Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Bibliography: leaves 328-332. / by Marcel Coderch i Collell. / Ph.D.

The focus of this study is the prediction of trajectories of solid particles injected into either a cylindrically- shaped solid rocket motor (SRM) or a bidirectional vortex chamber (BV). The Lagrangian particle trajectory is assumed to be governed by drag, virtual mass, Magnus, Saffman lift, and gravity forces in a Stokes flow regime. For the conditions in a solid rocket motor, it is determined that either the drag or gravity forces will dominate depending on whether the sidewall injection velocity is high (drag) or low (gravity). Using a one-way coupling paradigm in a solid rocket motor, the effects of particle size, sidewall injection velocity, and particle-to-gas density ratio are examined. The particle size and sidewall injection velocity are found to have a greater impact on particle trajectories than the density ratio. Similarly, for conditions associated with a bidirectional vortex engine, it is determined that the drag force dominates. Using a one-way particle tracking Lagrangian model, the effects of particle size, geometric inlet parameter, particle-to-gas density ratio, and initial particle velocity are examined. All but the initial particle velocity are found to have a significant impact on particle trajectories. The proposed models can assist in reducing slag retention and identifying fuel injection configurations that will ensure proper confinement of combusting droplets to the inner vortex in solid rocket motors and bidirectional vortex engines, respectively.

This thesis is concerned with the small-time asymptotics and expansions of call option prices, when the log-return processes of the underlying stock prices follow several Levy-based models. To be specific, we derive the time-to-maturity asymptotic behavior for both at-the-money (ATM), out-of-the-money (OTM) and in-the-money (ITM) call-option prices under several jump-diffusion models and stochastic volatility models with Levy jumps. In the OTM and ITM cases, we consider a general stochastic volatility model with independent Levy jumps, while in the ATM case, we consider the pure-jump CGMY model with or without an independent Brownian component. An accurate modeling of the option market and asset prices requires a mixture of a continuous diffusive component and a jump component. In this thesis, we first model the log-return process of a risk asset with a jump diffusion model by combining a stochastic volatility model with an independent pure-jump Levy process. By assuming smoothness conditions on the Levy density away from the origin and a small-time large deviation principle on the stochastic volatility model, we derive the small-time expansions, of arbitrary polynomial order, in time-t, for the tail distribution of the log-return process, and for the call-option price which is not at-the-money. Moreover, our approach allows for a unified treatment of more general payoff functions. As a consequence of our tail expansions, the polynomial expansion in t of the transition density is also obtained under mild conditions. The asymptotic behavior of the ATM call-option prices is more complicated to obtain, and, in general, is given by fractional powers of t, which depends on different choices of the underlying log-return models. Here, we focus on the CGMY model, one of the most popular tempered stable models used in financial modeling. A novel second-order approximation for ATM option prices under the pure-jump CGMY Levy model is derived, and then extended to a model with an additional independent Brownian component. The third-order asymptotic behavior of the ATM option prices as well as the asymptotic behavior of the corresponding Black-Scholes implied volatilities are also addressed.

The paper considers systems of linear first-order ODEs with a randomly perturbed system matrix and stationary additive noise. For the description of the long-term behavior of such systems it is necessary to study their stationary solutions. We deal with conditions for the existence of stationary solutions as well as with their representations and the computation of their moment functions. Assuming small perturbations of the system matrix we apply perturbation techniques to find series representations of the stationary solutions and give asymptotic expansions for their first- and second-order moment functions. We illustrate the findings with a numerical example of a scalar ODE, for which the moment functions of the stationary solution still can be computed explicitly. This allows the assessment of the goodness of the approximations found from the derived asymptotic expansions.

We develop expansion methods as a new computational approach towards high-dimensional partial differential equations (PDEs), particularly of such type as arising in the valuation of financial derivatives. The proposed methods are extended from [41] and use principal component analysis (PCA) of the underlying process in combination with a Taylor expansion of the value function into solutions to low-dimensional PDEs. They enable calculation of highly accurate approximate solutions with computational complexity polynomial in the number of dimensions for PDEs with a low number of dominant principal components. For the case of PDEs with constant coefficients, we show existence of expansion solutions and prove theoretical error bounds. We give a precise characterisation of when our methods can be applied and construct specific examples of a first and second order version. We provide numerical results showing that the empirically observed convergence speeds are in agreement with the theoretical predictions. For the case of PDEs with varying coefficients, we give a heuristic motivation using the Parametrix approach and empirically test the methods' accuracy for a range of variable parameter stock models. We demonstrate the applicability of our expansion methods to real-world securities pricing problems by considering path-dependent and early-exercise options in the LIBOR market model. Using the example of Bermudan swaptions and Ratchet floors, which are considered difficult benchmark problems, we give a careful analysis of the numerical accuracy and computational complexity. We are able to demonstrate that for problems with medium to high dimensionality, around 60-100, and moderate time horizons, the presented PDE methods deliver results comparable in accuracy to benchmark state-of-the-art Monte Carlo methods in similar or (significantly) faster run time.

In the paper asymptotic expansions for second-order moments of integral functionals of a class of random functions are considered. The random functions are assumed to be $\epsilon$-correlated, i.e. the values are not correlated excluding a $\epsilon$-neighbourhood of each point. The asymptotic expansions are derived for $\epsilon \to 0$. With the help of a special weak assumption there are found easier expansions as in the case of general weakly correlated functions.

Dans cette thèse, nous étudions plusieurs problèmes de mathématiques financières liés à la valorisation des produits dérivés. Par différentes approches asymptotiques, nous développons des méthodes pour calculer des approximations précises du prix de certains types d’options dans des cas où il n’existe pas de formule explicite.Dans le premier chapitre, nous nous intéressons à la valorisation des options dont le payoff dépend de la trajectoire du sous-jacent par méthodes de Monte-Carlo, lorsque le sous-jacent est modélisé par un processus affine à volatilité stochastique. Nous prouvons un principe de grandes déviations trajectoriel en temps long, que nous utilisons pour calculer, en utilisant le lemme de Varadhan, un changement de mesure asymptotiquement optimal, permettant de réduire significativement la variance de l’estimateur de Monte-Carlo des prix d’options.Le second chapitre considère la valorisation par méthodes de Monte-Carlo des options dépendant de plusieurs sous-jacents, telles que les options sur panier, dans le modèle à volatilité stochastique de Wishart, qui généralise le modèle Heston. En suivant la même approche que dans le précédent chapitre, nous prouvons que le processus vérifie un principe de grandes déviations en temps long, que nous utilisons pour réduire significativement la variance de l’estimateur de Monte-Carlo des prix d’options, à travers un changement de mesure asymptotiquement optimal. En parallèle, nous utilisons le principe de grandes déviations pour caractériser le comportement en temps long de la volatilité implicite Black-Scholes des options sur panier.Dans le troisième chapitre, nous étudions la valorisation des options sur variance réalisée, lorsque la volatilité spot est modélisée par un processus de diffusion à volatilité constante. Nous utilisons de récents résultats asymptotiques sur les densités des diffusions hypo-elliptiques pour calculer une expansion de la densité de la variance réalisée, que nous intégrons pour obtenir l’expansion du prix des options, puis de leur volatilité implicite Black-Scholes.Le dernier chapitre est consacré à la valorisation des dérivés de taux d’intérêt dans le modèle Lévy de marché Libor qui généralise le modèle de marché Libor classique (log-normal) par l’ajout de sauts. En écrivant le premier comme une perturbation du second et en utilisant la représentation de Feynman-Kac, nous calculons explicitement l’expansion asymptotique du prix des dérivés de taux, en particulier, des caplets et des swaptions. / In this thesis, we study several mathematical finance problems, related to the pricing of derivatives. Using different asymptotic approaches, we develop methods to calculate accurate approximations of the prices of certain types of options in cases where no explicit formulas are available.In the first chapter, we are interested in the pricing of path-dependent options, with Monte-Carlo methods, when the underlying is modelled as an affine stochastic volatility model. We prove a long-time trajectorial large deviations principle. We then combine it with Varadhan’s Lemma to calculate an asymptotically optimal measure change, that allows to reduce significantly the variance of the Monte-Carlo estimator of option prices.The second chapter considers the pricing with Monte-Carlo methods of options that depend on several underlying assets, such as basket options, in the Wishart stochastic volatility model, that generalizes the Heston model. Following the approach of the first chapter, we prove that the process verifies a long-time large deviations principle, that we use to reduce significantly the variance of the Monte-Carlo estimator of option prices, through an asymptotically optimal measure change. In parallel, we use the large deviations property to characterize the long-time behaviour of the Black-Scholes implied volatility of basket options.In the third chapter, we study the pricing of options on realized variance, when the spot volatility is modelled as a diffusion process with constant volatility. We use recent asymptotic results on densities of hypo-elliptic diffusions to calculate an expansion of the density of realized variance, that we integrate to obtain an expansion of option prices and their Black-Scholes implied volatility.The last chapter is dedicated to the pricing of interest rate derivatives in the Levy Libor market model, that generaliszes the classical (log-normal) Libor market model by introducing jumps. Writing the first model as a perturbation of the second and using the Feynman-Kac representation, we calculate explicit expansions of the prices of interest rate derivatives and, in particular, caplets and swaptions

La thèse porte sur l'apport de la méthode de Laplace pour l'approximation du filtre bayésien dans des modèles de Markov cachés généraux, c'est-à-dire dans un cadre séquentiel, avec comme domaine d'application privilégié la poursuite de cibles mobiles. A la base, la méthode de Laplace est une méthode asymptotique pour le calcul d'intégrales, c'est-à-dire dans un cadre statique, valide en théorie dès que la fonction à intégrer présente un maximum de plus en plus significatif, lequel apporte la contribution essentielle au résultat. En pratique, cette méthode donne des résultats souvent très précis même en dehors de ce cadre de validité théorique. Les deux contributions principales de la thèse sont les suivantes. Premièrement, nous avons utilisé la méthode de Laplace en complément du filtrage particulaire : on sait en effet que les méthodes de Monte Carlo séquentielles basées sur l'échantillonnage pondéré sont mises en difficulté quand la fonction de pondération (ici la fonction de vraisemblance) est trop localisée, par exemple quand la variance du bruit d'observation est trop faible, or c'est précisément là le domaine où la méthode de Laplace est efficace et justifiée théoriquement, d'où l'idée naturelle de combiner les deux points de vue. Nous proposons ainsi un algorithme associant la méthode de Laplace et le filtrage particulaire, appelé le Laplace particle filter. Deuxièmement, nous avons analysé l'approximation du filtre bayésien grâce à la méthode de Laplace seulement (c'est-à-dire sans génération d'échantillons aléatoires) : il s'agit ici de contrôler la propagation de l'erreur d'approximation d'un pas de temps au pas de temps suivant, dans un cadre asymptotique approprié, par exemple quand le bruit d'observation tend vers zéro, ou quand le bruit d'état et le bruit d'observation tendent conjointement (et à la même vitesse) vers zéro, ou plus généralement quand l'information contenue dans le système tend vers l'infini, avec une interprétation en terme d'identifiabilité. / The thesis deals with the contribution of the Laplace method to the approximation of the Bayesian filter in hidden Markov models with continuous state--space, i.e. in a sequential framework, with target tracking as the main application domain. Originally, the Laplace method is an asymptotic method used to compute integrals, i.e. in a static framework, valid in theory as soon as the function to be integrated exhibits an increasingly dominating maximum point, which brings the essential contribution to the integral. The two main contributions of the thesis are the following. Firstly, we have combined the Laplace method and particle filters: indeed, it is well-known that sequential Monte Carlo methods based on importance sampling are inefficient when the weighting function (here, the likelihood function) is too much spatially localized, e.g. when the variance of the observation noise is too small, whereas this is precisely the situation where the Laplace method is efficient and theoretically justified, hence the natural idea of combining the two approaches. We thus propose an algorithm associating the Laplace method and particle filtering, called the Laplace particle filter. Secondly, we have analyzed the approximation of the Bayesian filter based on the Laplace method only (i.e. without any generation of random samples): the objective has been to control the propagation of the approximation error from one time step to the next time step, in an appropriate asymptotic framework, e.g. when the variance of the observation noise goes to zero, or when the variances of the model noise and of the observation noise jointly go (with the same rate) to zero, or more generally when the information contained in the system goes to infinity, with an interpretation in terms of identifiability.

This thesis presents results in two apparently disparate mathematical fields which can both be examined -- and even united -- by means of pure analysis. Fractional calculus is the study of differentiation and integration to non-integer orders. Dating back to Leibniz, this idea was considered by many great mathematical figures, and in recent decades it has been used to model many real-world systems and processes, but a full development of the mathematical theory remains incomplete. Many techniques for partial differential equations (PDEs) can be extended to fractional PDEs too. Three chapters below cover my results in this area: establishing the elliptic regularity theorem, Malgrange-Ehrenpreis theorem, and unified transform method for fractional PDEs. Each one is analogous to a known result for classical PDEs, but the proof in the general fractional scenario requires new ideas and modifications. Fractional derivatives and integrals are not uniquely defined: there are many different formulae, each of which has its own advantages and disadvantages. The most commonly used is the classical Riemann-Liouville model, but others may be preferred in different situations, and now new fractional models are being proposed and developed each year. This creates many opportunities for new research, since each time a model is proposed, its mathematical fundamentals need to be examined and developed. Two chapters below investigate some of these new models. My results on the Atangana-Baleanu model proposed in 2016 have already had a noticeable impact on research in this area. Furthermore, this model and the results concerning it can be extended to more general fractional models which also have certain desirable properties of their own. Fractional calculus and zeta functions have rarely been united in research, but one chapter below covers a new formula expressing the Lerch zeta function as a fractional derivative of an elementary function. This result could have many ramifications in both fields, which are yet to be explored fully. Zeta functions are very important in analytic number theory: the Riemann zeta function relates to the distribution of the primes, and this field contains some of the most persistent open problems in mathematics. Since 2012, novel asymptotic techniques have been applied to derive new results on the growth of the Riemann zeta function. One chapter below modifies some of these techniques to prove asymptotics to all orders for the Hurwitz zeta function. Many new ideas are required, but the end result is more elegant than the original one for Riemann zeta, because some of the new methodologies enable different parts of the argument to be presented in a more unified way. Several related problems involve asymptotics arbitrarily near a stationary point. Ideally it should be possible to find uniform asymptotics which provide a smooth transition between the integration by parts and stationary phase methods. One chapter below solves this problem for a particular integral which arises in the analysis of zeta functions.

Dans cette thèse, nous menons à l'aide de la méthode des équations intégrales des analyses de sensitivité de solutions ou de spectres de l'équation de conductivité par rapport aux variations géométriques ou de paramètres de l'équation. En particulier, nous considérons le problème de conductivité dans des milieux à forts contrastes, le problème de perturbation du bord d'une inclusion de conductivité, le problème de valeurs propres du Laplacien dans des domaines perturbés et le problème d'ouverture de gap dans le spectre des cristaux photoniques.