résultats de simulation, qui offrent une meilleure évaluation et compréhension des phénomènes d'érosion / This study focuses on the numerical modelling of the interfacial erosion occurring at a cohesive soil undergoing an incompressible flow process. The model assumes that the erosion velocity is driven by a fluid shear stress at the water/soil interface. The numerical modelling is based on the eulerian approach: a penalization procedure is used to compute Navier-Stokes equations around soil obstacle, with a fictitious domain method, in order to avoid body- fitted unstructured meshes. The water/soil interface’s evolution is described by a Level Set function coupled to a threshold erosion law.Because we use adaptive mesh refinement, we develop a Discrete Duality Finite Volume scheme (DDFV), which allows non-conforming and non-structured meshes. The penalization method, used to take into account a free velocity in the soil with non-body-fitted mesh, introduces an inaccurate shear stress at the interface. We propose two approaches to compute accurately the erosion velocity of this free boundary problem. The ability of the model to predict the interfacial erosion of soils is confirmed by presenting several simulations that provide better evaluation and comprehension of erosion phenomena.

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.

In this work we address to the problems of the loan origination decision-making systems. In accordance with the basic principles of the loan origination process we considered the main rules of a clients parameters estimation, a change-point problem for the given data and a disorder moment detection problem for the real-time observations. In the first part of the work the main principles of the parameters estimation are given. Also the change-point problem is considered for the given sample in the discrete and continuous time with using the Maximum likelihood method. In the second part of the work the disorder moment detection problem for the real-time observations is considered as a disorder problem for a non-homogeneous Poisson process. The corresponding optimal stopping problem is reduced to the free-boundary problem with a complete analytical solution for the case when the intensity of defaults increases. Thereafter a scheme of the real time detection of a disorder moment is given.

Cette thèse est divisée en deux parties. Dans la première partie, on considère le système de réaction-diffusion-advection (Pε), qui est un modèle d'haptotaxie, mécanisme lié à la dissémination de tumeurs cancéreuses. Le résultat principal concerne la convergence de la solution du systeme (Pε) vers la solution d'un problème à frontière libre (P0) qui est bien défini. Dans la seconde partie, on considère une classe générale d'équations elliptiques du type Hénon:−∆u = |x|^{α} f(u) dans Ω ⊂ R^N avec α > -2. On examine deux cas classiques : f(u) = e^u, |u|^{p−1} u et deux autres cas : f(u) = u^{p}_{+} puis f(u) nonlinéarité générale. En étudiant les solutions stables en dehors d'un ensemble compact (en particulier, solutions stables et solutions avec indice de Morse fini) avec différentes méthodes, on obtient des résultats de classification. / This thesis is divided into two main parts. In the first part, we consider an example of reaction-diffusion-taxis system (Pε), which is a haptotaxis model - a mechanism about the spread of cancer cells. The main result concerns the convergence of the solution of System (Pε) to the solution of a free boundary problem (P0), where system (P0) is well-posed. In the second part, we consider a general class of Hénon type elliptic equations : −∆u = |x|^{α} f(u) in Ω ⊂ R^Nwith α > −2. We investigate two classical cases f(u) = e^u, |u|^{p−1} u and two others cases f(u) = u^{p}_{+} , f(u) is a general function. By studying the solutions which are stable outside a compact set (in particular, stable solutions and finite Morse index solutions) with different methods, we establish some classification results.

This thesis is in the field of non-linear partial differential equations (PDE), focusing on problems which show some type of phase-transition. A single phase Hele-Shaw flow models a Newtoninan fluid which is being injected in the space between two narrowly separated parallel planes. The time evolution of the space that the fluid occupies can be modelled by a semi-linear PDE. This is a problem within the field of free boundary problems. In the multi-phase problem we consider the time-evolution of a system of phases which interact according to the principle that the joint boundary which emerges when two phases meet is fixed for all future times. The problem is handled by introducing a parameterized equation which is regularized and penalized. The penalization is non-local in time and tracks the history of the system, penalizing the joint support of two different phases in space-time. The main result in the first paper is the existence theory of a weak solution to the parameterized equations in a Bochner space using the implicit function theorem. The family of solutions to the parameterized problem is uniformly bounded allowing us to extract a weakly convergent subsequence for the case when the penalization tends to infinity. The second problem deals with a parameterized highly oscillatory quasi-linear elliptic equation in divergence form. As the regularization parameter tends to zero the equation gets a jump in the conductivity which occur at the level set of a locally periodic function, the obstacle. As the oscillations in the problem data increases the solution to the equation experiences high frequency jumps in the conductivity, resulting in the corresponding solutions showing an effective global behaviour. The global behavior is related to the so called homogenized solution. We show that the parameterized equation has a weak solution in a Sobolev space and derive bounds on the solutions used in the analysis for the case when the regularization is lost. Surprisingly, the limiting problem in this case includes an extra term describing the interaction between the solution and the obstacle, not appearing in the case when obstacle is the zero level-set. The oscillatory nature of the problem makes standard numerical algorithms computationally expensive, since the global domain needs to be resolved on the micro scale. We develop a multi scale method for this problem based on the heterogeneous multiscale method (HMM) framework and using a finite element (FE) approach to capture the macroscopic variations of the solutions at a significantly lower cost. We numerically investigate the effect of the obstacle on the homogenized solution, finding empirical proof that certain choices of obstacles make the limiting problem have a form structurally different from that of the parameterized problem. / <p>QC 20180222</p>

We consider a variational formulation of a Bernoulli-type free boundary problem for the Laplacian operator with discontinuous boundary data. We show the existence of a weak solution to the problem. Moreover, we show that the solution has symmetry properties inherited by symmetric data. These results are achieved through the use of comparison arguments, the celebrated method of moving planes, and several elaborated techniques from existing literature. / Vi studerar ett Bernoulli frirandsproblem för Laplaceoperatorn med diskontinuerliga randdata. Detta görs via en variationsformulering av problemet. Vi visar att en svag lösning existerar för problemet. Utöver det visar vi bland annat att den svaga lösningen har symmetriegenskaper. Dessa resultat uppnås genom jämförelseargument, den välkända "moving-plane” metoden, samt flera utarbetade tekniker från befintlig litteratur.

過去的三十年間由於評價美式選擇權所產生的自由邊界問題已經有相當的研究成果。本論文將證明自由邊界問題的解為遞增函數。更進一步提出自由邊界凹性的嚴謹証明。利用我們的結論可以得知美式選擇權的最佳履約邊界對時間而言為嚴格遞減的凹函數。這個結果對可用來求導最佳履約邊界的漸近解。 對於美式交換選擇權，我們將其自由邊界問題轉換成單變數的積分方程，同時提供一個永續型美式交換選擇權的評價公式。對於有限時間的美式交換選擇權的最佳履約邊界，我們將提供一個接近到期日的漸近解並發展一個數值方法求其數值解。數值計算的結果顯示漸近解在接近到期日時與數值解非常接近。 對於評價美式選擇權，我們提出使用混合整數非線性規劃(MINLP)的模型，這個模型的最佳解同時提供賣方的完全避險策略、買方的最佳交易策略與美式選擇權的公平價格。因為求算MINLP模型的解需耗用大量的計算時間，我們證明此模型和其非線性規劃的寬鬆問題有相同的最佳解，所以只需求算寬鬆問題即可。觀察數值結果亦顯示非線性規劃的寬鬆問題可以大幅的降低計算的時間。此外，當市場的價格低於公平價格時，我們提出一個最小化賣方期望損失的數學規劃模型，此模型的解提供賣方最小化其期望損失的避險策略。 / In the past three decades, a great deal of effort has been made on solving the free boundary problem (FBP) arising from American option valuation problems. In this dissertation, we show that the solutions, the price and the free boundary, of this FBP are increasing functions. Furthermore, we provide a rigorous verification that the free boundary of this problem is concave. Our results imply that the optimal exercise boundary of an American call is a strictly decreasing concave function of time. These results will provide a useful information to obtain an asymptotic formula for the optimal exercise boundary. For pricing of American exchange options (AEO), we convert the associated FBP into a single variable integral equation (IE) and provide a formula for valuating the perpetual AEO. For the finite horizon AEO, we propose an asymptotic solution as time is near to expiration and develop a numerical method for its optimal exercise boundary. Compared with the computational results, the values of our asymptotic solution are close to the computational results as time is near to expiration. For valuating American options, we develop a mixed integer nonlinear programming (MINLP) model. The solution of the MINLP model provides a hedging portfolio for writers, the optimal trading strategy for buyers, and the fair price for American options at the same time. We show that it can be solved by its nonlinear programming (NLP) relaxation. The numerical results reveal that the use of NLP relaxation reduces the computation time rapidly. Moreover, when the market price is less than the fair price, we propose a minimum expected loss model. The solution of this model provides a hedging strategy that minimizes the expected loss for the writer.