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Dissertations / Theses on the topic 'Stochastic differential equations'
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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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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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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 données<br>This 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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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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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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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, respectivement<br>This 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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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.<br>Science, Faculty of<br>Mathematics, Department of<br>Graduate
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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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<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>
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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)<br>y(t)}<br>t is in [0<br>1]} with values in Rd and Rd×<br>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.<br>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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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.<br>This thesis was scanned as part of an electronic thesis pilot project.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (p. 149-155).<br>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.<br>by Alessio Spantini.<br>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.<br>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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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.
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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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Ahlip, Rehez Ajmal. "Stability & filtering of stochastic systems." Thesis, Queensland University of Technology, 1997.
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Banerjee, Paromita. "Numerical Methods for Stochastic Differential Equations and Postintervention in Structural Equation Models." Case Western Reserve University School of Graduate Studies / OhioLINK, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=case1597879378514956.
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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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Zerihun, Tadesse G. "Nonlinear Techniques for Stochastic Systems of Differential Equations." Scholar Commons, 2013. http://scholarcommons.usf.edu/etd/4970.
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Two of the most well-known nonlinear methods for investigating nonlinear dynamic processes in sciences and engineering are nonlinear variation of constants parameters and comparison method. Knowing the existence of solution process, these methods provide a very powerful tools for investigating variety of problems, for example, qualitative and quantitative properties of solutions, finding error estimates between solution processes of stochastic system and the corresponding nominal system, and inputs for the designing engineering and industrial problems. The aim of this work is to systematically develop mathematical tools to undertake the mathematical frame-work to investigate a complex nonlinear nonstationary stochastic systems of differential equations.
A complex nonlinear nonstationary stochastic system of differential equations are decomposed into nonlinear systems of stochastic perturbed and
unperturbed differential equations. Using this type of decomposition, the fundamental properties of solutions of nonlinear stochastic unperturbed
systems of differential equations are investigated(1). The fundamental properties are used to find the representation of solution process of nonlinear stochastic complex perturbed system in terms of solution process of nonlinear stochastic unperturbed system(2).
Employing energy function method and the fundamental properties of It\^{o}-Doob type stochastic auxiliary system of differential equations, we
establish generalized variation of constants formula for solution process of perturbed stochastic system of differential equations(3). Results
regarding deviation of solution of perturbed system with respect to solution of nominal system of stochastic differential equations are developed(4). The obtained results are used to study the qualitative properties of perturbed stochastic system of differential equations(5). Examples are given to illustrate the usefulness of the results.
Employing energy function method and the fundamental properties of It\^{o}-Doob type stochastic auxiliary system of differential equations, we establish
generalized variational comparison theorems in the context of stochastic and deterministic differential for solution processes of perturbed stochastic system of differential equations(6). Results regarding deviation of solutions with respect to nominal stochastic system are also developed(7). The obtained results are used to study the qualitative properties of perturbed stochastic system(8). Examples are given to illustrate the usefulness of the results.
A simple dynamical model of the effect of radiant flux density and CO_2
concentration on the rate of photosynthesis in light, dark
and enzyme reactions are analyzed(9). The coupled system of dynamic equations are solved numerically for some values
of rate constant and radiant flux density. We used Matlab to solve the
system numerically. Moreover, with the assumption that dynamic model of CO_2 concentration
is studied.
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Pilipenko, Andrey. "An introduction to stochastic differential equations with reflection." Universität Potsdam, 2014. http://opus.kobv.de/ubp/volltexte/2014/7078/.
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These lecture notes are intended as a short introduction to diffusion processes on a domain with a reflecting boundary for graduate students, researchers in stochastic analysis and interested readers. Specific results on stochastic differential equations with reflecting boundaries such as existence and uniqueness, continuity and Markov properties, relation to partial differential equations and submartingale problems are given. An extensive list of references to current literature is included. This book has its origins in a mini-course the author gave at the University of Potsdam and at the Technical University of Berlin in Winter 2013.
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Pokern, Yvo. "Fitting stochastic differential equations to molecular dynamics data." Thesis, University of Warwick, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.439586.
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Li, Qiming. "N-dimension numerical solution of stochastic differential equations." Thesis, University of Edinburgh, 2007. http://hdl.handle.net/1842/12417.
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We introduce an order γ(γ>1/2) strong scheme and an improved weak scheme for the numerical approximation of solutions to stochastic differential equations (SDEs), driven by N Weinner processes. The strong scheme, called the ¾ Scheme, which is dependent on a differently constructed Brownian path, involves the area terms to bring better asymptotic accuracy than any numerical method based on classic constructed N-Dimension Brownian path. We demonstrate how to construct such a Brownian path, besides how to subdivide the Brownian path to get a sequence of approximations which converges pathwise as h tends to 0. We prove that the convergence of such method is guaranteed if the time step size h tends to 0. We also present the Improved Weak Euler Scheme (IWES), whose sample error is much smaller than the classic Weak Euler Scheme’s. The method reduces computation load and the sample error, which is generated during the Monte-Carlo(MC) approximation, by balancing the times of Euler iteration and MC simulation. A further improved IWES can be achieved by reusing the Brownian path. We prove that the extra sample error from reusing Brownian path is negligible in the latter method.
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Luo, Ye. "Random periodic solutions of stochastic functional differential equations." Thesis, Loughborough University, 2014. https://dspace.lboro.ac.uk/2134/16112.
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In this thesis, we study the existence of random periodic solutions for both nonlinear dissipative stochastic functional differential equations (SFDEs) and semilinear nondissipative SFDEs in C([-r,0],R^d). Under some sufficient conditions for the existence of global semiflows for SFDEs, by using pullback-convergence technique to SFDE, we obtain a general theorem about the existence of random periodic solutions. By applying coupled forward-backward infinite horizon integral equations method, we perform the argument of the relative compactness of Wiener-Sobolev spaces in C([0,τ],C([-r,0]L²(Ω))) and the generalized Schauder's fixed point theorem to show the existence of random periodic solutions.
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Wang, Xince. "Quasilinear PDEs and forward-backward stochastic differential equations." Thesis, Loughborough University, 2015. https://dspace.lboro.ac.uk/2134/17383.
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In this thesis, first we study the unique classical solution of quasi-linear second order parabolic partial differential equations (PDEs). For this, we study the existence and uniqueness of the $L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{d}) \otimes L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{k})\otimes L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{k\times d})$ valued solution of forward backward stochastic differential equations (FBSDEs) with finite horizon, the regularity property of the solution of FBSDEs and the connection between the solution of FBSDEs and the solution of quasi-linear parabolic PDEs. Then we establish their connection in the Sobolev weak sense, in order to give the weak solution of the quasi-linear parabolic PDEs. Finally, we study the unique weak solution of quasi-linear second order elliptic PDEs through the stationary solution of the FBSDEs with infinite horizon.
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Zhou, Yiqian. "Stability of stochastic differential equations in infinite dimensions." Thesis, University of Liverpool, 2012. http://livrepository.liverpool.ac.uk/10513/.
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In engineering, physics and economics, many dynamical systems involving with stochastic components and random noise are often modeled by stochastic models. The stochastic effects of these models are often used to describe the uncertainty about the operating systems. Motivated by the development of analysis and theory of stochastic processes, as well as the studies of natural sciences, the theory of stochastic differential equations in infinite dimensional spaces evolves gradually into a branch of modern analysis. Many qualitative properties of such systems have been studied in the past few decades, among which, investigation of stability of such systems is often regarded as the first characteristic of the dynamical systems or models. In general, this thesis is mainly concerned with the studies of the stability property of stochastic differential equations in Hilbert spaces. Chapter 1 is an introduction to a brief history of stochastic differential equations in infinite dimensions, together with an overview of the studies. Chapter 2 is a presentation of preliminaries to some basic stochastic analysis. In Chapter 3, we study the stability in distribution of mild solutions to stochastic delay differential equations with Poisson jumps. Firstly, we use approximation of strong solutions to pass on the stability of strong solutions to the mild ones. Then, by constructing a suitable metric between the transition probability functions of mild solutions, we obtain the desired stability result under some suitable conditions. In Chapter 4, we investigate the stochastic partial delay differential equations with Markovian switching and Poisson jumps. By estimating the coefficients of energy equality, both the exponential stability and almost sure exponential stability of energy solutions to the equations are obtained. In Chapter 5, we study the relationship among strong, weak and mild solutions to the stochastic functional differential equations of neutral type. Finally, in Chapter 6, we study the asymptotic stability of two types of equations, impulsive stochastic delay differential equations with Poisson jumps and stochastic evolution equations with Poisson jumps. By employing the fixed point theorem, we derive the desired stability results under some criteria.
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Xiong, Sheng. "Stochastic Differential Equations: Some Risk and Insurance Applications." Diss., Temple University Libraries, 2011. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/133166.
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Mathematics<br>Ph.D.<br>In this dissertation, we have studied diffusion models and their applications in risk theory and insurance. Let Xt be a d-dimensional diffusion process satisfying a system of Stochastic Differential Equations defined on an open set G Rd, and let Ut be a utility function of Xt with U0 = u0. Let T be the first time that Ut reaches a level u^*. We study the Laplace transform of the distribution of T, as well as the probability of ruin, psileft(u_{0}right)=Prleft{ T<inftyright} , and other important probabilities. A class of exponential martingales is constructed to analyze the asymptotic properties of all probabilities. In addition, we prove that the expected discounted penalty function, a generalization of the probability of ultimate ruin, satisfies an elliptic partial differential equation, subject to some initial boundary conditions. Two examples from areas of actuarial work to which martingales have been applied are given to illustrate our methods and results: 1. Insurer's insolvency. 2. Terrorism risk. In particular, we study insurer's insolvency for the Cram'{e}r-Lundberg model with investments whose price follows a geometric Brownian motion. We prove the conjecture proposed by Constantinescu and Thommann.<br>Temple University--Theses
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Schmid, Matthias J. A. "A New Control Paradigm for Stochastic Differential Equations." Thesis, State University of New York at Buffalo, 2017. http://pqdtopen.proquest.com/#viewpdf?dispub=10285670.
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<p> This study presents a novel comprehensive approach to the control of dynamic systems under uncertainty governed by stochastic differential equations (SDEs). Large Deviations (LD) techniques are employed to arrive at a control law for a large class of nonlinear systems minimizing sample path deviations. Thereby, a paradigm shift is suggested from point-in-time to sample path statistics on function spaces. </p><p> A suitable formal control framework which leverages embedded Freidlin-Wentzell theory is proposed and described in detail. This includes the precise definition of the control objective and comprises an accurate discussion of the adaptation of the Freidlin-Wentzell theorem to the particular situation. The new control design is enabled by the transformation of an ill-posed control objective into a well-conditioned sequential optimization problem. </p><p> A direct numerical solution process is presented using quadratic programming, but the emphasis is on the development of a closed-form expression reflecting the asymptotic deviation probability of a particular nominal path. This is identified as the key factor in the success of the new paradigm. An approach employing the second variation and the differential curvature of the effective action is suggested for small deviation channels leading to the Jacobi field of the rate function and the subsequently introduced Jacobi field performance measure. This closed-form solution is utilized in combination with the supplied parametrization of the objective space. For the first time, this allows for an LD based control design applicable to a large class of nonlinear systems. Thus, Minimum Large Deviations (MLD) control is effectively established in a comprehensive structured framework. The construction of the new paradigm is completed by an optimality proof for the Jacobi field performance measure, an interpretive discussion, and a suggestion for efficient implementation. </p><p> The potential of the new approach is exhibited by its extension to scalar systems subject to state-dependent noise and to systems of higher order. The suggested control paradigm is further advanced when a sequential application of MLD control is considered. This technique yields a nominal path corresponding to the minimum total deviation probability on the entire time domain. It is demonstrated that this sequential optimization concept can be unified in a single objective function which is revealed to be the Jacobi field performance index on the entire domain subject to an endpoint deviation. The emerging closed-form term replaces the previously required nested optimization and, thus, results in a highly efficient application-ready control design. This effectively substantiates Minimum Path Deviation (MPD) control. </p><p> The proposed control paradigm allows the specific problem of stochastic cost control to be addressed as a special case. This new technique is employed within this study for the stochastic cost problem giving rise to Cost Constrained MPD (CCMPD) as well as to Minimum Quadratic Cost Deviation (MQCD) control. An exemplary treatment of a generic scalar nonlinear system subject to quadratic costs is performed for MQCD control to demonstrate the elementary expandability of the new control paradigm. </p><p> This work concludes with a numerical evaluation of both MPD and CCMPD control for three exemplary benchmark problems. Numerical issues associated with the simulation of SDEs are briefly discussed and illustrated. The numerical examples furnish proof of the successful design. </p><p> This study is complemented by a thorough review of statistical control methods, stochastic processes, Large Deviations techniques and the Freidlin-Wentzell theory, providing a comprehensive, self-contained account. The presentation of the mathematical tools and concepts is of a unique character, specifically addressing an engineering audience.</p><p>
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Kasonga, Raphael Abel Carleton University Dissertation Mathematics. "Asymptotic parameter estimation theory for stochastic differential equations." Ottawa, 1986.
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Jeisman, Joseph Ian. "Estimation of the parameters of stochastic differential equations." Thesis, Queensland University of Technology, 2006. https://eprints.qut.edu.au/16205/1/Joseph_Jesiman_Thesis.pdf.
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Stochastic di®erential equations (SDEs) are central to much of modern finance theory and have been widely used to model the behaviour of key variables such as the instantaneous short-term interest rate, asset prices, asset returns and their volatility. The explanatory and/or predictive power of these models depends crucially on the particularisation of the model SDE(s) to real data through the choice of values for their
parameters. In econometrics, optimal parameter estimates are generally considered to be those that maximise the likelihood of the sample. In the context of the estimation of the parameters of SDEs, however, a closed-form expression for the likelihood function is rarely available and hence exact maximum-likelihood (EML) estimation is
usually infeasible. The key research problem examined in this thesis is the development
of generic, accurate and computationally feasible estimation procedures based on the ML principle, that can be implemented in the absence of a closed-form expression for the likelihood function. The overall recommendation to come out of the thesis is that an estimation procedure based on the finite-element solution of a reformulation of the Fokker-Planck equation in terms of the transitional cumulative distribution function(CDF) provides the best balance across all of the desired characteristics. The recommended approach involves the use of an interpolation technique proposed in this thesis which greatly reduces the required computational effort.
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Jeisman, Joseph Ian. "Estimation of the parameters of stochastic differential equations." Queensland University of Technology, 2006. http://eprints.qut.edu.au/16205/.
Full textAbstract:
Stochastic di®erential equations (SDEs) are central to much of modern finance theory and have been widely used to model the behaviour of key variables such as the instantaneous short-term interest rate, asset prices, asset returns and their volatility. The explanatory and/or predictive power of these models depends crucially on the particularisation of the model SDE(s) to real data through the choice of values for their parameters. In econometrics, optimal parameter estimates are generally considered to be those that maximise the likelihood of the sample. In the context of the estimation of the parameters of SDEs, however, a closed-form expression for the likelihood function is rarely available and hence exact maximum-likelihood (EML) estimation is usually infeasible. The key research problem examined in this thesis is the development of generic, accurate and computationally feasible estimation procedures based on the ML principle, that can be implemented in the absence of a closed-form expression for the likelihood function. The overall recommendation to come out of the thesis is that an estimation procedure based on the finite-element solution of a reformulation of the Fokker-Planck equation in terms of the transitional cumulative distribution function(CDF) provides the best balance across all of the desired characteristics. The recommended approach involves the use of an interpolation technique proposed in this thesis which greatly reduces the required computational effort.
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Xu, Lina. "Simulation methods for stochastic differential equations in finance." Thesis, Queensland University of Technology, 2019. https://eprints.qut.edu.au/134388/1/Lina_Xu_Thesis.pdf.
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This thesis resolves a number of econometric problems relating to the use of stochastic differential equations based on computer-intensive simulation methods. Stochastic differential equations play an important role in modern finance. They have been used to model the trajectories of key variables such as short-term interest rates and the volatility of financial assets. The central theme of the thesis is the use of Hermite polynomials to approximate the transitional probability distribution functions of stochastic differential equations. Based on these approximations, a new method is proposed for simulating solutions to these equations and new testing procedures are developed to examine the fit of the equations to observed data.
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Lythe, Grant David. "Stochastic slow-fast dynamics." Thesis, University of Cambridge, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.338108.
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Li, Wuchen. "A study of stochastic differential equations and Fokker-Planck equations with applications." Diss., Georgia Institute of Technology, 2016. http://hdl.handle.net/1853/54999.
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Fokker-Planck equations, along with stochastic differential equations, play vital roles in physics, population modeling, game theory and optimization (finite or infinite dimensional). In this thesis, we study three topics, both theoretically and computationally, centered around them. In part one, we consider the optimal transport for finite discrete states, which are on a finite but arbitrary graph. By defining a discrete 2-Wasserstein metric, we derive Fokker-Planck equations on finite graphs as gradient flows of free energies. By using dynamical viewpoint, we obtain an exponential convergence result to equilibrium. This derivation provides tools for many applications, including numerics for nonlinear partial differential equations and evolutionary game theory. In part two, we introduce a new stochastic differential equation based framework for optimal control with constraints. The framework can efficiently solve several real world problems in differential games and Robotics, including the path-planning problem. In part three, we introduce a new noise model for stochastic oscillators. With this model, we prove global boundedness of trajectories. In addition, we derive a pair of associated Fokker-Planck equations.