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1
Curry, Charles. "Algebraic structures in stochastic differential equations". Thesis, Heriot-Watt University, 2014. http://hdl.handle.net/10399/2791.
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We define a new numerical integration scheme for stochastic differential equations driven by Levy processes with uniformly lower mean square remainder than that of the scheme of the same strong order of convergence obtained by truncating the stochastic Taylor series. In doing so we generalize recent results concerning stochastic differential equations driven by Wiener processes. The aforementioned works studied integration schemes obtained by applying an invertible mapping to the stochastic Taylor series, truncating the resulting series and applying the inverse of the original mapping. The shuffle Hopf algebra and its associated convolution algebra play important roles in the their analysis, arising from the combinatorial structure of iterated Stratonovich integrals. It was recently shown that the algebra generated by iterated It^o integrals of independent Levy processes is isomorphic to a quasi-shuffle algebra. We utilise this to consider map-truncate-invert schemes for Levy processes. To facilitate this, we derive a new form of stochastic Taylor expansion from those of Wagner & Platen, enabling us to extend existing algebraic encodings of integration schemes. We then derive an alternative method of computing map-truncate-invert schemes using a single step, resolving diffculties encountered at the inversion step in previous methods.
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Dabrowski, Yoann. "Free entropies, free Fisher information, free stochastic differential equations, with applications to Von Neumann algebras". Thesis, Paris Est, 2010. http://www.theses.fr/2010PEST1015.
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Ce travail étend nos connaissances des entropies libres et des équations différentielles stochastiques (EDS) libres dans trois directions. Dans un premier temps, nous montrons que l'algèbre de von Neumann engendrée par au moins deux autoadjoints ayant une information de Fisher finie n'a pas la propriété $Gamma$ de Murray et von Neumann. C'est un analogue d'un résultat de Voiculescu pour l'entropie microcanonique libre. Dans un second temps, nous étudions des EDS libres à coefficients opérateurs non-bornés (autrement dit des sortes d' EDP stochastiques libres ). Nous montrons la stationnarité des solutions dans des cas particuliers. Nous en déduisons un calcul de la dimension entropique libre microcanonique dans le cas d'une information de Fisher lipschitzienne. Dans un troisième et dernier temps, nous introduisons une méthode générale de résolutions d'EDS libres stationnaires, s'appuyant sur un analogue non-commutatif d'un espace de chemins. En définissant des états traciaux sur cet analogue, nous construisons des dilatations markoviennes de nombreux semigroupes complètement markoviens sur une algèbre de von Neumann finie, en particulier de tous les semigroupes symétriques. Pour des semigroupes particuliers, par exemple dès que le générateur s'écrit sous une forme divergence pour une dérivation à valeur dans la correspondance grossière, ces dilatations résolvent des EDS libres. Entre autres applications, nous en déduisons une inégalité de Talagrand pour l'entropie non-microcanonique libre (relative à une sous-algèbre et une application complètement positive). Nous utilisons aussi ces déformations dans le cadre des techniques de déformations/rigidité de PopaThis works extends our knowledge of free entropies, free Fisher information and free stochastic differential equations in three directions. First, we prove that if a $W^{*}$-probability space generated by more than 2 self-adjoints with finite non-microstates free Fisher information doesn't have property $Gamma$ of Murray and von Neumann (especially is not amenable). This is an analogue of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy. Second, we study a general free stochastic differential equation with unbounded coefficients (``stochastic PDE"), and prove stationarity of solutions in well-chosen cases. This leads to a computation of microstates free entropy dimension in case of Lipschitz conjugate variable. Finally, we introduce a non-commutative path space approach to solve general stationary free Stochastic differential equations. By defining tracial states on a non-commutative analogue of a path space, we construct Markov dilations for a class of conservative completely Markov semigroups on finite von Neumann algebras. This class includes all symmetric semigroups. For well chosen semigroups (for instance with generator any divergence form operator associated to a derivation valued in the coarse correspondence) those dilations give rise to stationary solutions of certain free SDEs. Among applications, we prove a non-commutative Talagrand inequality for non-microstate free entropy (relative to a subalgebra $B$ and a completely positive map $eta:Bto B$). We also use those new deformations in conjunction with Popa's deformation/rigidity techniques, to get absence of Cartan subalgebra results
3
Ding, Jie. "Structural and fluid analysis for large scale PEPA models, with applications to content adaptation systems". Thesis, University of Edinburgh, 2010. http://hdl.handle.net/1842/7975.
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The stochastic process algebra PEPA is a powerful modelling formalism for concurrent systems, which has enjoyed considerable success over the last decade. Such modelling can help designers by allowing aspects of a system which are not readily tested, such as protocol validity and performance, to be analysed before a system is deployed. However, model construction and analysis can be challenged by the size and complexity of large scale systems, which consist of large numbers of components and thus result in state-space explosion problems. Both structural and quantitative analysis of large scale PEPA models suffers from this problem, which has limited wider applications of the PEPA language. This thesis focuses on developing PEPA, to overcome the state-space explosion problem, and make it suitable to validate and evaluate large scale computer and communications systems, in particular a content adaption framework proposed by the Mobile VCE. In this thesis, a new representation scheme for PEPA is proposed to numerically capture the structural and timing information in a model. Through this numerical representation, we have found that there is a Place/Transition structure underlying each PEPA model. Based on this structure and the theories developed for Petri nets, some important techniques for the structural analysis of PEPA have been given. These techniques do not suffer from the state-space explosion problem. They include a new method for deriving and storing the state space and an approach to finding invariants which can be used to reason qualitatively about systems. In particular, a novel deadlock-checking algorithm has been proposed to avoid the state-space explosion problem, which can not only efficiently carry out deadlock-checking for a particular system but can tell when and how a system structure lead to deadlocks. In order to avoid the state-space explosion problem encountered in the quantitative analysis of a large scale PEPA model, a fluid approximation approach has recently been proposed, which results in a set of ordinary differential equations (ODEs) to approximate the underlying CTMC. This thesis presents an improved mapping from PEPA to ODEs based on the numerical representation scheme, which extends the class of PEPA models that can be subjected to fluid approximation. Furthermore, we have established the fundamental characteristics of the derived ODEs, such as the existence, uniqueness, boundedness and nonnegativeness of the solution. The convergence of the solution as time tends to infinity for several classes of PEPA models, has been proved under some mild conditions. For general PEPA models, the convergence is proved under a particular condition, which has been revealed to relate to some famous constants of Markov chains such as the spectral gap and the Log-Sobolev constant. This thesis has established the consistency between the fluid approximation and the underlying CTMCs for PEPA, i.e. the limit of the solution is consistent with the equilibrium probability distribution corresponding to a family of underlying density dependent CTMCs. These developments and investigations for PEPA have been applied to both qualitatively and quantitatively evaluate the large scale content adaptation system proposed by the Mobile VCE. These analyses provide an assessment of the current design and should guide the development of the system and contribute towards efficient working patterns and system optimisation.
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Tribastone, Mirco. "Scalable analysis of stochastic process algebra models". Thesis, University of Edinburgh, 2010. http://hdl.handle.net/1842/4629.
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The performance modelling of large-scale systems using discrete-state approaches is fundamentally hampered by the well-known problem of state-space explosion, which causes exponential growth of the reachable state space as a function of the number of the components which constitute the model. Because they are mapped onto continuous-time Markov chains (CTMCs), models described in the stochastic process algebra PEPA are no exception. This thesis presents a deterministic continuous-state semantics of PEPA which employs ordinary differential equations (ODEs) as the underlying mathematics for the performance evaluation. This is suitable for models consisting of large numbers of replicated components, as the ODE problem size is insensitive to the actual population levels of the system under study. Furthermore, the ODE is given an interpretation as the fluid limit of a properly defined CTMC model when the initial population levels go to infinity. This framework allows the use of existing results which give error bounds to assess the quality of the differential approximation. The computation of performance indices such as throughput, utilisation, and average response time are interpreted deterministically as functions of the ODE solution and are related to corresponding reward structures in the Markovian setting. The differential interpretation of PEPA provides a framework that is conceptually analogous to established approximation methods in queueing networks based on meanvalue analysis, as both approaches aim at reducing the computational cost of the analysis by providing estimates for the expected values of the performance metrics of interest. The relationship between these two techniques is examined in more detail in a comparison between PEPA and the Layered Queueing Network (LQN) model. General patterns of translation of LQN elements into corresponding PEPA components are applied to a substantial case study of a distributed computer system. This model is analysed using stochastic simulation to gauge the soundness of the translation. Furthermore, it is subjected to a series of numerical tests to compare execution runtimes and accuracy of the PEPA differential analysis against the LQN mean-value approximation method. Finally, this thesis discusses the major elements concerning the development of a software toolkit, the PEPA Eclipse Plug-in, which offers a comprehensive modelling environment for PEPA, including modules for static analysis, explicit state-space exploration, numerical solution of the steady-state equilibrium of the Markov chain, stochastic simulation, the differential analysis approach herein presented, and a graphical framework for model editing and visualisation of performance evaluation results.
5
Bringuier, Hugo. "Marches quantiques ouvertes". Thesis, Toulouse 3, 2018. http://www.theses.fr/2018TOU30064/document.
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Cette thèse est consacrée à l'étude de modèles stochastiques associés aux systèmes quantiques ouverts. Plus particulièrement, nous étudions les marches quantiques ouvertes qui sont les analogues quantiques des marches aléatoires classiques. La première partie consiste en une présentation générale des marches quantiques ouvertes. Nous présentons les outils mathématiques nécessaires afin d'étudier les systèmes quantiques ouverts, puis nous exposons les modèles discrets et continus des marches quantiques ouvertes. Ces marches sont respectivement régies par des canaux quantiques et des opérateurs de Lindblad. Les trajectoires quantiques associées sont quant à elles données par des chaînes de Markov et des équations différentielles stochastiques avec sauts. La première partie s'achève avec la présentation de quelques pistes de recherche qui sont le problème de Dirichlet pour les marches quantiques ouvertes et les théorèmes asymptotiques pour les mesures quantiques non destructives. La seconde partie rassemble les articles rédigés durant cette thèse. Ces articles traîtent les sujets associés à l'irréductibilité, à la dualité récurrence-transience, au théorème central limite et au principe de grandes déviations pour les marches quantiques ouvertes à temps continuThis thesis is devoted to the study of stochastic models derived from open quantum systems. In particular, this work deals with open quantum walks that are the quantum analogues of classical random walks. The first part consists in giving a general presentation of open quantum walks. The mathematical tools necessary to study open quan- tum systems are presented, then the discrete and continuous time models of open quantum walks are exposed. These walks are respectively governed by quantum channels and Lindblad operators. The associated quantum trajectories are given by Markov chains and stochastic differential equations with jumps. The first part concludes with discussions over some of the research topics such as the Dirichlet problem for open quantum walks and the asymptotic theorems for quantum non demolition measurements. The second part collects the articles written within the framework of this thesis. These papers deal with the topics associated to the irreducibility, the recurrence-transience duality, the central limit theorem and the large deviations principle for continuous time open quantum walks
6
Trenn, Stephan. "Distributional differential algebraic equations". Ilmenau Univ.-Verl, 2009. http://d-nb.info/99693197X/04.
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Bahar, Arifah. "Applications of stochastic differential equations and stochastic delay differential equations in population dynamics". Thesis, University of Strathclyde, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.415294.
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Dareiotis, Anastasios Constantinos. "Stochastic partial differential and integro-differential equations". Thesis, University of Edinburgh, 2015. http://hdl.handle.net/1842/14186.
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In this work we present some new results concerning stochastic partial differential and integro-differential equations (SPDEs and SPIDEs) that appear in non-linear filtering. We prove existence and uniqueness of solutions of SPIDEs, we give a comparison principle and we suggest an approximation scheme for the non-local integral operators. Regarding SPDEs, we use techniques motivated by the work of De Giorgi, Nash, and Moser, in order to derive global and local supremum estimates, and a weak Harnack inequality.
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Abourashchi, Niloufar. "Stability of stochastic differential equations". Thesis, University of Leeds, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.509828.
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Zhang, Qi. "Stationary solutions of stochastic partial differential equations and infinite horizon backward doubly stochastic differential equations". Thesis, Loughborough University, 2008. https://dspace.lboro.ac.uk/2134/34040.
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In this thesis we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. For this, we prove the existence and uniqueness of the L2ρ (Rd; R1) × L2ρ (Rd; Rd) valued solutions of BDSDEs with Lipschitz nonlinear term on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary weak solutions (independent of any initial value) of SPDEs. Also the L2ρ (Rd; R1) × L2ρ (Rd; Rd) valued BDSDE with non-Lipschitz term is considered. Moreover, we verify the time and space continuity of solutions of real-valued BDSDEs, so obtain the stationary stochastic viscosity solutions of real-valued SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.
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Trenn, Stephan [Verfasser]. "Distributional differential algebraic equations / von Stephan Trenn". Ilmenau : Univ.-Verl, 2009. http://d-nb.info/998021652/34.
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Mu, Tingshu. "Backward stochastic differential equations and applications : optimal switching, stochastic games, partial differential equations and mean-field". Thesis, Le Mans, 2020. http://www.theses.fr/2020LEMA1023.
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Cette thèse est relative aux Equations Différentielles Stochastique Rétrogrades (EDSRs) réfléchies avec deux obstacles et leurs applications aux jeux de switching de somme nulle, aux systèmes d’équations aux dérivées partielles, aux problèmes de mean-field. Il y a deux parties dans cette thèse. La première partie porte sur le switching optimal stochastique et est composée de deux travaux. Dans le premier travail, nous montrons l’existence de la solution d’un système d’EDSR réfléchies à obstacles bilatéraux interconnectés dans le cadre probabiliste général. Ce problème est lié à un jeu de switching de somme nulle. Ensuite nous abordons la question de l’unicité de la solution. Et enfin nous appliquons les résultats obtenus pour montrer que le système d’EDP associé à une unique solution au sens viscosité, sans la condition de monotonie habituelle. Dans le second travail, nous considérons aussi un système d’EDSRs réfléchies à obstacles bilatéraux interconnectés dans le cadre markovien. La différence avec le premier travail réside dans le fait que le switching ne s’opère pas de la même manière. Cette fois-ci quand le switching est opéré, le système est mis dans l’état suivant importe peu lequel des joueurs décide de switcher. Cette différence est fondamentale et complique singulièrement le problème de l’existence de la solution du système. Néanmoins, dans le cadre markovien nous montrons cette existence et donnons un résultat d’unicité en utilisant principalement la méthode de Perron. Ensuite, le lien avec un jeu de switching spécifique est établi dans deux cadres. Dans la seconde partie nous étudions les EDSR réfléchies unidimensionnelles à deux obstacles de type mean-field. Par la méthode du point fixe, nous montrons l’existence et l’unicité de la solution dans deux cadres, en fonction de l’intégrabilité des donnéesThis thesis is related to Doubly Reflected Backward Stochastic Differential Equations (DRBSDEs) with two obstacles and their applications in zero-sum stochastic switching games, systems of partial differential equations, mean-field problems.There are two parts in this thesis. The first part deals with optimal stochastic switching and is composed of two works. In the first work we prove the existence of the solution of a system of DRBSDEs with bilateral interconnected obstacles in a probabilistic framework. This problem is related to a zero-sum switching game. Then we tackle the problem of the uniqueness of the solution. Finally, we apply the obtained results and prove that, without the usual monotonicity condition, the associated PDE system has a unique solution in viscosity sense. In the second work, we also consider a system of DRBSDEs with bilateral interconnected obstacles in the markovian framework. The difference between this work and the first one lies in the fact that switching does not work in the same way. In this second framework, when switching is operated, the system is put in the following state regardless of which player decides to switch. This difference is fundamental and largely complicates the problem of the existence of the solution of the system. Nevertheless, in the Markovian framework we show this existence and give a uniqueness result by the Perron’s method. Later on, two particular switching games are analyzed.In the second part we study a one-dimensional Reflected BSDE with two obstacles of mean-field type. By the fixed point method, we show the existence and uniqueness of the solution in connection with the integrality of the data
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Saravi, Masoud. "Numerical solution of linear ordinary differential equations and differential-algebraic equations by spectral methods". Thesis, Open University, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.446280.
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This thesis involves the implementation of spectral methods, for numerical solution of linear Ordinary Differential Equations (ODEs) and linear Differential-Algebraic Equations (DAEs). First we consider ODEs with some ordinary problems, and then, focus on those problems in which the solution function or some coefficient functions have singularities. Then, by expressing weak and strong aspects of spectral methods to solve these kinds of problems, a modified pseudospectral method which is more efficient than other spectral methods is suggested and tested on some examples. We extend the pseudo-spectral method to solve a system of linear ODEs and linear DAEs and compare this method with other methods such as Backward Difference Formulae (BDF), and implicit Runge-Kutta (RK) methods using some numerical examples. Furthermore, by using appropriatec hoice of Gauss-Chebyshev-Radapuo ints, we will show that this method can be used to solve a linear DAE whenever some of coefficient functions have singularities by providing some examples. We also used some problems that have already been considered by some authors by finite difference methods, and compare their results with ours. Finally, we present a short survey of properties and numerical methods for solving DAE problems and then we extend the pseudo-spectral method to solve DAE problems with variable coefficient functions. Our numerical experience shows that spectral and pseudo-spectral methods and their modified versions are very promising for linear ODE and linear DAE problems with solution or coefficient functions having singularities. In section 3.2, a modified method for solving an ODE is introduced which is new work. Furthermore, an extension of this method for solving a DAE or system of ODEs which has been explained in section 4.6 of chapter four is also a new idea and has not been done by anyone previously. In all chapters, wherever we talk about ODE or DAE we mean linear.
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Rassias, Stamatiki. "Stochastic functional differential equations and applications". Thesis, University of Strathclyde, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.486536.
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The general truth that the principle of causality, that is, the future state of a system is independent of its past history, cannot support all the cases under consideration, leads to the introduction of the FDEs. However, the strong need of modelling real life problems, demands the inclusion of stochasticity. Thus, the appearance of the SFDEs (special case of which is the SDDEs) is necessary and definitely unavoidable. It has been almost a century since Langevin's model that the researchers incorporate noise terms into their work. Two of the main research interests are linked with the existence and uniqueness of the solution of the pertinent SFDE/SDDE which describes the problem under consideration, and the qualitative behaviour of the solution. This thesis, explores the SFDEs and their applications. According to the scientific literature, Ito's work (1940) contributed fundamentally into the formulation and study of the SFDEs. Khasminskii (1969), introduced a powerful test for SDEs to have non-explosion solutions without the satisfaction of the linear growth condition. Mao (2002), extended the idea so as to approach the SDDEs. However, Mao's test cannot be applied in specific types of SDDEs. Through our research work we establish an even more general Khasminskii-type test for SDDEs which covers a wide class of highly non-linear SDDEs. Following the proof of the non-explosion of the pertinent solution, we focus onto studying its qualitative behaviour by computing some moment and almost sure asymptotic estimations. In an attempt to apply and extend our theoretical results into real life problems we devote a big part of our research work into studying two very interesting problems that arise : from the area of the population dynamks and from·a problem related to the physical phenomenon of ENSO (EI Nino - Southern Oscillation)
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Hofmanová, Martina. "Degenerate parabolic stochastic partial differential equations". Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2013. http://tel.archives-ouvertes.fr/tel-00916580.
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In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyperbolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochastic forcing and study their approximations in the sense of Bhatnagar-Gross-Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkhod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition.
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Rajotte, Matthew. "Stochastic Differential Equations and Numerical Applications". VCU Scholars Compass, 2014. http://scholarscompass.vcu.edu/etd/3383.
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We will explore the topic of stochastic differential equations (SDEs) first by developing a foundation in probability theory and It\^o calculus. Formulas are then derived to simulate these equations analytically as well as numerically. These formulas are then applied to a basic population model as well as a logistic model and the various methods are compared. Finally, we will study a model for low dose anthrax exposure which currently implements a stochastic probabilistic uptake in a deterministic differential equation, and analyze how replacing the probablistic uptake with an SDE alters the dynamics.
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Nie, Tianyang. "Stochastic differential equations with constraints on the state : backward stochastic differential equations, variational inequalities and fractional viability". Thesis, Brest, 2012. http://www.theses.fr/2012BRES0047.
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Le travail de thèse est composé de trois thèmes principaux : le premier étudie l'existence et l'unicité pour des équations différentielles stochastiques (EDS) progressives-rétrogrades fortement couplées avec des opérateurs sous-différentiels dans les deux équations, dans l’équation progressive ainsi que l’équation rétrograde, et il discute également un nouveau type des inégalités variationnelles partielles paraboliques associées, avec deux opérateurs sous-différentiels, l’un agissant sur le domaine de l’état, l’autre sur le co-domaine. Le second thème est celui des EDS rétrogrades sans ainsi qu’avec opérateurs sous-différentiels, régies par un mouvement brownien fractionnaire avec paramètre de Hurst H> ½. Il étend de manière rigoureuse les résultats de Hu et Peng (SICON, 2009) aux inégalités variationnelles stochastiques rétrogrades. Enfin, le troisième thème met l’accent sur la caractérisation déterministe de la viabilité pour les EDS régies par un mouvement brownien fractionnaire. Ces trois thèmes de recherche mentionnés ci-dessus ont en commun d’étudier des EDS avec contraintes sur le processus d’état. Chacun des trois sujets est basé sur une publication et des manuscrits soumis pour publication, respectivementThis PhD thesis is composed of three main topics: The first one studies the existence and the uniqueness for fully coupled forward-backward stochastic differential equations (SDEs) with subdifferential operators in both the forward and the backward equations, and it discusses also a new type of associated parabolic partial variational inequalities with two subdifferential operators, one acting over the state domain and the other over the co-domain. The second topic concerns the investigation of backward SDEs without as well as with subdifferential operator, both driven by a fractional Brownian motion with Hurst parameter H> 1/2. It extends in a rigorous manner the results of Hu and Peng (SICON, 2009) to backward stochastic variational inequalities. Finally, the third topic focuses on a deterministic characterisation of the viability for SDEs driven by a fractional Brownian motion. The three research topics mentioned above have in common to study SDEs with state constraints. The discussion of each of the three topics is based on a publication and on submitted manuscripts, respectively
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Reich, Sebastian. "Differential-algebraic equations and applications in circuit theory". Universität Potsdam, 1992. http://opus.kobv.de/ubp/volltexte/2010/4664/.
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Technical and physical systems, especially electronic circuits, are frequently modeled as a system of differential and nonlinear implicit equations. In the literature such systems of equations are called differentialalgebraic equations (DAEs). It turns out that the numerical and analytical properties of a DAE depend on an integer called the index of the problem. For example, the well-known BDF method of Gear can be applied, in general, to a DAE only if the index does not exceed one. In this paper we give a geometric interpretation of higherindex DAEs and indicate problems arising in connection with such DAEs by means of several examples.Die mathematische Modellierung technisch physikalischer Systeme wie elektrische Netzwerke, führt häufig auf ein System von Differentialgleichungen und nichtlinearen impliziten Gleichungen sogenannten Algebrodifferentialgleichungen (ADGL). Es zeigt sich, daß die numerischen und analytischen Eigenschaften von ADGL durch den Index des Problems charakterisiert werden können. Insbesondere können die bekannten Integrationsformeln von Gear im allgemeinen nur auf ADGL mit dem Index eins angewendet werden. In diesem Beitrag wird eine geometrische Interpretation von ADGL mit einem höheren Index gegeben sowie auf Probleme im Zusammenhang mit derartigen ADGL an Hand verschiedener Beispiele hingewiesen.
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Reich, Sebastian. "On a geometrical interpretation of differential-algebraic equations". Universität Potsdam, 1990. http://opus.kobv.de/ubp/volltexte/2010/4668/.
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The subject of this paper is the relation of differential-algebraic equations (DAEs) to vector fields on manifolds. For that reason, we introduce the notion of a regular DAE as a DAE to which a vector field uniquely corresponds. Furthermore, a technique is described which yields a family of manifolds for a given DAE. This socalled family of constraint manifolds allows in turn the formulation of sufficient conditions for the regularity of a DAE. and the definition of the index of a regular DAE. We also state a method for the reduction of higher-index DAEs to lowsr-index ones that can be solved without introducing additional constants of integration. Finally, the notion of realizability of a given vector field by a regular DAE is introduced, and it is shown that any vector field can be realized by a regular DAE. Throughout this paper the problem of path-tracing is discussed as an illustration of the mathematical phenomena.
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Tidefelt, Henrik. "Differential-algebraic equations and matrix-valued singular perturbation". Doctoral thesis, Linköpings universitet, Reglerteknik, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-51653.
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With the arrival of modern component-based modeling tools for dynamic systems, the differential-algebraic equation form is increasing in popularity as it is general enough to handle the resulting models. However, if uncertainty is allowed in the equations — no matter how small — this thesis stresses that such equations generally become ill-posed. Rather than deeming the general differential-algebraic structure useless up front due to this reason, the suggested approach to the problem is to ask what assumptions that can be made in order to obtain well-posedness. Here, “well-posedness” is used in the sense that the uncertainty in the solutions should tend to zero as the uncertainty in the equations tends to zero. The main theme of the thesis is to analyze how the uncertainty in the solution to a differential-algebraic equation depends on the uncertainty in the equation. In particular, uncertainty in the leading matrix of linear differential-algebraic equations leads to a new kind of singular perturbation, which is referred to as “matrix-valued singular perturbation”. Though a natural extension of existing types of singular perturbation problems, this topic has not been studied in the past. As it turns out that assumptions about the equations have to be made in order to obtain well-posedness, it is stressed that the assumptions should be selected carefully in order to be realistic to use in applications. Hence, it is suggested that any assumptions (not counting properties which can be checked by inspection of the uncertain equations) should be formulated in terms of coordinate-free system properties. In the thesis, the location of system poles has been the chosen target for assumptions. Three chapters are devoted to the study of uncertain differential-algebraic equations and the associated matrix-valued singular perturbation problems. Only linear equations without forcing function are considered. For both time-invariant and time-varying equations of nominal differentiation index 1, the solutions are shown to converge as the uncertainties tend to zero. For time-invariant equations of nominal index 2, convergence has not been shown to occur except for an academic example. However, the thesis contains other results for this type of equations, including the derivation of a canonical form for the uncertain equations. While uncertainty in differential-algebraic equations has been studied in-depth, two related topics have been studied more passingly. One chapter considers the development of point-mass filters for state estimation on manifolds. The highlight is a novel framework for general algorithm development with manifold-valued variables. The connection to differential-algebraic equations is that one of their characteristics is that they have an underlying manifold-structure imposed on the solution. One chapter presents a new index closely related to the strangeness index of a differential-algebraic equation. Basic properties of the strangeness index are shown to be valid also for the new index. The definition of the new index is conceptually simpler than that of the strangeness index, hence making it potentially better suited for both practical applications and theoretical developments.
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Weickert, J. "Navier-Stokes equations as a differential-algebraic system". Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800942.
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Nonsteady Navier-Stokes equations represent a differential-algebraic system of strangeness index one after any spatial discretization. Since such systems are hard to treat in their original form, most approaches use some kind of index reduction. Processing this index reduction it is important to take care of the manifolds contained in the differential-algebraic equation (DAE). We investigate for several discretization schemes for the Navier-Stokes equations how the consideration of the manifolds is taken into account and propose a variant of solving these equations along the lines of the theoretically best index reduction. Applying this technique, the error of the time discretisation depends only on the method applied for solving the DAE.
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Hendriks, Peter Anne. "Algebraic aspects of linear differential and difference equations". [S.l. : [Groningen] : s.n.] ; [University Library Groningen] [Host], 1996. http://irs.ub.rug.nl/ppn/153769580.
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Tidefelt, Henrik. "Structural algorithms and perturbations in differential-algebraic equations". Licentiate thesis, Linköping : Department of Electrical Engineering, Linköpings universitet, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-9011.
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Wong, Kwok-kin, i 黃國堅. "Exact meromorphic solutions of complex algebraic differential equations". Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2012. http://hub.hku.hk/bib/B48330218.
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For any given complex algebraic ordinary differential equation (ODE), one major task of both pure and applied mathematicians is to find explicit meromorphic solutions due to their extensive applications in science.
In 2010, Conte and Ng in [12] proposed a new technique for solving complex algebraic ODEs. The method consists of an idea due to Eremenko in [20] and the subequation method of Conte and Musette, which was first proposed in [9].
Eremenko’s idea is to make use of the Nevanlinna theory to analyze the value distribution and growth rate of the solutions, from which one would be able to show that in some cases, all the meromorphic solutions of the studied differential equation are in a class of functions called “class W”, which consists of elliptic functions and their degenerates. The establishment of solutions is then achieved by the subequation method. The main idea is to build subequations which have solutions that also satisfy the original differential equation, hoping that the subequations will be easier to solve.
As in [12], the technique has been proven to be very successful in obtaining explicit particular meromorphic solutions as well as giving complete classification of meromorphic solutions. In this thesis, the necessary theoretical background, including the Nevanlinna theory and the subequation method, will be developed. The technique will then be applied to obtain all meromorphic stationary wave solutions of the real cubic Swift-Hohenberg equation (RCSH). This last part is joint work with Conte and Ng and will appear in Studies in Applied Mathematics [13].
RCSH is important in several studies in physics and engineering problems. For instance, RCSH is used as modeling equation for Rayleigh- B?nard convection in hydrodynamics [43] as well as in pattern formation [16]. Among the explicit stationary wave solutions obtained by the technique used in this thesis, one of them appears to be new and could be written down as a rational function composite with Weierstrass elliptic function.published_or_final_version
Mathematics
Master
Master of Philosophy
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Kashiwara, Masaki D'Agnolo Andrea Schneiders Jean-Pierre. "Algebraic study of systems of partial differential equations /". Marseille (BP 67, 13274 Cedex 9) ; [Paris] : Société mathématique de France, 1995. http://catalogue.bnf.fr/ark:/12148/cb37168718p.
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Reiss, Markus. "Nonparametric estimation for stochastic delay differential equations". [S.l.] : [s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=964782480.
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Yalman, Hatice. "Change Point Estimation for Stochastic Differential Equations". Thesis, Växjö University, School of Mathematics and Systems Engineering, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:vxu:diva-5748.
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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.
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Leng, Weng San. "Backward stochastic differential equations and option pricing". Thesis, University of Macau, 2003. http://umaclib3.umac.mo/record=b1447308.
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Tunc, Vildan. "Two Studies On Backward Stochastic Differential Equations". Master's thesis, METU, 2012. http://etd.lib.metu.edu.tr/upload/12614541/index.pdf.
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Backward stochastic differential equations appear in many areas of research including mathematical finance, nonlinear partial differential equations, financial economics and stochastic control. The first existence and uniqueness result for nonlinear backward stochastic differential equations was given by Pardoux and Peng (Adapted solution of a backward stochastic differential equation. System and Control Letters, 1990). They looked for an adapted pair of processes {x(t)y(t)}
t is in [0
1]} with values in Rd and Rd×
k respectively, which solves an equation of the form: x(t) + int_t^1 f(s,x(s),y(s))ds + int_t^1 [g(s,x(s)) + y(s)]dWs = X. This dissertation studies this paper in detail and provides all the steps of the proofs that appear in this seminal paper. In addition, we review (Cvitanic and Karatzas, Hedging contingent claims with constrained portfolios. The annals of applied probability, 1993). In this paper, Cvitanic and Karatzas studied the following problem: the hedging of contingent claims with portfolios constrained to take values in a given closed, convex set K. Processes intimately linked to BSDEs naturally appear in the formulation of the constrained hedging problem. The analysis of Cvitanic and Karatzas is based on a dual control problem. One of the contributions of this thesis is an algorithm that numerically solves this control problem in the case of constant volatility. The algorithm is based on discretization of time. The convergence proof is also provided.
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Zettervall, Niklas. "Multi-scale methods for stochastic differential equations". Thesis, Umeå universitet, Institutionen för fysik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-53704.
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Standard Monte Carlo methods are used extensively to solve stochastic differential equations. This thesis investigates a Monte Carlo (MC) method called multilevel Monte Carlo that solves the equations on several grids, each with a specific number of grid points. The multilevel MC reduces the computational cost compared to standard MC. When using a fixed computational cost the variance can be reduced by using the multilevel method compared to the standard one. Discretization and statistical error calculations are also being conducted and the possibility to evaluate the errors coupled with the multilevel MC creates a powerful numerical tool for calculating equations numerically. By using the multilevel MC method together with the error calculations it is possible to efficiently determine how to spend an extended computational budget.Standard Monte Carlo metoder används flitigt för att lösa stokastiska differentialekvationer. Denna avhandling undersöker en Monte Carlo-metod (MC) kallad multilevel Monte Carlo som löser ekvationerna på flera olika rutsystem, var och en med ett specifikt antal punkter. Multilevel MC reducerar beräkningskomplexiteten jämfört med standard MC. För en fixerad beräkningskoplexitet kan variansen reduceras genom att multilevel MC-metoden används istället för standard MC-metoden. Diskretiserings- och statistiska felberäkningar görs också och möjligheten att evaluera de olika felen, kopplat med multilevel MC-metoden skapar ett kraftfullt verktyg för numerisk beräkning utav ekvationer. Genom att använda multilevel MC tillsammans med felberäkningar så är det möjligt att bestämma hur en utökad beräkningsbudget speneras så effektivt som möjligt.
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Matetski, Kanstantsin. "Discretisations of rough stochastic partial differential equations". Thesis, University of Warwick, 2016. http://wrap.warwick.ac.uk/81460/.
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This thesis consists of two parts, in both of which we consider approximations of rough stochastic PDEs and investigate convergence properties of the approximate solutions. In the first part we use the theory of (controlled) rough paths to define a solution for one-dimensional stochastic PDEs of Burgers type driven by an additive space-time white noise. We prove that natural numerical approximations of these equations converge to the solution of a corrected continuous equation and that their optimal convergence rate in the uniform topology (in probability) is arbitrarily close to 1/2 . In the second part of the thesis we develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in time. As an application, we show that the dynamical �43 model on the dyadic grid converges after renormalisation to its continuous counterpart. This result in particular implies that, as expected, the �43 measure is invariant for this equation and that the lifetime of its solutions is almost surely infinite for almost every initial condition.
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Hashemi, Seyed Naser. "Singular perturbations in coupled stochastic differential equations". Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/NQ65244.pdf.
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Hollingsworth, Blane Jackson Schmidt Paul G. "Stochastic differential equations a dynamical systems approach /". Auburn, Ala, 2008. http://repo.lib.auburn.edu/EtdRoot/2008/SPRING/Mathematics_and_Statistics/Dissertation/Hollingsworth_Blane_43.pdf.
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Matsikis, Iakovos. "High gain control of stochastic differential equations". Thesis, University of Exeter, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.403248.
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Althubiti, Saeed. "STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE MEMORY". OpenSIUC, 2018. https://opensiuc.lib.siu.edu/dissertations/1544.
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In this dissertation, we discuss the existence and uniqueness of Ito-type stochastic functional differential equations with infinite memory using fixed point theorem technique. We also address the properties of the solution which are an upper bound for the pth moments of the solution and the Lp-regularity. Then, we provide an analysis to show the local asymptotic L2-stability of the trivial solution using fixed point theorem technique, and we give an approximation of the solution using Euler-Maruyama method providing the global error followed by simulating examples.
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Spantini, Alessio. "Preconditioning techniques for stochastic partial differential equations". Thesis, Massachusetts Institute of Technology, 2013. http://hdl.handle.net/1721.1/82507.
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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2013.This thesis was scanned as part of an electronic thesis pilot project.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 149-155).
This thesis is about preconditioning techniques for time dependent stochastic Partial Differential Equations arising in the broader context of Uncertainty Quantification. State-of-the-art methods for an efficient integration of stochastic PDEs require the solution field to lie on a low dimensional linear manifold. In cases when there is not such an intrinsic low rank structure we must resort on expensive and time consuming simulations. We provide a preconditioning technique based on local time stretching capable to either push or keep the solution field on a low rank manifold with substantial reduction in the storage and the computational burden. As a by-product we end up addressing also classical issues related to long time integration of stochastic PDEs.
by Alessio Spantini.
S.M.
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Kolli, Praveen C. "Topics in Rank-Based Stochastic Differential Equations". Research Showcase @ CMU, 2018. http://repository.cmu.edu/dissertations/1205.
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In this thesis, we tackle two problems. In the first problem, we study fluctuations of a system of diffusions interacting through the ranks when the number of diffusions goes to infinity. It is known that the empirical cumulative distribution function of such diffusions converges to a non-random limiting cumulative distribution function which satisfies the porous medium PDE. We show that the fluctuations of the empirical cumulative distribution function around its limit are governed by a suitable SPDE. In the second problem, we introduce common noise that has a rank preserving structure into systems of diffusions interacting through the ranks and study the behaviour of such diffusion processes as the number of diffusions goes to infinity. We show that the limiting distribution function is no longer deterministic and furthermore, it satisfies a suitable SPDE. iii
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Prerapa, Surya Mohan. "Projection schemes for stochastic partial differential equations". Thesis, University of Southampton, 2009. https://eprints.soton.ac.uk/342800/.
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The focus of the present work is to develop stochastic reduced basis methods (SRBMs) for solving partial differential equations (PDEs) defined on random domains and nonlinear stochastic PDEs (SPDEs). SRBMs have been extended in the following directions: Firstly, an h-refinement strategy referred to as Multi-Element-SRBMs (ME-SRBMs) is developed for local refinement of the solution process. The random space is decomposed into subdomains where SRBMs are employed in each subdomain resulting in local response statistics. These local statistics are subsequently assimilated to compute the global statistics. Two types of preconditioning strategies namely global and local preconditioning strategies are discussed due to their merits such as degree of parallelizability and better convergence trends. The improved accuracy and convergence trends of ME-SRBMs are demonstrated by numerical investigation of stochastic steady state elasticity and stochastic heat transfer applications. The second extension involves the development of a computational approach employing SRBMs for solving linear elliptic PDEs defined on random domains. The key idea is to carry out spatial discretization of the governing equations using finite element (FE) methods and mesh deformation strategies. This results in a linear random algebraic system of equations whose coefficients of expansion can be computed nonintrusively either at the element or the global level. SRBMs are subsequently applied to the linear random algebraic system of equations to obtain the response statistics. We establish conditions that the input uncertainty model must satisfy to ensure the well-posedness of the problem. The proposed formulation is demonstrated on two and three dimensional model problems with uncertain boundaries undergoing steady state heat transfer. A large scale study involving a three-dimensional gas turbine model with uncertain boundary, has been presented in this context. Finally, a numerical scheme that combines SRBMs with the Picard iteration scheme is proposed for solving nonlinear SPDEs. The governing equations are linearized using the response process from the previous iteration and spatially discretized. The resulting linear random algebraic system of equations are solved to obtain the new response process which acts as a guess for the next iteration. These steps of linearization, spatial discretization, solving the system of equations and updating the current guess are repeated until the desired accuracy is achieved. The effectiveness and the limitations of the formulation are demonstrated employing numerical studies in nonlinear heat transfer and the one-dimensional Burger’s equation.
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Liu, Ge. "Statistical Inference for Multivariate Stochastic Differential Equations". The Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1562966204796479.
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Gauthier, Genevieve Carleton University Dissertation Mathematics and Statistics. "Multilevel bilinear system of stochastic differential equations". Ottawa, 1995.
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Zhang, Xiling. "On numerical approximations for stochastic differential equations". Thesis, University of Edinburgh, 2017. http://hdl.handle.net/1842/28931.
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This thesis consists of several problems concerning numerical approximations for stochastic differential equations, and is divided into three parts. The first one is on the integrability and asymptotic stability with respect to a certain class of Lyapunov functions, and the preservation of the comparison theorem for the explicit numerical schemes. In general, those properties of the original equation can be lost after discretisation, but it will be shown that by some suitable modification of the Euler scheme they can be preserved to some extent while keeping the strong convergence rate maintained. The second part focuses on the approximation of iterated stochastic integrals, which is the essential ingredient for the construction of higher-order approximations. The coupling method is adopted for that purpose, which aims at finding a random variable whose law is easy to generate and is close to the target distribution. The last topic is motivated by the simulation of equations driven by Lévy processes, for which the main difficulty is to generalise some coupling results for the one-dimensional central limit theorem to the multi-dimensional case.
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Reiß, Markus. "Nonparametric estimation for stochastic delay differential equations". Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2002. http://dx.doi.org/10.18452/14741.
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Sei (X(t), t>= -r) ein stationärer stochastischer Prozess, der die affine stochastische Differentialgleichung mit Gedächtnis dX(t)=L(X(t+s))dt+sigma dW(t), t>= 0, löst, wobei sigma>0, (W(t), t>=0) eine Standard-Brownsche Bewegung und L ein stetiges lineares Funktional auf dem Raum der stetigen Funktionen auf [-r,0], dargestellt durch ein endliches signiertes Maß a, bezeichnet. Wir nehmen an, dass eine Trajektorie (X(t), -r 0, konvergiert. Diese Rate ist schlechter als in vielen klassischen Fällen. Wir beweisen jedoch eine untere Schranke, die zeigt, dass keine Schätzung eine bessere Rate im Minimax-Sinn aufweisen kann. Für zeit-diskrete Beobachtungen von maximalem Abstand Delta konvergiert die Galerkin-Schätzung immer noch mit obiger Rate, sofern Delta is in etwa von der Ordnung T^(-1/2). Hingegen wird bewiesen, dass für festes Delta unabhängig von T die Rate sich signifikant verschlechtern muss, indem eine untere Schranke von T^(-s/(2s+6)) gezeigt wird. Außerdem wird eine adaptive Schätzung basierend auf Wavelet-Thresholding-Techniken für das assoziierte schlechtgestellte Problem konstruiert. Diese nichtlineare Schätzung erreicht die obige Minimax-Rate sogar für die allgemeinere Klasse der Besovräume B^s_(p,infinity) mit p>max(6/(2s+3),1). Die Restriktion p>=max(6/(2s+3),1) muss für jede Schätzung gelten und ist damit inhärent mit dem Schätzproblem verknüpft. Schließlich wird ein Hypothesentest mit nichtparametrischer Alternative vorgestellt, der zum Beispiel für das Testen auf Gedächtnis verwendet werden kann. Dieser Test ist anwendbar für eine L^2-Trennungsrate zwischen Hypothese und Alternative der Ordnung T^(-s/(2s+2.5)). Diese Rate ist wiederum beweisbar optimal für jede mögliche Teststatistik. Für die Beweise müssen die Parameterabhängigkeit der stationären Lösungen sowie die Abbildungseigenschaften der assoziierten Kovarianzoperatoren detailliert bestimmt werden. Weitere Resultate von allgemeinem Interessen beziehen sich auf die Mischungseigenschaft der stationären Lösung, eine Fallstudie zu exponentiellen Gewichtsfunktionen sowie der Approximation des stationären Prozesses durch autoregressive Prozesse in diskreter Zeit.Let (X(t), t>= -r) be a stationary stochastic process solving the affine stochastic delay differential equation dX(t)=L(X(t+s))dt+sigma dW(t), t>= 0, with sigma>0, (W(t), t>=0) a standard one-dimensional Brownian motion and with a continuous linear functional L on the space of continuous functions on [-r,0], represented by a finite signed measure a. Assume that a trajectory (X(t), -r 0. This rate is worse than those obtained in many classical cases. However, we prove a lower bound, stating that no estimator can attain a better rate of convergence in a minimax sense. For discrete time observations of maximal distance Delta, the Galerkin estimator still attains the above asymptotic rate if Delta is roughly of order T^(-1/2). In contrast, we prove that for observation intervals Delta, with Delta independent of T, the rate must deteriorate significantly by providing the rate estimate T^(-s/(2s+6)) from below. Furthermore, we construct an adaptive estimator by applying wavelet thresholding techniques to the corresponding ill-posed inverse problem. This nonlinear estimator attains the above minimax rate even for more general classes of Besov spaces B^s_(p,infinity) with p>max(6/(2s+3),1). The restriction p >= 6/(2s+3) is shown to hold for any estimator, hence to be inherently associated with the estimation problem. Finally, a hypothesis test with a nonparametric alternative is constructed that could for instance serve to decide whether a trajectory has been generated by a stationary process with or without time delay. The test works for an L^2-separation rate between hypothesis and alternative of order T^(-s/(2s+2.5)). This rate is again shown to be optimal among all conceivable tests. For the proofs, the parameter dependence of the stationary solutions has to be studied in detail and the mapping properties of the associated covariance operators have to be determined exactly. Other results of general interest concern the mixing properties of the stationary solution, a case study for exponential weight functions and the approximation of the stationary process by discrete time autoregressive processes.
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Nguyen, Cu Ngoc. "Stochastic differential equations with long-memory input". Thesis, Queensland University of Technology, 2001.
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Zangeneh, Bijan Z. "Semilinear stochastic evolution equations". Thesis, University of British Columbia, 1990. http://hdl.handle.net/2429/31117.
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Let H be a separable Hilbert space. Suppose (Ω, F, Ft, P) is a complete stochastic basis with a right continuous filtration and {Wt,t ∈ R} is an H-valued cylindrical Brownian motion with respect to {Ω, F, Ft, P). U(t, s) denotes an almost strong evolution operator generated by a family of unbounded closed linear operators on H. Consider the semilinear stochastic integral equation
[formula omitted]
where
• f is of monotone type, i.e., ft(.) = f(t, w,.) : H → H is semimonotone, demicon-tinuous, uniformly bounded, and for each x ∈ H, ft(x) is a stochastic process which satisfies certain measurability conditions.
• gs(.) is a uniformly-Lipschitz predictable functional with values in the space of Hilbert-Schmidt operators on H.
• Vt is a cadlag adapted process with values in H.
• X₀ is a random variable.
We obtain existence, uniqueness, boundedness of the solution of this equation. We show the solution of this equation changes continuously when one or all of X₀, f, g, and V are varied. We apply this result to find stationary solutions of certain equations, and to study the associated large deviation principles.
Let {Zt,t ∈ R} be an H-valued semimartingale. We prove an Ito-type inequality and a Burkholder-type inequality for stochastic convolution [formula omitted]. These are the main tools for our study of the above stochastic integral equation.Science, Faculty of
Mathematics, Department of
Graduate
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Seufer, Ingo. "Generalized inverses of differential-algebraic equations and their discretization". [S.l.] : [s.n.], 2006. http://deposit.ddb.de/cgi-bin/dokserv?idn=980230306.
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Reich, Sebastian. "On the local qualitative behavior of differential-algebraic equations". Universität Potsdam, 1995. http://opus.kobv.de/ubp/volltexte/2010/4673/.
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A theoretical famework for the investigation of the qualitative behavior of differential-algebraic equations (DAEs) near an equilibrium point is established. The key notion of our approach is the notion of regularity. A DAE is called regular locally around an equilibrium point if there is a unique vector field such that the solutions of the DAE and the vector field are in one-to-one correspondence in a neighborhood of this equili Drium point. Sufficient conditions for the regularity of an equilibrium point are stated. This in turn allows us to translate several local results, as formulated for vector fields, to DAEs that are regular locally around a g: ven equilibrium point (e.g. Local Stable and Unstable Manifold Theorem, Hopf theorem). It is important that ihese theorems are stated in terms of the given problem and not in terms of the corresponding vector field.
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Pätz, Torben [Verfasser]. "Segmentation of Stochastic Images using Stochastic Partial Differential Equations / Torben Pätz". Bremen : IRC-Library, Information Resource Center der Jacobs University Bremen, 2012. http://d-nb.info/1035219735/34.
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Moon, Kyoung-Sook. "Adaptive Algorithms for Deterministic and Stochastic Differential Equations". Doctoral thesis, KTH, Numerical Analysis and Computer Science, NADA, 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3586.
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Guillouzic, Steve. "Fokker-Planck approach to stochastic delay differential equations". Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/NQ58279.pdf.
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Sipiläinen, Eeva-Maria. "Pathwise view on solutions of stochastic differential equations". Thesis, University of Edinburgh, 1993. http://hdl.handle.net/1842/8202.
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The Ito-Stratonovich theory of stochastic integration and stochastic differential equations has several shortcomings, especially when it comes to existence and consistency with the theory of Lebesque-Stieltjes integration and ordinary differential equations. An attempt is made firstly, to isolate the path property, possessed by almost all Brownian paths, that makes the stochastic theory of integration work. Secondly, to construct a new concept of solutions for differential equations, which would have the required consistency and continuity properties, within a class of deterministic noise functions, large enough to include almost all Brownian paths. The algebraic structure of iterated path integrals for smooth paths leads to a formal definition of a solution for a differential equation in terms of generalized path integrals for more general noises. This suggests a way of constructing solutions to differential equations in a large class of paths as limits of operators. The concept of the driving noise is extended to include the generalized path integrals of the noise. Less stringent conditions on the Holder continuity of the path can be compensated by giving more of its iterated integrals. Sufficient conditions for the solution to exist are proved in some special cases, and it is proved that almost all paths of Brownian motion as well as some other stochastic processes can be included in the theory.