The market for mortgage backed securities (MBS) was active and fast growing from the issuance of the first MBS in 1981. This enabled financial firms to transform risky individual mortgages into liquid and tradable market instruments. The subprime mortgage crisis of 2007 shows the need for a better understanding and development of mathematical models for these securities. The aim of this thesis is to develop a model for MBS that is flexible enough to capture both regular and subprime MBS. The thesis considers two models, one for a single mortgage in an intensity based framework and the second for mortgage backed securities using a stochastic partial differential equation approach. In the model for a single mortgage, we capture the prepayment and default incentives of the borrower using intensity processes. Using the minimum of the two intensity processes, we develop a nonlinear equation for the mortgage rate and solve it numerically and present some case studies. In modelling of an MBS in a structural framework using stochastic PDEs (SPDEs), we consider a large number of individuals in a mortgage pool and assume that the wealth of each individual follows a stochastic process, driven by two Brownian mo- tions, one capturing the idiosyncratic noise of each individual and the second a common market factor. By defining the empirical measure of a large pool of these individuals we study the evolution of the limit empirical measure and derive an SPDE for the evolution of the density of the limit empirical measure. We numerically solve the SPDE to demonstrate its flexibility in different market environments. The calibration of the model to financial data is the focus of the final part of thesis. We discuss the different parameters and demonstrate how many can be fitted to observed data. Finally, for the key model parameters, we present a strategy to estimate them given observations of the loss function and use this to determine implied model parameters of ABX.HE.

We propose two alternative approaches to evaluate numerically credit basket derivatives in a N-name structural model where the number of entities, N, is large, and where the names are independent and identically distributed random variables conditional on common random factors. In the first framework, we treat a N-name model as a set of N Bernoulli random variables indicating a default or a survival. We show that certain expected functionals of the proportion L_N of variables in a given state converge at rate 1/N as N [right arrow - infinity]. Based on these results, we propose a multi-level simulation algorithm using a family of sequences with increasing length, to obtain estimators for these expected functionals with a mean-square error of epsilon ² and computational complexity of order epsilon⁻², independent of N. In particular, this optimal complexity order also holds for the infinite-dimensional limit. Numerical examples are presented for tranche spreads of basket credit derivatives. In the second framework, we extend the approximation of Bush et al. [13] to a structural jump-diffusion model with discretely monitored defaults. Under this approach, a N-name model is represented as a system of particles with an absorbing boundary that is active in a discrete time set, and the loss of a portfolio is given as the function of empirical measure of the system. We show that, for the infinite system, the empirical measure has a density with respect to the Lebesgue measure that satisfies a stochastic partial differential equation. Then, we develop an algorithm to efficiently estimate CDO index and tranche spreads consistent with underlying credit default swaps, using a finite difference simulation for the resulting SPDE. We verify the validity of this approximation numerically by comparison with results obtained by direct Monte Carlo simulation of the basket constituents. A calibration exercise assesses the flexibility of the model and its extensions to match CDO spreads from precrisis and crisis periods.

Cette thèse traite des Équations aux Dérivées Partielles Stochastiques Quasilinéaires. Elle est divisée en deux parties. La première partie concerne le problème d’obstacle pour les équations aux dérivées partielles stochastiques quasilinéaires et la deuxième partie est consacrée à l’étude des équations aux dérivées partielles stochastiques quasilinéaires dirigées par un G-mouvement brownien. Dans la première partie, on montre d’abord l’existence et l’unicité d’un problème d’obstacle pour les équations aux dérivées partielles stochastiques quasilinéaires (en bref OSPDE). Notre méthode est basée sur des techniques analytiques venant de la théorie du potentiel parabolique. La solution est exprimée comme une paire (u,v) où u est un processus prévisible continu qui prend ses valeurs dans un espace de Sobolev et v est une mesure régulière aléatoire satisfaisant la condition de Skohorod. Ensuite, on établit un principe du maximum pour la solution locale des équations aux dérivées partielles stochastiques quasilinéaires avec obstacle. La preuve est basée sur une version de la formule d’Itô et les estimations pour la partie positive d’une solution locale qui est négative sur le bord du domaine considéré. L’objectif de la deuxième partie est d’étudier l’existence et l’unicité de la solution des équations aux dérivées partielles stochastiques dirigées par G-mouvement brownien dans le cadre d’un espace muni d’une espérance sous-linéaire. On établit une formule d’Itô pour la solution et un théorème de comparaison. / This thesis deals with quasilinear Stochastic Partial Differential Equations (in short SPDE). It is divided into two parts, the first part concerns the obstacle problem for quasilinear SPDE and the second part solves quasilinear SPDE driven by G-Brownian motion. In the first part we begin with the existence and uniqueness result for the obstacle problem of quasilinear stochastic partial differential equations (in short OSPDE). Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair (u, v) where u is a predictable continuous process which takes values in a proper Sobolev space and v is a random regular measure satisfying minimal Skohorod condition. Then we prove a maximum principle for a local solution of quasilinear stochastic partial differential equations with obstacle. The proofs are based on a version of Itô’s formula and estimates for the positive part of a local solution which is negative on the lateral boundary. The objective of the second part is to study the well-posedness of stochastic partial differential equations driven by G-Brownian motion in the framework of sublinear expectation spaces. One can also establish an Itô formula for the solution and a comparison theorem.

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.

We consider linear and semilinear stochastic partial differential equations that in some sense can be viewed as being at the "endpoints" of the classical variational theory by Krylov and Rozovskii [25]. In terms of regularity of the coeffcients, the minimal assumption is boundedness and measurability, and a unique L2- valued solution is then readily available. We investigate its further properties, such as higher order integrability, boundedness, and continuity. The other class of equations considered here are the ones whose leading operators do not satisfy the strong coercivity condition, but only a degenerate version of it, and therefore are not covered by the classical theory. We derive solvability in Wmp spaces and also discuss their numerical approximation through finite different schemes.

The central theme of this thesis is the design of optimal balanced black-box stopping criteria in iterative solvers of symmetric positive-definite, symmetric indefinite, and nonsymmetric linear systems arising from finite element approximation of stochastic (parametric) partial differential equations. For a given stochastic and spatial approximation, it is known that iteratively solving the corresponding linear(ized) system(s) of equations to too tight algebraic error tolerance results in a wastage of computational resources without decreasing the usually unknown approximation error. In order to stop optimally-by avoiding unnecessary computations and premature stopping-algebraic error and a posteriori approximation error estimate must be balanced at the optimal stopping iteration. Efficient and reliable a posteriori error estimators do exist for close estimation of the approximation error in a finite element setting. But the algebraic error is generally unknown since the exact algebraic solution is not usually available. Obtaining tractable upper and lower bounds on the algebraic error in terms of a readily computable and monotonically decreasing quantity (if any) of the chosen iterative solver is the distinctive feature of the designed optimal balanced stopping strategy. Moreover, this work states the exact constants, that is, there are no user-defined parameters in the optimal balanced stopping tests. Hence, an iterative solver incorporating the optimal balanced stopping methodology that is presented here will be a black-box iterative solver. Typically, employing such a stopping methodology would lead to huge computational savings and in any case would definitely rule out premature stopping. The constants in the devised optimal balanced black-box stopping tests in MINRES solver for solving symmetric positive-definite and symmetric indefinite linear systems can be estimated cheaply on-the- fly. The contribution of this thesis goes one step further for the nonsymmetric case in the sense that it not only provides an optimal balanced black-box stopping test in a memory-expensive Krylov solver like GMRES but it also presents an optimal balanced black-box stopping test in memory-inexpensive Krylov solvers such as BICGSTAB(L), TFQMR etc. Currently, little convergence theory exists for the memory-inexpensive Krylov solvers and hence devising stopping criteria for them is an active field of research. Also, an optimal balanced black-box stopping criterion is proposed for nonlinear (Picard or Newton) iterative method that is used for solving the finite dimensional Navier-Stokes equations. The optimal balanced black-box stopping methodology presented in this thesis can be generalized for any iterative solver of a linear(ized) system arising from numerical approximation of a partial differential equation. The only prerequisites for this purpose are the existence of a cheap and tight a posteriori error estimator for the approximation error along with cheap and tractable bounds on the algebraic error.

L’objectif de cette thèse est l’étude d’un schéma numérique pour l’approximation des solutions d’équations différentielles doublement stochastiques rétrogrades (EDDSR). Durant les deux dernières décennies, plusieurs méthodes ont été proposées afin de permettre la résolution numérique des équations différentielles stochastiques rétrogrades standards. Dans cette thèse, on propose une extension de l’une de ces méthodes au cas doublement stochastique. Notre méthode numérique nous permet d’attaquer une large gamme d’équations aux dérivées partielles stochastiques (EDPS) nonlinéaires. Ceci est possible par le biais de leur représentation probabiliste en termes d’EDDSRs. Dans la dernière partie, nous étudions une nouvelle méthode des particules dans le cadre des études de protection en neutroniques. / The purpose of this thesis is to study a numerical method for backward doubly stochastic differential equations (BDSDEs in short). In the last two decades, several methods were proposed to approximate solutions of standard backward stochastic differential equations. In this thesis, we propose an extension of one of these methods to the doubly stochastic framework. Our numerical method allows us to tackle a large class of nonlinear stochastic partial differential equations (SPDEs in short), thanks to their probabilistic interpretation. In the last part, we study a new particle method in the context of shielding studies.

Random fields are frequently used in computational simulations of real-life processes. In particular, in this work they are used in modeling of flow and transport in porous media. Porous media as they arise in geological formations are intrinsically deterministic but there is significant uncertainty involved in determination of their properties such as permeability, porosity and diffusivity. In many situations description of properties of the porous media is aided by a limited number of observations at fixed points. These observations constrain the randomness of the field and lead to conditional simulations. In this work we propose a method of simulating the random fields which respect the observed data. An advantage of our method is that in the case that additional data becomes available it can be easily incorporated into subsequent representations. The proposed method is based on infinite series representations of random fields. We provide truncation error estimates which bound the discrepancy between the truncated series and the random field. We additionally provide the expansions for some processes that have not yet appeared in the literature. There are several approaches to efficient numerical computations for partial differential equations with random parameters. In this work we compare the solutions of flow and transport equations obtained by conditional simulations with Monte Carlo (MC) and stochastic collocation (SC) methods. Due to its simplicity MC method is one of the most popular methods used for the solution of stochastic equations. However, it is computationally expensive. The SC method is functionally similar to the MC method but it provides the faster convergence of the statistical moments of the solutions through the use of the carefully chosen collocation points at which the flow and transport equations are solved. We show that for both methods the conditioning on measurements helps to reduce the uncertainty of the solutions of the flow and transport equations. This especially holds in the neighborhood of the conditioning points. Conditioning reduces the variances of solutions helping to quantify the uncertainty in the output of the flow and transport equations. / Graduation date: 2013