We consider a class of optimization problems in which the aim is to find some optimizing probability distributions. One particular example is optimal design. We first review the optimal design theory, and determine the optimality conditions using directional derivatives. We then construct optimal designs for various polynomial regression models by finding the analytic solutions and by using a class of algorithms. We consider a practical problem, namely a radiation-dosage example, and discuss important aspects of optimal design throughout this example. We also construct optimal designs for various polynomial regression models with more than one design variable. We consider another practical problem, namely a vocabulary-growth study. We then construct D-optimal and c-optimal designs for various models with and without the interaction term and the second order terms in design variables. We also develop strategies for constructing designs by using the properties of the directional derivatives.

We consider a class of optimization problems in which the aim is to find some optimizing probability distributions. One particular example is optimal design. We first review the optimal design theory, and determine the optimality conditions using directional derivatives. We then construct optimal designs for various polynomial regression models by finding the analytic solutions and by using a class of algorithms. We consider a practical problem, namely a radiation-dosage example, and discuss important aspects of optimal design throughout this example. We also construct optimal designs for various polynomial regression models with more than one design variable. We consider another practical problem, namely a vocabulary-growth study. We then construct D-optimal and c-optimal designs for various models with and without the interaction term and the second order terms in design variables. We also develop strategies for constructing designs by using the properties of the directional derivatives.

Une première partie de cette thèse est dédiée à l’étude de la régularité de la densité de transport sigma dans le problème de Monge entre deux mesures f^+ et f^- sur un domaine Omega. Tout d’abord, on étudie la question de la sommabilité L^p de cette densité de transport entre une mesure f^+ et sa projection sur le bord (P)# f^+, qui ne découle pas en fait des résultats connus (dus à De Pascale - Evans - Pratelli - Santambrogio) sur la densité de transport entre deux densités L^p, comme dans notre cas la mesure cible est singulière. Par une méthode de symétrisation, dès que Omega est convexe ou satisfait une condition de boule uniforme extérieure, nous prouvons les estimations L^p (si f^+ in L^p, alors sigma in L^p). En plus, nous analysons le cas où on paye des coûts supplémentaires g^± sur le bord, en prouvant que la densité de transport est dans L^p dès que f^± in L^p, Omega satisfait une condition de boule uniforme extérieure et, g^± sont lambda^± Lipschitiziens avec lambda^± < 1 et semi-concaves. Ensuite, on s’attaque à la régularité d’ordre supérieur (W^{1,p}, C^{0,alpha}, BV · · ·) de la densité de transport sigma entre deux densités régulières f^+ et f^-. Plus précisément, nous fournissons une famille de contre-exemples à la régularité supérieure: nous prouvons que la régularité W^{1,p} des mesures source et cible, f^+ et f^-, n’implique pas que la densité de transport est W^{1,p}, de même pour la régularité BV, et même f^± in C^infty n’implique pas que sigma est dans W^{1,p}, pour p grand. Ensuite, nous étudions la sommabilité L^p de la densité de transport entre deux mesures f^+ et f^- concentrées sur le bord. Plus précisément, nous prouvons que si f^+ et f^- sont dans L^p(partialOmega), alors la densité de transport sigma entre eux est dans L^p(Omega) dès que Omega est uniformément convexe et p leq 2; de plus, nous introduisons un contre-exemple montrant que ce résultat n’est plus vrai si p > 2. Cela fournit des résultats de régularité W^{1,p} sur la solution u du problème de gradient minimal avec donnée au bord g dans des domaines uniformément convexes (si g in W^{1,p}(partialOmega) alors u in W^{1,p}(Omega)).Dans une deuxième partie, nous étudions un problème de contrôle optimal motivé par un modèle de jeux à champ moyen. D’abord, nous montrons des résultats de différentiabilité et semi-concavité sur la fonction valeur associée au problème de contrôle (le résultat de semi-concavité est optimal en ce qui concerne les hypothèses sur la régularité en temps). Ensuite, nous démontrons que la densité des agents rho_t, dans le modèle MFG considéré, est dans L^p dès que la densité initiale rho_0 in L^p. En plus, nous arrivons à prouver l’existence d’un équilibre pour le problème MFG considéré dans un cas où la dynamique n’est pas régulière.Dernièrement, nous considérons le problème stationnaire associé au problème MFG. Nous montrons que la densité d’équilibre n’est rien d’autre que la densité de transport entre une densité source f et sa projection sur le bord en utilisant une métrique Riemannienne non-uniforme comme coût de transport. Cela nous permet de démontrer que la densité d’équilibre rho est dans L^p dès que la densité source f in L^p. Par conséquent, nous arrivons à prouver aussi l’existence d’un équilibre stationnaire dans un cas où la dynamique n’est pas régulière. / A first part of this thesis is dedicated to the study of the regularity of the transport density sigma in the Monge problem between two measures f^+ and f^- on a domain Omega. First, we study the question of L^p summability of this transport density between a measure f^+ and its projection on the boundary (P)# f^+, which does not actually follow from the known results (due to De Pascale, Evans, Pratelli, Santambrogio) on the transport density between two L^p densities, as in our case the target measure is singular. By a symmetrization trick, if Omega is convex or satisfies a uniform exterior ball condition, we prove the L^p estimates (if f^+ in L^p, then sigma in L^p). In addition, we analyze the case where we pay additional costs g^± on the boundary, proving that the transport density sigma is in L^p as soon as f^± in L^p, Omega satisfies a uniform exterior ball condition and, g^± are lambda^± Lip with lambda^± < 1 and semi-concave. Then we attack the higher-order regularity (W^{1,p}, C^{0,alpha}, BV · · ·) of the transport density sigma between two regular densities f^+ and f^-. More precisely, we provide a family of counter-examples to the higher regularity: we prove that the W^{1,p} regularity of the source and target measures, f^+ and f^-, does not imply that the transport density is in W^{1,p}, the same for the BV regularity, and even f^± in C^infty does not imply that sigma is in W^{1,p}, for large p. Next, we study the L^p summability of the transport density between two measures, f^+ and f^-, concentrated on the boundary. More precisely, we prove that if f^+ and f^- are in L^p(partialOmega), then the transport density sigma between them is in L^p(Omega) as soon as Omega is uniformly convex and p leq 2; moreover, we introduce a counter-example showing that this result is no longer true if p > 2. This provides W^{1,p} regularity results on the solution u of the least gradient problem with boundary datum g in uniformly convex domains (if g in W^{1,p}(partialOmega) then u in W^{1,p}(Omega)).In a second part, we study an optimal control problem motivated by a model of mean field games. First, we show differentiability and semi-concavity results on the value function associated with the control problem (the semi-concavity result will be sharp in regards to the hypotheses on the regularity in time). Then we show that the density of agents rho_t, in the considered MFG model, is in L^p as soon as the initial density rho_0 in L^p. In addition, we prove existence of an equilibrium for the considered MFG problem in a case where the dynamic is non-regular.Lastly, we consider the stationary problem associated with the MFG model. We show that the equilibrium density is nothing but the transport density between a source density f and its projection on the boundary using a non-uniform Riemannian metric as a transport cost. This allows us to show that the equilibrium density rho is in L^p as soon as the source density f in L^p. Therefore, we also prove existence of a stationary equilibrium in a case where the dynamic is non-regular.

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The goal of this paper is to extend the results of Bayraktar and Young (2006) on minimizing an individual's probability of lifetime ruin; i.e. the probability that the individual goes bankrupt before dying. We consider a scenario in which the individual is allowed to invest in both a domestic bank account and a foreign bank account, with the exchange rate between the two currencies being modeled by geometric Brownian motion. Additionally, we impose the restriction that the individual is not allowed to borrow money, and assume that the individual's wealth is consumed at a constant rate. We derive formulas for the minimum probability of ruin as well as the individual's optimal investment strategy. We also give a few numerical examples to illustrate these results.

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A near-optimal feedback guidance algorithm is applied to several different applications in the Circular-Restricted Three Body Problem and in proximity operations in LEO modeled by Keplerian motion. In both scenarios gravitational perturbations are introduced in order to assess the algorithm's robustness. Two forms of the guidance algorithm are studied: a zero-effort miss/zero-effort velocity feedback control law and a zero-effort miss/zero-effort velocity feedback control law augmented with a sliding mode. Both guidance laws have previously been applied to the problems of planetary landing, asteroid intercept, and close-proximity maneuvers near an asteroid. This study is motivated by the growing interest in spacecraft autonomy for proximity operations and in cases where a high frequency of open-loop commanded maneuvers is not practical. Results demonstrate that nominal zero-effort miss/zero-effort velocity feedback guidance is suboptimal in all test cases, but performance can be improved through the addition of waypoints and tuning of guidance law parameters. Additionally, the application of a sliding-mode is shown to overcome limitations introduced by gravitational perturbations in some instances.

Degree: Master of Science Department: Science / The superharmonic characterization of the value function is proved, under the assumption that an optimal stopping time exists. The fair price of an American contingent claim is established as an optimal stopping problem. The price of the perpetual Russian option is derived, using the dual martingale measure to reduce the dimension of the problem. American barrier options are discussed, and the solution to the perpetual American up-and-out put is derived. The price of the American put on a finite time horizon is shown to be the price of the European put plus an early exercise premium, through the use of a local time-space formula. The optimal stopping boundary is characterised as the unique increasing solution of a non-linear integral equation. Finally, the integral representation of the price of an American floating strike Asian call with arithmetic averaging is derived.