We prove strong convergence of order 1/4 - E for arbitrarily small E > 0 of the Euler-Maruyama method for multidimensional stochastic differential equations (SDEs) with discontinuous drift and degenerate diffusion coefficient. The proof is based on estimating the difference between the Euler-Maruyama scheme and another numerical method, which is constructed by applying the Euler-Maruyama scheme to a transformation of the SDE we aim to solve.

We investigate a certain epidemics model, with and without noise. Some parameter analysis is performed together with computer simulations. The model was presented in Iacus (2008).

A stochastic differential equationdriven by a Brownian motion where the dispersion is determined by a parameter is considered. The parameter undergoes a change at a certain time point. Estimates of the time change point and the parameter, before and after that time, is considered.The estimates were presented in Lacus 2008. Two cases are considered: (1) the drift is known, (2) the drift is unknown and the dispersion space-independent. Applications to Dow-Jones index 1971-1974  and Goldmann-Sachs closings 2005-- May 2009 are given.

<p>We investigate a certain epidemics model, with and without noise. Some parameter analysis is performed together with computer simulations. The model was presented in Iacus (2008).</p>

<p>A stochastic differential equationdriven by a Brownian motion where the dispersion is determined by a parameter is considered. The parameter undergoes a change at a certain time point. Estimates of the time change point and the parameter, before and after that time, is considered.The estimates were presented in Lacus 2008. Two cases are considered: (1) the drift is known, (2) the drift is unknown and the dispersion space-independent. Applications to Dow-Jones index 1971-1974  and Goldmann-Sachs closings 2005-- May 2009 are given.</p>

This thesis concerns modelling of Coulomb collisions in toroidal plasma with Monte Carlo operators, which is important for many applications such as heating, current drive and collisional transport in fusion plasmas. Collisions relax the distribution functions towards local isotropic ones and transfer power to the background species when they are perturbed e.g. by wave-particle interactions or injected beams. The evolution of the distribution function in phase space, due to the Coulomb scattering on background ions and electrons and the interaction with RF waves, can be obtained by solving a Fokker-Planck equation.The coupling between spatial and velocity coordinates in toroidal plasmas correlates the spatial diffusion with the pitch angle scattering by Coulomb collisions. In many applications the diffusion coefficients go to zero at the boundaries or in a part of the domain, which makes the SDE singular. To solve such SDEs or equivalent diffusion equations with Monte Carlo methods, we have proposed a new method, the hybrid method, as well as an adaptive method, which selects locally the faster method from the drift and diffusion coefficients. The proposed methods significantly reduce the computational efforts and improves the convergence. The radial diffusion changes rapidly when crossing the trapped-passing boundary creating a boundary layer. To solve this problem two methods are proposed. The first one is to use a non-standard drift term in the Monte Carlo equation. The second is to symmetrize the flux across the trapped passing boundary. Because of the coupling between the spatial and velocity coordinates drift terms associated with radial gradients in density, temperature and fraction of the trapped particles appear. In addition an extra drift term has been included to relax the density profile to a prescribed one. A simplified RF-operator in combination with the collision operator has been used to study the relaxation of a heated distribution function. Due to RF-heating the density of thermal ions is reduced by the formation of a high-energy tail in the distribution function. The Coulomb collisions tries to restore the density profile and thus generates an inward diffusion of thermal ions that results in a peaking of the total density profile of resonant ions. / <p>QC 20130415</p>

The objective of this thesis is to develop a novel methodology of fractional stochastic dynamics to study stochastic stability of viscoelastic systems under stochastic loadings. Numerous structures in civil engineering are driven by dynamic forces, such as seismic and wind loads, which can be described satisfactorily only by using probabilistic models, such as white noise processes, real noise processes, or bounded noise processes. Viscoelastic materials exhibit time-dependent stress relaxation and creep; it has been shown that fractional calculus provide a unique and powerful mathematical tool to model such a hereditary property. Investigation of stochastic stability of viscoelastic systems with fractional calculus frequently leads to a parametrized family of fractional stochastic differential equations of motion. Parametric excitation may cause parametric resonance or instability, which is more dangerous than ordinary resonance as it is characterized by exponential growth of the response amplitudes even in the presence of damping. The Lyapunov exponents and moment Lyapunov exponents provide not only the information about stability or instability of stochastic systems, but also how rapidly the response grows or diminishes with time. Lyapunov exponents characterizes sample stability or instability. However, this sample stability cannot assure the moment stability. Hence, to obtain a complete picture of the dynamic stability, it is important to study both the top Lyapunov exponent and the moment Lyapunov exponent. Unfortunately, it is very difficult to obtain the accurate values of theses two exponents. One has to resort to numerical and approximate approaches. The main contributions of this thesis are: (1) A new numerical simulation method is proposed to determine moment Lyapunov exponents of fractional stochastic systems, in which three steps are involved: discretization of fractional derivatives, numerical solution of the fractional equation, and an algorithm for calculating Lyapunov exponents from small data sets. (2) Higher-order stochastic averaging method is developed and applied to investigate stochastic stability of fractional viscoelastic single-degree-of-freedom structures under white noise, real noise, or bounded noise excitation. (3) For two-degree-of-freedom coupled non-gyroscopic and gyroscopic viscoelastic systems under random excitation, the Stratonovich equations of motion are set up, and then decoupled into four-dimensional Ito stochastic differential equations, by making use of the method of stochastic averaging for the non-viscoelastic terms and the method of Larionov for viscoelastic terms. An elegant scheme for formulating the eigenvalue problems is presented by using Khasminskii and Wedig’s mathematical transformations from the decoupled Ito equations. Moment Lyapunov exponents are approximately determined by solving the eigenvalue problems through Fourier series expansion. Stability boundaries, critical excitations, and stability index are obtained. The effects of various parameters on the stochastic stability of the system are discussed. Parametric resonances are studied in detail. Approximate analytical results are confirmed by numerical simulations.

Cette thèse traite des équations différentielles stochastiques rétrogrades réfléchies du second ordre dans une filtration générale . Nous avons traité tout d'abord la réflexion à une barrière inférieure puis nous avons étendu le résultat dans le cas d'une barrière supérieure. Notre contribution consiste à démontrer l'existence et l'unicité de la solution de ces équations dans le cadre d'une filtration générale sous des hypothèses faibles. Nous remplaçons la régularité uniforme par la régularité de type Borel. Le principe de programmation dynamique pour le problème de contrôle stochastique robuste est donc démontré sous les hypothèses faibles c'est à dire sans régularité sur le générateur, la condition terminal et la barrière. Dans le cadre des Équations Différentielles Stochastiques Rétrogrades (EDSRs ) standard, les problèmes de réflexions à barrières inférieures et supérieures sont symétriques. Par contre dans le cadre des EDSRs de second ordre, cette symétrie n'est plus valable à cause des la non linéarité de l'espérance sous laquelle est définie notre problème de contrôle stochastique robuste non dominé. Ensuite nous un schéma d'approximation numérique d'une classe d'EDSR de second ordre réfléchies. En particulier nous montrons la convergence de schéma et nous testons numériquement les résultats obtenus. / This thesis deals with the second-order reflected backward stochastic differential equations (2RBSDEs) in general filtration. In the first part , we consider the reflection with a lower obstacle and then extended the result in the case of an upper obstacle . Our main contribution consists in demonstrating the existence and the uniqueness of the solution of these equations defined in the general filtration under weak assumptions. We replace the uniform regularity by the Borel regularity(through analytic measurability). The dynamic programming principle for the robust stochastic control problem is thus demonstrated under weak assumptions, that is to say without regularity on the generator, the terminal condition and the obstacle. In the standard Backward Stochastic Differential Equations (BSDEs) framework, there is a symmetry between lower and upper obstacles reflection problem. On the contrary, in the context of second order BSDEs, this symmetry is no longer satisfy because of the nonlinearity of the expectation under which our robust stochastic non-dominated stochastic control problem is defined. In the second part , we get a numerical approximation scheme of a class of second-order reflected BSDEs. In particular we show the convergence of our scheme and we test numerically the results.

En physique statistique, l’information macroscopique d’intérêt pour les systèmes considérés peut être dé-duite à partir de moyennes sur des configurations microscopiques réparties selon des mesures de probabilitéµ caractérisant l’état thermodynamique du système. En raison de la haute dimensionnalité du système (quiest proportionnelle au nombre de particules), les configurations sont le plus souvent échantillonnées en util-isant des trajectoires d’équations différentielles stochastiques ou des chaînes de Markov ergodiques pourla mesure de Boltzmann-Gibbs µ, qui décrit un système à température constante. Un processus stochas-tique classique permettant d’échantillonner cette mesure est la dynamique de Langevin. En pratique, leséquations de la dynamique de Langevin ne peuvent pas être intégrées analytiquement, la solution est alorsapprochée par un schéma numérique. L’analyse numérique de ces schémas de discrétisation est maintenantbien maîtrisée pour l’énergie cinétique quadratique standard. Une limitation importante des estimateurs desmoyennes sontleurs éventuelles grandes erreurs statistiques.Sous certaines hypothèsessur lesénergies ciné-tique et potentielle, il peut être démontré qu’un théorème de limite central est vrai. La variance asymptotiquepeut être grande en raison de la métastabilité du processus de Langevin, qui se produit dès que la mesure deprobabilité µ est multimodale.Dans cette thèse, nous considérons la discrétisation de la dynamique de Langevin modifiée qui améliorel’échantillonnage de la distribution de Boltzmann-Gibbs en introduisant une fonction cinétique plus généraleà la place de la formulation quadratique standard. Nous avons en fait deux situations en tête : (a) La dy-namique de Langevin Adaptativement Restreinte, où l’énergie cinétique s’annule pour les faibles moments,et correspond à l’énergie cinétique standard pour les forts moments. L’intérêt de cette dynamique est que lesparticules avec une faible énergie sont restreintes. Le gain vient alors du fait que les interactions entre lesparticules restreintes ne doivent pas être mises à jour. En raison de la séparabilité des positions et des mo-ments marginaux de la distribution, les moyennes des observables qui dépendent de la variable de positionsont égales à celles calculées par la dynamique de Langevin standard. L’efficacité de cette méthode résidedans le compromis entre le gain de calcul et la variance asymptotique des moyennes ergodiques qui peutaugmenter par rapport à la dynamique standards car il existe a priori plus des corrélations dans le tempsen raison de particules restreintes. De plus, étant donné que l’énergie cinétique est nulle sur un ouvert, ladynamique de Langevin associé ne parvient pas à être hypoelliptique. La première tâche de cette thèse est deprouver que la dynamique de Langevin avec une telle énergie cinétique est ergodique. L’étape suivante con-siste à présenter une analyse mathématique de la variance asymptotique de la dynamique AR-Langevin. Afinde compléter l’analyse de ce procédé, on estime l’accélération algorithmique du coût d’une seule itération,en fonction des paramètres de la dynamique. (b) Nous considérons aussi la dynamique de Langevin avecdes énergies cinétiques dont la croissance est plus que quadratique à l’infini, dans une tentative de réduire lamétastabilité. La liberté supplémentaire fournie par le choix de l’énergie cinétique doit être utilisée afin deréduire la métastabilité de la dynamique. Dans cette thèse, nous explorons le choix de l’énergie cinétique etnous démontrons une convergence améliorée des moyennes ergodiques sur un exemple de faible dimension.Un des problèmes avec les situations que nous considérons est la stabilité des régimes discrétisés. Afind’obtenir une méthode de discrétisation faiblement cohérente d’ordre 2 (ce qui n’est plus trivial dans le casde l’énergie cinétique générale), nous nous reposons sur les schémas basés sur des méthodes de Metropolis. / In statistical physics, the macroscopic information of interest for the systems under consideration can beinferred from averages over microscopic configurations distributed according to probability measures µcharacterizing the thermodynamic state of the system. Due to the high dimensionality of the system (whichis proportional to the number of particles), these configurations are most often sampled using trajectories ofstochastic differential equations or Markov chains ergodic for the probability measure µ, which describesa system at constant temperature. One popular stochastic process allowing to sample this measure is theLangevin dynamics. In practice, the Langevin dynamics cannot be analytically integrated, its solution istherefore approximated with a numerical scheme. The numerical analysis of such discretization schemes isby now well-understood when the kinetic energy is the standard quadratic kinetic energy.One important limitation of the estimators of the ergodic averages are their possibly large statisticalerrors.Undercertainassumptionsonpotentialandkineticenergy,itcanbeshownthatacentrallimittheoremholds true. The asymptotic variance may be large due to the metastability of the Langevin process, whichoccurs as soon as the probability measure µ is multimodal.In this thesis, we consider the discretization of modified Langevin dynamics which improve the samplingof the Boltzmann–Gibbs distribution by introducing a more general kinetic energy function U instead of thestandard quadratic one. We have in fact two situations in mind:(a) Adaptively Restrained (AR) Langevin dynamics, where the kinetic energy vanishes for small momenta,while it agrees with the standard kinetic energy for large momenta. The interest of this dynamics isthat particles with low energy are restrained. The computational gain follows from the fact that theinteractions between restrained particles need not be updated. Due to the separability of the positionand momenta marginals of the distribution, the averages of observables which depend on the positionvariable are equal to the ones computed with the standard Langevin dynamics. The efficiency of thismethod lies in the trade-off between the computational gain and the asymptotic variance on ergodic av-erages which may increase compared to the standard dynamics since there are a priori more correlationsin time due to restrained particles. Moreover, since the kinetic energy vanishes on some open set, theassociated Langevin dynamics fails to be hypoelliptic. In fact, a first task of this thesis is to prove thatthe Langevin dynamics with such modified kinetic energy is ergodic. The next step is to present a math-ematical analysis of the asymptotic variance for the AR-Langevin dynamics. In order to complementthe analysis of this method, we estimate the algorithmic speed-up of the cost of a single iteration, as afunction of the parameters of the dynamics.(b) We also consider Langevin dynamics with kinetic energies growing more than quadratically at infinity,in an attempt to reduce metastability. The extra freedom provided by the choice of the kinetic energyshould be used in order to reduce the metastability of the dynamics. In this thesis, we explore thechoice of the kinetic energy and we demonstrate on a simple low-dimensional example an improvedconvergence of ergodic averages.An issue with the situations we consider is the stability of discretized schemes. In order to obtain aweakly consistent method of order 2 (which is no longer trivial for a general kinetic energy), we rely on therecently developped Metropolis schemes.

Title: Stochastic evolution equations with multiplicative fractional noise Author: Jana Šnupárková Departement: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc. Supervisor's e-mail address: maslow@karlin.mff.cuni.cz Abstract: The fractional Gaussian noise is a formal derivative of a fractional Brownian motion with Hurst parameter H ∈ (0, 1). An explicit formula for a solution to stochastic differential equations with a multiplicative fractional Gaussian noise in a separable Hilbert space is given. The large time behaviour of the solution is studied. In addition, equations of this type with a nonlinear perturbation of a drift part are investigated in the case H > 1/2. Keywords: Fractional Brownian Motion, Stochastic Differential Equations in Hilbert Space, Explicit Formula for Solution