Dissertations / Theses on the topic 'Stochastic convergence'
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Xiong, Xiaoping. "Stochastic optimization algorithms and convergence /." College Park, Md. : University of Maryland, 2005. http://hdl.handle.net/1903/2360.
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Thesis (Ph. D.) -- University of Maryland, College Park, 2005.Thesis research directed by: Business and Management. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.
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Suzuki, Kohei. "Convergence of stochastic processes on varying metric spaces." 京都大学 (Kyoto University), 2016. http://hdl.handle.net/2433/215281.
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Greensmith, Evan, and evan greensmith@gmail com. "Policy Gradient Methods: Variance Reduction and Stochastic Convergence." The Australian National University. Research School of Information Sciences and Engineering, 2005. http://thesis.anu.edu.au./public/adt-ANU20060106.193712.
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In a reinforcement learning task an agent must learn a policy for performing actions so as to perform well in a given environment. Policy gradient methods consider a parameterized class of policies, and using a policy from the class, and a trajectory through the environment taken by the agent using this policy, estimate the performance of the policy with respect to the parameters. Policy gradient methods avoid some of the problems of value function methods, such as policy degradation, where inaccuracy in the value function leads to the choice of a poor policy. However, the estimates produced by policy gradient methods can have high variance.¶
In Part I of this thesis we study the estimation variance of policy gradient algorithms, in particular, when augmenting the estimate with a baseline, a common method for reducing estimation variance, and when using actor-critic methods. A baseline adjusts the reward signal supplied by the environment, and can be used to reduce the variance of a policy gradient estimate without adding any bias. We find the baseline that minimizes the variance. We also consider the class of constant baselines, and find the constant baseline that minimizes the variance. We compare this to the common technique of adjusting the rewards by an estimate of the performance measure. Actor-critic methods usually attempt to learn a value function accurate enough to be used in a gradient estimate without adding much bias. In this thesis we propose that in learning the value function we should also consider the variance. We show how considering the variance of the gradient estimate when learning a value function can be beneficial, and we introduce a new optimization criterion for selecting a value function.¶
In Part II of this thesis we consider online versions of policy gradient algorithms, where we update our policy for selecting actions at each step in time, and study the convergence of the these online algorithms. For such online gradient-based algorithms, convergence results aim to show that the gradient of the performance measure approaches zero. Such a result has been shown for an algorithm which is based on observing trajectories between visits to a special state of the environment. However, the algorithm is not suitable in a partially observable setting, where we are unable to access the full state of the environment, and its variance depends on the time between visits to the special state, which may be large even when only few samples are needed to estimate the gradient. To date, convergence results for algorithms that do not rely on a special state are weaker. We show that, for a certain algorithm that does not rely on a special state, the gradient of the performance measure approaches zero. We show that this continues to hold when using certain baseline algorithms suggested by the results of Part I.
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Greensmith, Evan. "Policy gradient methods : variance reduction and stochastic convergence /." View thesis entry in Australian Digital Theses Program, 2005. http://thesis.anu.edu.au/public/adt-ANU20060106.193712/index.html.
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Sapozhnikov, Artyom Vasilyevich. "Existence of moments and convergence rates in stochastic networks." Thesis, Heriot-Watt University, 2005. http://hdl.handle.net/10399/256.
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Schiopu-Kratina, I. (Ioana). "General tightness conditions and weak convergence of point processes." Thesis, McGill University, 1985. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=71994.
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In this dissertation, we consider two aspects of the theory of weak convergence of cadlag processes.We first give a necessary and sufficient condition for the tightness of a sequence of cadlag processes (chapters 2,3) which generalizes Rebolledo's condition (see 13 ). It is a stochastic condition in the sense that stopping times rather than deterministic times are used in the statement.
We then discuss the predictability of the limit of a sequence of predictable processes (chapters 4-6). For a convergent sequence of point processes we show that, if the sequence of compensators converges, then the limit of compensators is the compensator of the limit of point processes (chapters 4,5).
Finally, we prove in Chapter 6 that extended weak convergence of a sequence of increasing predictable processes ensures the predictability of the limit.
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Schmitz, Abe Klaus E. "Pricing exotic options using improved strong convergence." Thesis, University of Oxford, 2008. http://ora.ox.ac.uk/objects/uuid:5a9fb837-238f-46a7-976a-fe3bae0e7b09.
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Today, better numerical approximations are required for multi-dimensional SDEs to improve on the poor performance of the standard Monte Carlo integration. With this aim in mind, the material in the thesis is divided into two main categories, stochastic calculus and mathematical finance. In the former, we introduce a new scheme or discrete time approximation based on an idea of Paul Malliavin where, for some conditions, a better strong convergence order is obtained than the standard Milstein scheme without the expensive simulation of the Lévy Area. We demonstrate when the conditions of the 2−Dimensional problem permit this and give an exact solution for the orthogonal transformation (θ Scheme or Orthogonal Milstein Scheme). Our applications are focused on continuous time diffusion models for the volatility and variance with their discrete time approximations (ARV). Two theorems that measure with confidence the order of strong and weak convergence of schemes without an exact solution or expectation of the system are formally proved and tested with numerical examples. In addition, some methods for simulating the double integrals or Lévy Area in the Milstein approximation are introduced. For mathematical finance, we review evidence of non-constant volatility and consider the implications for option pricing using stochastic volatility models. A general stochastic volatility model that represents most of the stochastic volatility models that are outlined in the literature is proposed. This was necessary in order to both study and understand the option price properties. The analytic closed-form solution for a European/Digital option for both the Square Root Model and the 3/2 Model are given. We present the Multilevel Monte Carlo path simulation method which is a powerful tool for pricing exotic options. An improved/updated version of the ML-MC algorithm using multi-schemes and a non-zero starting level is introduced. To link the contents of the thesis, we present a wide variety of pricing exotic option examples where considerable computational savings are demonstrated using the new θ Scheme and the improved Multischeme Multilevel Monte Carlo method (MSL-MC). The computational cost to achieve an accuracy of O(e) is reduced from O(e−3) to O(e−2) for some applications.
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Moon, Kyoung-Sook. "Convergence rates of adaptive algorithms for deterministic and stochastic differential equations." Licentiate thesis, KTH, Numerical Analysis and Computer Science, NADA, 2001. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-1382.
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von, Schwerin Erik. "Convergence rates of adaptive algorithms for stochastic and partial differential equations." Licentiate thesis, KTH, Numerical Analysis and Computer Science, NADA, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-302.
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Schwerin, Erik von. "Convergence rates of adaptive algorithms for stochastic and partial differential equations /." Stockholm, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-302.
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Denz, Markus [Verfasser]. "Convergence results for stochastic particle systems with social interaction / Markus Denz." Mainz : Universitätsbibliothek Mainz, 2014. http://d-nb.info/1046966782/34.
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Binotto, Giulia. "Contributions to stochastic analysis." Doctoral thesis, Universitat de Barcelona, 2018. http://hdl.handle.net/10803/565571.
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The aim of this dissertation is to present some new results on stochastic analysis. It consists on three works that deal with two Gaussian processes: the Brownian motion and the fractional Brownian motion with Hurst parameter H less than 1/2.
In the first work we construct a family of processes, from a single Poisson process and a sequence of independent random variables with common Bernoulli distribution, that converges in law to a complex Brownian motion. We find realizations of these processes that converge almost surely to the complex Brownian motion, uniformly on the unit time interval, and we derive the rate of convergence.
In the second work, we establish the weak convergence, in the topology of the Skorohod space, of the symmetric Riemann sums for functionals of the fractional Brownian motion when the Hurst parameter takes a critical value that depends on the chosen measure. As a consequence, we derive a change-of-variable formula in distribution, where the correction term is a stochastic integral with respect to a Brownian motion that is independent of the fractional Brownian motion.
The last work is devoted to prove that, when the delay goes to zero, the solution of delay differential equations driven by a Hölder continuous function of order in (1/3,1/2) converges with the supremum norm to the solution of the equation without delay.L’objectiu d’aquesta tesi és presentar alguns resultats innovadors en el camp de l’anàlisi estocàstica. Proposem tres treballs que tracten amb dos processos Gaussians: el moviment Brownià i el moviment Brownià fraccionari amb paràmetre de Hurst menor que 1/2. En el primer treball, construïm una família de processos, a partir d’un procés de Poisson i d’una seqüència de variables aleatòries independents amb distribució de Bernoulli, que convergeix en llei cap a un moviment Brownià complex. Trobem realitzacions d’aquests processos que convergeixen quasi segurament a un moviment Brownià complex, uniformement a l’interval de temps unitat. En derivem també la velocitat de convergència. En el segon treball, determinem la convergència feble, en la topologia de l’espai de Skorohod, de les sumes de Riemann simètriques per funcionals del moviment Brownià fraccionari quan el paràmetre de Hurst pren un valor crític que depèn de la mesura considerada. Com a conseqüència, derivem una fórmula de canvi de variable en distribució, on el terme de correcció és una integral estocàstica amb respecte a un moviment Brownià independent del moviment Brownià fraccionari. En l’últim treball demostrem que, quan el retard tendeix a zero, la solució d’equacions diferencials amb retard dirigides per una funció Hölder contínua amb ordre a (1/3,1/2) convergeix en la norma del suprem a la solució d’equacions sense retard.
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Fouque, Jean-Pierre. "Sur trois aspects de la théorie des processus stochastiques." Paris 6, 1986. http://www.theses.fr/1986PA066401.
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Evans, Josephine Angela Holly. "Deterministic and stochastic approaches to relaxation to equilibrium for particle systems." Thesis, University of Cambridge, 2019. https://www.repository.cam.ac.uk/handle/1810/282876.
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This work is about convergence to equilibrium problems for equations coming from kinetic theory. The bulk of the work is about Hypocoercivity. Hypocoercivity is the phenomenon when a semigroup shows exponentially relaxation towards equilibrium without the corresponding coercivity (dissipativity) inequality on the Dirichlet form in the natural space, i.e. a lack of contractivity. In this work we look at showing hypocoercivity in weak measure distances, and using probabilistic techniques. First we review the history of convergence to equilibrium for kinetic equations, particularly for spatially inhomogeneous kinetic theory (Boltzmann and Fokker-Planck equations) which motivates hypocoercivity. We also review the existing work on showing hypocoercivity using probabilistic techniques. We then present three different ways of showing hypocoercivity using stochastic tools. First we study the kinetic Fokker-Planck equation on the torus. We give two different coupling strategies to show convergence in Wasserstein distance, $W_2$. The first relies on explicitly solving the stochastic differential equation. In the second we couple the driving Brownian motions of two solutions with different initial data, in a well chosen way, to show convergence. Next we look at a classical tool to show convergence to equilibrium for Markov processes, Harris's theorem. We use this to show quantitative convergence to equilibrium for three Markov jump processes coming from kinetic theory: the linear relaxation/BGK equation, the linear Boltzmann equation, and a jump process which is similar to the kinetic Fokker-Planck equation. We show convergence to equilibrium for these equations in total variation or weighted total variation norms. Lastly, we revisit a version of Harris's theorem in Wasserstein distance due to Hairer and Mattingly and use this to show quantitative hypocoercivity for the kinetic Fokker-Planck equation with a confining potential via Malliavin calculus. We also look at showing hypocoercivity in relative entropy. In his seminal work work on hypocoercivity Villani obtained results on hypocoercivity in relative entropy for the kinetic Fokker-Planck equation. We review this and subsequent work on hypocoercivity in relative entropy which is restricted to diffusions. We show entropic hypocoercivity for the linear relaxation Boltzmann equation on the torus which is a non-local collision equation. Here we can work around issues arising from the fact that the equation is not in the H\"{o}rmander sum of squares form used by Villani, by carefully modulating the entropy with hydrodynamical quantities. We also briefly review the work of others to show a similar result for a close to quadratic confining potential and then show hypocoercivity for the linear Boltzmann equation with close to quadratic confining potential using similar techniques. We also look at convergence to equilibrium for Kac's model coupled to a non-equilibrium thermostat. Here the equation is directly coercive rather than hypocoercive. We show existence and uniqueness of a steady state for this model. We then show that the solution will converge exponentially fast towards this steady state both in the GTW metric (a weak measure distance based on Fourier transforms) and in $W_2$. We study how these metrics behave with the dimension of the state space in order to get rates of convergence for the first marginal which are uniform in the number of particles.
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Leoff, Elisabeth [Verfasser]. "Stochastic Filtering in Regime-Switching Models: Econometric Properties, Discretization and Convergence / Elisabeth Leoff." München : Verlag Dr. Hut, 2017. http://d-nb.info/1126297348/34.
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Ellis, Tom. "Coalescing stochastic flows driven by Poisson random measure and convergence to the Brownian web." Thesis, University of Cambridge, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.609747.
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Mirebrahimi, Seyedmeghdad. "Interacting stochastic systems with individual and collective reinforcement." Thesis, Poitiers, 2019. http://www.theses.fr/2019POIT2274/document.
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L'urne de Polya est l'exemple typique de processus stochastique avec renforcement. La limite presque sûre (p.s.) en temps existe, est aléatoire et non dégénérée. L'urne de Friedman est une généralisation naturelle dont la limite (proportion asymptotique en temps) n'est plus aléatoire. De nombreux modèles aléatoires sont fondés sur des processus de renforcement comme pour la conception d'essais cliniques au design adaptatif, en économie, ou pour des algorithmes stochastiques à des fins d'optimisation ou d'estimation non paramétrique. Dans ce mémoire, inspirés par de nombreux articles récents, nous introduisons une nouvelle famille de systèmes (finis) de processus de renforcement où l'interaction se traduit par un phénomène de renforcement collectif additif, de type champ moyen. Les deux taux de renforcement (l'un spécifique à chaque composante, l'autre collectif et commun à toutes les composantes) sont possiblement différents. Nous prouvons deux types de résultats mathématiques. Différents régimes de paramètres doivent être considérés : type de la règle (brièvement, Polya/Friedman), taux du renforcement. Nous prouvons l'existence d'une limite p.s. coommune à toutes les composantes du système (synchronisation). La nature de la limite (aléatoire/déterministe) est étudiée en fonction du régime de paramètres. Nous étudions également les fluctuations en prouvant des théorèmes centraux de la limite. Les changements d'échelle varient en fonction du régime considéré. Différentes vitesses de convergence sont ainsi établiesThe Polya urn is the paradigmatic example of a reinforced stochastic process. It leads to a random (non degenerated) almost sure (a.s.) time-limit.The Friedman urn is a natural generalization whose a.s. time-limit is not random anymore. Many stochastic models for applications are based on reinforced processes, like urns with their use in adaptive design for clinical trials or economy, stochastic algorithms with their use in non parametric estimation or optimisation. In this work, in the stream of previous recent works, we introduce a new family of (finite) systems of reinforced stochastic processes, interacting through an additional collective reinforcement of mean field type. The two reinforcement rules strengths (one componentwise, one collective) are tuned through (possibly) different rates. In the case the reinforcement rates are like 1/n, these reinforcements are of Polya or Friedman type as in urn contexts and may thus lead to limits which may be random or not. We state two kind of mathematical results. Different parameter regimes needs to be considered: type of reinforcement rule (Polya/Friedman), strength of the reinforcement. We study the time-asymptotics and prove that a.s. convergence always holds. Moreover all the components share the same time-limit (synchronization). The nature of the limit (random/deterministic) according to the parameters' regime is considered. We then study fluctuations by proving central limit theorems. Scaling coefficients vary according to the regime considered. This gives insights into the different rates of convergence
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Gersch, Oliver. "Convergence in distribution of random closed sets and applications in stability theory of stochastic optimisation." [S.l.] : [s.n.], 2007. http://deposit.ddb.de/cgi-bin/dokserv?idn=983617856.
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Lucic, Vladimir. "On uniqueness and weak convergence of solutions for the stochastic differential equations of nonlinear filtering." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/NQ60554.pdf.
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Liu, Wei. "Asymptotic properties and finite time convergence of classical and modified methods for stochastic differential equations." Thesis, University of Strathclyde, 2013. http://oleg.lib.strath.ac.uk:80/R/?func=dbin-jump-full&object_id=22727.
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As few stochastic differential equations have explicit solutions, the numerical schemes are studied to approximate the underlying solution. The fast development in computer science in recent years has made large scale simulations available, then the numerical analysis for stochastic differential equations has been blooming in past decades. However, the study on numerical solutions is still far behind the study on the underlying solutions. This thesis is devoted to mathematically rigorous investigation on the numerical solutions. Among all those attractive mysteries in the numerical analysis of stochastic differential equations, one of the popular problems is that if the numerical solutions can reproduce different properties of the underlying solutions. In thesis, we present some interesting results on this topic, which includes the asymptotic moment boundedness, the stationary distribution and the almost sure stability. The methods considered in this part are two classical methods , the explicit Euler-Maruyama method and the backward Euler-Maruyama method, and one modified method, the Euler-Maruyama method with random variable step size, which is first introduced in this thesis. Another main focus of numerical analysis is the finite time convergence. Our work on this topic is to modify the explicit Euler-Maruyama method and investigate the strong convergence (in the L2 sense) of it. Our investigation first goes to reproduce the asymptotic boundedness in small moment of the underlying solutions. The explicit Euler-Maruyama method is shown to be able to achieve this goal if both the drift coefficient and the diffusion coefficient are global Lipschitz. But with the global Lipschitz condition on the drift coefficient violated, a counter example indicates the failure of the explicit Euler-Maruyama method. A natural replacement, the backward Euler-Maruyama method, then is considered and successfully reproduce the asymptotic boundedness. In the case of small moment, we are only able to reproduce the boundedness property qualitatively so far. To answer another close related question that if we could reproduce the upper bound quantitatively, we strengthen the conditions and show that for the case of second moment the upper bound of the underlying solution can be reproduced as well. As the moment boundedness is key to the existence and uniqueness of the stationary distribution, we next study this property for the numerical solution. Since the backward Euler-Maruyama method has better performance than the explicit Euler-Maruyama method, in this part we only discuss the backward Euler-Maruyama method. The coefficient related sufficient conditions are given for the existence and uniqueness of the stationary distribution of the backward Euler-Maruyama method. Then the numerical stationary distribution is proved to converge to the stationary distribution of the underlying solution as step size vanishes. These results largely extend the existing works to cover wider range of stochastic differential equations. The almost sure stability is one of the hottest topics and many papers have studied the reproduction of this property by different kinds of classical methods. Therefore, we seek to study this property by one modified method, the Euler-Maruyama method with random variable step size. To our best knowledge, this is the first work to apply the random variable step size to the analysis of the almost sure stability of the explicit Euler-Maruyama method. One of our key contributions is that we show that the time variable is a stopping time, which were ignored by many researchers, and only under this circumstance the rest results hold. Compare with those fixed step size or nonrandom variable step size methods, the Euler-Maruyama method with random variable step size is shown to be able to reproduce the almost sure stability with much weaker conditions. As the strong convergence of the classical methods has already been widely studied and the recent works have shown the good performance of the modified classical methods, we present our findings in this area by introducing the stopped Euler method and show the strong convergence of it to the underlying solution with the rate a half. Briefly, the stopped Euler method is the classical Euler-Maruyama method equipped with the stopping time technique. The stopping time is originally employed to preserve the non-negativity of the numerical solution, and it turns out that the non-negativity in return enables the strong convergence of the method with the rate arbitrarily close to a half. Compare with the explicit Euler-Maruyama method, the stopped Euler method can cover some highly non-linear stochastic differential equations.
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Szyszkowicz, B. (Barbara) Carleton University Dissertation Mathematics. "Weak convergence of stochastic processes in weighted metrics and their applications to contiguous changepoint analysis." Ottawa, 1992.
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Bernergård, Zandra. "Connection between discrete time random walks and stochastic processes by Donsker's Theorem." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-48719.
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In this paper we will investigate the connection between a random walk and a continuous time stochastic process. Donsker's Theorem states that a random walk under certain conditions will converge to a Wiener process. We will provide a detailed proof of this theorem which will be used to prove that a geometric random walk converges to a geometric Brownian motion.
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Hu, Liujia. "Convergent algorithms in simulation optimization." Diss., Georgia Institute of Technology, 2015. http://hdl.handle.net/1853/54883.
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It is frequently the case that deterministic optimization models could be made more practical by explicitly incorporating uncertainty. The resulting stochastic optimization problems are in general more difficult to solve than their deterministic counterparts, because the objective function cannot be evaluated exactly and/or because there is no explicit relation between the objective function and the corresponding decision variables. This thesis develops random search algorithms for solving optimization problems with continuous decision variables when the objective function values can be estimated with some noise via simulation. Our algorithms will maintain a set of sampled solutions, and use simulation results at these solutions to guide the search for better solutions. In the first part of the thesis, we propose an Adaptive Search with Resampling and Discarding (ASRD) approach for solving continuous stochastic optimization problems. Our ASRD approach is a framework for designing provably convergent algorithms that are adaptive both in seeking new solutions and in keeping or discarding already sampled solutions. The framework is an improvement over the Adaptive Search with Resampling (ASR) method of Andradottir and Prudius in that it spends less effort on inferior solutions (the ASR method does not discard already sampled solutions). We present conditions under which the ASRD method is convergent almost surely and carry out numerical studies aimed at comparing the algorithms. Moreover, we show that whether it is beneficial to resample or not depends on the problem, and analyze when resampling is desirable. Our numerical results show that the ASRD approach makes substantial improvements on ASR, especially for difficult problems with large numbers of local optima. In traditional simulation optimization problems, noise is only involved in the objective functions. However, many real world problems involve stochastic constraints. Such problems are more difficult to solve because of the added uncertainty about feasibility. The second part of the thesis presents an Adaptive Search with Discarding and Penalization (ASDP) method for solving continuous simulation optimization problems involving stochastic constraints. Rather than addressing feasibility separately, ASDP utilizes the penalty function method from deterministic optimization to convert the original problem into a series of simulation optimization problems without stochastic constraints. We present conditions under which the ASDP algorithm converges almost surely from inside the feasible region, and under which it converges to the optimal solution but without feasibility guarantee. We also conduct numerical studies aimed at assessing the efficiency and tradeoff under the two different convergence modes. Finally, in the third part of the thesis, we propose a random search method named Gaussian Search with Resampling and Discarding (GSRD) for solving simulation optimization problems with continuous decision spaces. The method combines the ASRD framework with a sampling distribution based on a Gaussian process that not only utilizes the current best estimate of the optimal solution but also learns from past sampled solutions and their objective function observations. We prove that our GSRD algorithm converges almost surely, and carry out numerical studies aimed at studying the effects of utilizing the Gaussian sampling strategy. Our numerical results show that the GSRD framework performs well when the underlying objective function is multi-modal. However, it takes much longer to sample solutions, especially in higher dimensions.
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Thamrongrat, Nopporn [Verfasser], and Mark [Akademischer Betreuer] Podolskij. "Stable Convergence in Statistical Inference and Numerical Approximation of Stochastic Processes / Nopporn Thamrongrat ; Betreuer: Mark Podolskij." Heidelberg : Universitätsbibliothek Heidelberg, 2016. http://d-nb.info/1180615751/34.
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Fischer, Manfred M., and Peter Stumpner. "Income Distribution Dynamics and Cross-Region Convergence in Europe. Spatial filtering and novel stochastic kernel representations." WU Vienna University of Economics and Business, 2007. http://epub.wu.ac.at/3969/1/SSRN%2Did981148.pdf.
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This paper suggests an empirical framework for analysing income distribution
dynamics and cross-region convergence in the European Union of 27 member states, 1995-
2003. The framework lies in the research tradition that allows the state income space to be
continuous, puts emphasis on both shape and intra-distribution dynamics and uses stochastic
kernels for studying transition dynamics and implied long-run behaviour. In this paper
stochastic kernels are described by conditional density functions, estimated by a product
kernel estimator of conditional density and represented by means of novel visualisation tools.
The technique of spatial filtering is used to account for spatial effects, in order to avoid
misguided inferences and interpretations caused by the presence of spatial autocorrelation in
the income distributions. The results reveal a slow catching-up of the poorest regions and a
process of polarisation, with a small group of very rich regions shifting away from the rest of
the cross-section. This is well evidenced by both, the unfiltered and the filtered ergodic
density view. Differences exist in detail, and these emphasise the importance to properly deal
with the spatial autocorrelation problem. (authors' abstract)
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Arwini, Saleh. "Improving the convergence rate of seismic history matching with a proxy derived method to aid stochastic sampling." Thesis, Heriot-Watt University, 2013. http://hdl.handle.net/10399/2651.
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History matching is a very important activity during the continued development and management of petroleum reservoirs. Time-lapse (4D) seismic data provide information on the dynamics of fluids in reservoirs, relating variations of seismic signal to saturation and pressure changes. This information can be integrated with history matching to improve convergence towards a simulation model that predicts available data. The main aim of this thesis is to develop a method to speed up the convergence rate of assisted seismic history matching using proxy derived gradient method. Stochastic inversion algorithms often rely on simple assumptions for selecting new models by random processes. In this work, we improve the way that such approaches learn about the system they are searching and thus operate more efficiently. To this end, a new method has been developed called NA with Proxy derived Gradients (NAPG). To improve convergence, we use a proxy model to understand how parameters control the misfit and then use a global stochastic method with these sensitivities to optimise the search of the parameter space. This leads to an improved set of final reservoir models. These in turn can be used more effectively in reservoir management decisions. To validate the proposed approach, we applied the new approach on a number of analytical functions and synthetic cases. In addition, we demonstrate the proposed method by applying it to the UKCS Schiehallion field. The results show that the new method speeds up the rate of convergence by a factor of two to three generally. The performance of NAPG is much improved by updating the regression equation coefficients instead of keeping it fixed. In addition, we found that the initial number of models to start NAPG or NA could be reduced by using Experimental Design instead of using random initialization. Ultimately, with all of these approaches combined, the number of models required to find a good match reduced by an order of magnitude. We have investigated the criteria for stopping the SHM loop, particularly the use of a proxy model to help. More research is needed to complete this work but the approach is promising. Quantifying parameter uncertainty using NA and NAPG was studied using the NA-Bayes approach (NAB). We found that NAB is very sensitive to misfit magnitude but otherwise NA and NAPG produce similar uncertainty measures.
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Petersson, Mikael. "Comparison of modes of convergence in a particle system related to the Boltzmann equation." Thesis, Linköpings universitet, Matematisk statistik, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-61303.
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The distribution of particles in a rarefied gas in a vessel can be described by the Boltzmann equation. As an approximation of the solution to this equation, Caprino, Pulvirenti and Wagner [3] constructed a random N-particle system. In the equilibrium case, they prove in [3] that the L1-distance between the density function of k particles in the N-particle process and the k-fold product of the solution to the stationary Boltzmann equation is of order 1/N. They do this in order to show that the N-particle system converges to the system described by the stationary Boltzmann equation as the number of particles tends to infinity. This is different from the standard approach of describing convergence of an N-particle system. Usually, convergence in distribution of random measures or weak convergence of measures over the space of probability measures is used. The purpose of the present thesis is to compare different modes of convergence of the N-particle system as N tends to infinity assuming stationarity.
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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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Sloma, Przemyslaw. "Contribution to the weak convergence of empirical copula process : contribution to the stochastic claims reserving in general insurance." Thesis, Paris 6, 2014. http://www.theses.fr/2014PA066563/document.
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Dans la première partie de la thèse, nous nous intéressons à la convergence faible du processus empirique pondéré des copules. Nous fournissons la condition suffisante pour que cette convergence ait lieu vers un processus Gaussien limite. Nos résultats sont obtenus dans un espace de Banach L^p. Nous donnons des applications statistiques de ces résultats aux tests d'adéquation (tests of goodness of fit) pour les copules. Une attention spéciale est portée aux tests basées sur des statistiques de type Cramér-von Mises.Dans un second temps, nous étudions le problème de provisionnement stochastique pour une compagnie d'assurance non-vie. Les méthodes stochastiques sont utilisées afin d'évaluer la variabilité des réserves. Le point de départ pour cette thèse est une incohérence entre les méthodes utilisées en pratique et celles publiées dans la littérature. Pour remédier à cela, nous présentons un outil général de provisionnement stochastique à horizon ultime (Chapitre 3) et à un an (Chapitre 4), basé sur la méthode Chain LadderThe aim of this thesis is twofold. First, we concentrate on the study of weak convergence of weighted empirical copula processes. We provide sufficient conditions for this convergence to hold to a limiting Gaussian process. Our results are obtained in the framework of convergence in the Banach space $L^{p}$ ($1\leq p <\infty $). Statistical applications to goodness of fit (GOF) tests for copulas are given to illustrate these results. We pay special attention to GOF tests based on Cramér-von Mises type statistics. Second, we discuss the problem of stochastic claims reserving in general non-life insurance. Stochastic models are needed in order to assess the variability of the claims reserve. The starting point of this thesis is an observed inconsistency between the approaches used in practice and that suggested in the literature. To fill this gap, we present a general tool for measuring the uncertainty of reserves in the framework of ultimate (Chapter 3) and one-year time horizon (Chapter 4), based on the Chain-Ladder method
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Ouyang, Hua. "Optimal stochastic and distributed algorithms for machine learning." Diss., Georgia Institute of Technology, 2013. http://hdl.handle.net/1853/49091.
Full textAbstract:
Stochastic and data-distributed optimization algorithms have received lots of attention from the machine learning community due to the tremendous demand from the large-scale learning and the big-data related optimization. A lot of stochastic and deterministic learning algorithms are proposed recently under various application scenarios. Nevertheless, many of these algorithms are based on heuristics and their optimality in terms of the generalization error is not sufficiently justified. In this talk, I will explain the concept of an optimal learning algorithm, and show that given a time budget and proper hypothesis space, only those achieving the lower bounds of the estimation error and the optimization error are optimal. Guided by this concept, we investigated the stochastic minimization of nonsmooth convex loss functions, a central problem in machine learning. We proposed a novel algorithm named Accelerated Nonsmooth Stochastic Gradient Descent, which exploits the structure of common nonsmooth loss functions to achieve optimal convergence rates for a class of problems including SVMs. It is the first stochastic algorithm that can achieve the optimal O(1/t) rate for minimizing nonsmooth loss functions. The fast rates are confirmed by empirical comparisons with state-of-the-art algorithms including the averaged SGD. The Alternating Direction Method of Multipliers (ADMM) is another flexible method to explore function structures. In the second part we proposed stochastic ADMM that can be applied to a general class of convex and nonsmooth functions, beyond the smooth and separable least squares loss used in lasso. We also demonstrate the rates of convergence for our algorithm under various structural assumptions of the stochastic function: O(1/sqrt{t}) for convex functions and O(log t/t) for strongly convex functions. A novel application named Graph-Guided SVM is proposed to demonstrate the usefulness of our algorithm. We also extend the scalability of stochastic algorithms to nonlinear kernel machines, where the problem is formulated as a constrained dual quadratic optimization. The simplex constraint can be handled by the classic Frank-Wolfe method. The proposed stochastic Frank-Wolfe methods achieve comparable or even better accuracies than state-of-the-art batch and online kernel SVM solvers, and are significantly faster. The last part investigates the problem of data-distributed learning. We formulate it as a consensus-constrained optimization problem and solve it with ADMM. It turns out that the underlying communication topology is a key factor in achieving a balance between a fast learning rate and computation resource consumption. We analyze the linear convergence behavior of consensus ADMM so as to characterize the interplay between the communication topology and the penalty parameters used in ADMM. We observe that given optimal parameters, the complete bipartite and the master-slave graphs exhibit the fastest convergence, followed by bi-regular graphs.
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Cozma, Andrei. "Numerical methods for foreign exchange option pricing under hybrid stochastic and local volatility models." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:44a27fbc-1b7a-4f1a-bd2d-abeb38bf1ff7.
Full textAbstract:
In this thesis, we study the FX option pricing problem and put forward a 4-factor hybrid stochastic-local volatility model. The model, which describes the dynamics of an exchange rate, its volatility and the domestic and foreign short rates, allows for a perfect calibration to European options and has a good hedging performance. Due to the high-dimensionality of the problem, we propose a Monte Carlo simulation scheme that combines the full truncation Euler scheme for the stochastic volatility component and the stochastic short rates with the log-Euler scheme for the exchange rate. We analyze exponential integrability properties of Euler discretizations for the square-root process driving the stochastic volatility and the short rates, properties which play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a large class of stochastic differential equations in finance, including the ones studied in this thesis. Hence, we prove the strong convergence of the exchange rate approximations and the convergence of Monte Carlo estimators for a number of vanilla and exotic options. Then, we calibrate the model to market data and discuss its fitness for pricing FX options. Next, due to the relatively slow convergence of the Monte Carlo method in the number of simulations, we examine a variance reduction technique obtained by mixing Monte Carlo and finite difference methods via conditioning. We consider a purely stochastic version of the model and price vanilla and exotic options by simulating the paths of the volatility and the short rates, and then evaluating the "inner" Black-Scholes-type expectation by means of a partial differential equation. We prove the convergence of numerical approximations and carry out a theoretical variance reduction analysis. Finally, we illustrate the efficiency of the method through a detailed quantitative assessment.
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Berrouane, Salah. "Les lois limites des k-iemes valeurs de record et leurs concomitants." Paris 6, 1986. http://www.theses.fr/1986PA066388.
Full textAbstract:
Nous généraliserons tous les résultats des lois limites des k-iemes valeurs de record (k >ou= 1). Nous énoncerons deux résultats concernant la structure poissonienne des k-iemes valeurs de record. Nous étudierons en détail la stabilité en probabilité ; et nous énoncerons un résultat concernant les records issus d'une fonction de répartition normale. Nous utiliserons le processus de winner w(t) pour l'approximation forte des k-iemes valeurs de record. Une application du théorème de Komlos, Major et Tusnady donne un résultat concernant la stabilité forte. En second lieu, nous définissons les concomitants des k-iemes valeurs de record. Nous déterminerons la distribution générale des concomitants des k-iemes valeurs de record, dans le cas général, et dans le cas de la régression linéaire simple. Nous étudierons les lois limites des concomitants des k-iemes valeurs de record dans le modèle de la régression linéaire simple, et dans le cas général.
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Pleis, Jan [Verfasser], Andreas [Akademischer Betreuer] Rößler, and Andreas [Akademischer Betreuer] Neuenkirch. "Lp and pathwise convergence of the Milstein scheme for stochastic delay differential equations / Jan Pleis ; Akademische Betreuer: Andreas Rößler, Andreas Neuenkirch." Lübeck : Zentrale Hochschulbibliothek Lübeck, 2021. http://d-nb.info/122492598X/34.
Full text
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Koukkous, Abdellatif. "Comportement hydrodynamique de différents processus de zéro range." Rouen, 1997. http://www.theses.fr/1997ROUES051.
Full textAbstract:
Cette thèse comprend trois parties. Dans la première, nous étudions le comportement asymptotique d'un processus de zéro range symétrique en volume fini dans un environnement aléatoire. Nous démontrons que, pour presque tout environnement, la mesure empirique converge en probabilité vers l'unique solution faible d'une équation de diffusion non linéaire indépendante de l'environnement. Dans la seconde partie, réalisée en collaboration avec G. Gielis et C. Landim, nous abordons le problème des fluctuations à l'équilibre pour le processus de zéro range avec environnement aléatoire. Il s'agit d'obtenir un résultat de type théorème central limite pour le champ de densité, autrement dit de montrer que le champ des fluctuations de la densité converge en loi vers un champ gaussien généralisé. Nous établissons le principe de Boltzmann-Gibbs et la convergence faible du champ de fluctuations de la densité du processus de zéro range en environnement aléatoire vers un processus d'Ornstein-Uhlenbeck généralisé dont l'évolution est décrite par linéarisation de l'équation hydrodynamique autour d'une densité fixée en présence d'un bruit blanc. Dans la dernière partie, réalisée en collaboration avec O. Benois et C. Landim, nous donnons une nouvelle interprétation des corrections de Navier-Stokes à l'équation hydrodynamique d'un système asymétrique de particules en interaction. Nous considérons un système dont la mesure initiale est associée à un profil constant dans la direction de la dérive. Nous montrons que, sous une renormalisation diffusive, le comportement du processus est décrit par une équation parabolique non linéaire dont le coefficient de diffusion coïncide avec le coefficient de diffusion de l'équation hydrodynamique de la version symétrique du processus.
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Evans, Lawrence C. 1949. "A strong maximum principle for reaction-diffusion systems and a weak convergence scheme for reflected stochastic differential equations by Lawrence Christopher Evans." Thesis, Massachusetts Institute of Technology, 2010. http://hdl.handle.net/1721.1/59784.
Full textAbstract:
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 125-126).
This thesis consists of two results. The first result is a strong maximum principle for certain parabolic systems of equations, which, for illustrative purposes, I consider as reaction-diffusion systems. Using the theory of viscosity solutions, I give a proof which extends the previous theorem to no longer require any regularity assumptions on the boundary of the convex set in which the system takes its values. The second result is an approximation scheme for reflected stochastic differential equations (SDE) of the Stratonovich type. This is a joint result with Professor Daniel W. Stroock. We show that the distribution of the solution to such a reflected SDE is the weak limit of the distribution of the solutions of the reflected SDEs one gets by replacing the driving Brownian motion by its N-dyadic linear interpolation. In particular, we can infer geometric properties of the solutions to a Stratonovich reflected SDE from those of the solutions to the approximating reflected SDE.
Ph.D.
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Bouamaine, Abdelhalim. "Analyse factorielle séquentielle par approximation stochastique." Nancy 1, 1986. http://www.theses.fr/1986NAN10177.
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Résumé : Dans cette thèse est défini un processus d'approximation stochastique. Nous étudions la convergence presque sure de ce processus et nous l'appliquons à l'estimation séquentielle des facteurs en analyse factorielle inférentielle
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Bascompte, Viladrich David. "Models for bacteriophage systems, Weak convergence of Gaussian processes and L2 modulus of Brownian local time." Doctoral thesis, Universitat Autònoma de Barcelona, 2013. http://hdl.handle.net/10803/129911.
Full textAbstract:
En aquesta memòria es tracten tres problemes diferents. En el Capítol 1 es
construeixen dues famílies de processos que convergeixen, en el sentit de les distribucions
en dimensió finita, cap a dos processos Gaussians independents. El
Capítol 2 està dedicat a l’estudi d’un model de tractament amb bacteriòfags per
infeccions bacterianes. Finalment, en el Capítol 3, estudiem alguns aspectes del
L2 mòdul de continuïtat del temps local del Brownià.
En el primer capítol considerem dos processos Gaussians independents que es
poden representar en termes d’una integral estocàstica d’un nucli determinista
respecte el procés de Wiener, i construïm, a partir d’un únic procés de Poisson,
dues famílies de processos que convergeixen, en el sentit de les distribucions en
dimensió finita, cap a aquests processos Gaussians.
Utilitzarem aquest resultat per a provar resultats de convergència en llei cap a
altres processos, com ara el moviment Brownià sub-fraccionari.
En el Capítol 2 construïm i estudiem diferents model que pretenen estudiar
el comportament d’un tractament amb bacteriòfags en certs animals de granja.
Aquest problema ha estat motivat pel Grup de Biologia Molecular del Departament
de Genètica i Microbiologia de la Universitat Autònoma de Barcelona.
Començant per un model bàsic, n’estudiarem diferent variacions, primer des
d’un punt de vista determinista, trobant diversos resultat sobre els equilibris i
l’estabilitat, i després en un context amb soroll, produint resultats de concentració.
Finalment, en el Capítol 3 estudiarem la descomposició en caos de Wiener del
L2 mòdul de continuïtat del temps local del Brownià. Més concretament, trobarem
un Teorema Central del Límit per a cada element del caos de Wiener del L2 mòdul
de continuïtat del temps local del Brownià. Aquest resultat ens proporciona un
exemple d’una família de variables que convergeix en llei cap a una distribució
Normal, però que els elements del seu caos d’ordre parell no convergeixen.In this dissertation three different problems are treated. In Chapter 1 we construct two families of processes that converge, in the sense of the finite dimensional distributions, towards two independent Gaussian processes. Chapter 2 is devoted to the study of a model of bacteriophage treatments for bacterial infections. Finally, in Chapter 3 we study some aspects of the L2 modulus of continuity of Brownian local time. In the first chapter we consider two independent Gaussian processes that can be represented in terms of a stochastic integral of a deterministic kernel with respect to the Wiener process and we construct, from a single Poisson process, two families of processes that converge, in the sense of the finite dimensional distributions, towards these Gaussian processes. We will use this result to prove convergence in law results towards some other processes, like sub-fractional Brownian motion. In Chapter 2 we construct and study several models that pretend to study how will behave a treatment of bateriophages in some farm animals. This problem has been brought to our attention by the Molecular Biology Group of the Department of Genetics and Microbiology at the Universitat Autònoma de Barcelona. Starting from a basic model, we will study several variations, first from a deterministic point of view, finding several results on equilibria and stability, and later in a noisy context, producing concentration type results. Finally, in Chapter 3 we shall study the decomposition on Wiener chaos of the L2 modulus of continuity of the Brownian local time. More precisely, we shall find a Central Limit Theorem for each Wiener chaos element of the L2 modulus of continuity of the Brownian local time. This result provides us with an example of a family of random variables that is convergent in law to a Normal distribution, but its chaos elements of even order do not converge.
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Pieczynski, Wojciech. "Sur diverses applications de la décantation des lois de probabilité dans la théorie générale de l'estimation statistique." Paris 6, 1986. http://www.theses.fr/1986PA066064.
Full textAbstract:
On cherche à construire des estimateurs convergents dans le cas des V. A. Non nécessairement indépendantes et équidistribuées. La méthode de la décantation est particulièrement adaptée car elle permet la construction explicite de tels estimateurs et donne des renseignements sur leur vitesse de convergence.
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Kponsou, Messan. "Estimation spectrale dans les processus à n-accroissement stationnaires." Rouen, 1997. http://www.theses.fr/1997ROUES071.
Full textAbstract:
Nous considérons un processus à accroissements aléatoires stationnaires d'ordre n. Tout d'abord nous donnons quelques propriétés de ce processus. Dans la première partie, le processus est à temps discret et nous estimons sa densité spectrale d'ordre n à partir d'un échantillon en temps discret. Nous donnons une condition suffisante de convergence uniforme presque complète de l'estimateur vers la densité spectrale. Dans la seconde partie, le processus est à temps continu et nous estimons sa densité spectrale d'ordre n à partir d'un échantillon en temps continu. Nous donnons également une condition suffisante de convergence uniforme presque complète de l'estimateur vers la densité spectrale. Dans la troisième partie, nous estimons la densité spectrale d'ordre n du processus à temps continu à partir d'un échantillonnage alias-free de taille aléatoire. La quatrième partie concerne la comparaison de deux méthodes échantillonnage : échantillonnage poissonnien et non poissonnien. Dans la dernière partie, nous calculons les estimateurs du chapitre 3 à partir de données simulées.
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Leobacher, Gunther, and Michaela Szölgyenyi. "Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient." Springer Nature, 2018. http://dx.doi.org/10.1007/s00211-017-0903-9.
Full textAbstract:
We prove strong convergence of order 1/4 - E for arbitrarily small E > 0 of
the Euler-Maruyama method for multidimensional stochastic differential equations
(SDEs) with discontinuous drift and degenerate diffusion coefficient. The proof is
based on estimating the difference between the Euler-Maruyama scheme and another
numerical method, which is constructed by applying the Euler-Maruyama scheme to
a transformation of the SDE we aim to solve.
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Alt, Jean-Christian. "Sur le comportement asymptotique presque sur des sommes de variables aleatoires a valeurs vectorielles." Université Louis Pasteur (Strasbourg) (1971-2008), 1988. http://www.theses.fr/1988STR13017.
Full textAbstract:
Cette these se compose de trois articles consacres a l'etude de la loi forte des grands nombres pour des variables aleatoires a valeurs dans un espace de banach. Nous placant dans des cadres empruntes aux theoremes classiques du calcul des probabilites sur la droite reelle, nous etendons en particulier a la dimension infinie le theoreme de prokhorov et la loi du logarithme itere de kolmogorov. Nos resultats font apparaitre que les substituts naturels des variances du cas reel sont des variances faibles, et non les moments forts d'ordre deux precedemment utilises pour l'etude de la loi forte des grands nombres vectorielle. Les demonstrations utilisent essentiellement des techniques provenant de l'etude des fonctions aleatoires: randomisation par des variables aleatoires gaussiennes ou des variables de rademacher, inegalites de deviation gaussienne de c. Borell, inegalite isoperimetrique de m. Talagrand
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Cronvald, Kristofer. "An introduction to Multilevel Monte Carlo with applications to options." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-166671.
Full textAbstract:
A standard problem in mathematical finance is the calculation of the price of some financial derivative such as various types of options. Since there exists analytical solutions in only a few cases it will often boil down to estimating the price with Monte Carlo simulation in conjunction with some numerical discretization scheme. The upside of using what we can call standard Monte Carlo is that it is relative straightforward to apply and can be used for a wide variety of problems. The downside is that it has a relatively slow convergence which means that the computational cost or complexity can be very large. However, this slow convergence can be improved upon by using Multilevel Monte Carlo instead of standard Monte Carlo. With this approach it is possible to reduce the computational complexity and cost of simulation considerably. The aim of this thesis is to introduce the reader to the Multilevel Monte Carlo method with applications to European and Asian call options in both the Black-Scholes-Merton (BSM) model and in the Heston model. To this end we first cover the necessary background material such as basic probability theory, estimators and some of their properties, the stochastic integral, stochastic processes and Ito’s theorem. We introduce stochastic differential equations and two numerical discretizations schemes, the Euler–Maruyama scheme and the Milstein scheme. We define strong and weak convergence and illustrate these concepts with examples. We also describe the standard Monte Carlo method and then the theory and implementation of Multilevel Monte Carlo. In the applications part we perform numerical experiments where we compare standard Monte Carlo to Multilevel Monte Carlo in conjunction with the Euler–Maruyama scheme and Milsteins scheme. In the case of a European call in the BSM model, using the Euler–Maruyama scheme, we achieved a cost O(ε-2(log ε)2) to reach the desired error in accordance with theory in comparison to the O(ε-3) cost for standard Monte Carlo. When using Milsteins scheme instead of the Euler–Maruyama scheme it was possible to reduce the cost in terms of the number of simulations needed to achieve the desired error even further. By using Milsteins scheme, a method with greater order of strong convergence than Euler–Maruyama, we achieved the O(ε-2) cost predicted by the complexity theorem compared to the standard Monte Carlo cost of order O(ε-3). In the final numerical experiment we applied the Multilevel Monte Carlo method together with the Euler–Maruyama scheme to an Asian call in the Heston model. In this case, where the coefficients of the Heston model do not satisfy a global Lipschitz condition, the study of strong or weak convergence is much harder. The numerical experiments suggested that the strong convergence was slightly slower compared to what was found in the case of a European call in the BSM model. Nevertheless, we still achieved substantial savings in computational cost compared to using standard Monte Carlo.
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Hall, Eric Joseph. "Accelerated numerical schemes for deterministic and stochastic partial differential equations of parabolic type." Thesis, University of Edinburgh, 2013. http://hdl.handle.net/1842/8038.
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First we consider implicit finite difference schemes on uniform grids in time and space for second order linear stochastic partial differential equations of parabolic type. Under sufficient regularity conditions, we prove the existence of an appropriate asymptotic expansion in powers of the the spatial mesh and hence we apply Richardson's method to accelerate the convergence with respect to the spatial approximation to an arbitrarily high order. Then we extend these results to equations where the parabolicity condition is allowed to degenerate. Finally, we consider implicit finite difference approximations for deterministic linear second order partial differential equations of parabolic type and give sufficient conditions under which the approximations in space and time can be simultaneously accelerated to an arbitrarily high order.
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Olivero, Quinteros Héctor Cristian. "Strong convergence of a milstein scheme for a CEV-like SDE and some contributions to the analysis of the stochastic Morris-Lecar neuron model." Tesis, Universidad de Chile, 2016. http://repositorio.uchile.cl/handle/2250/143568.
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Doctor en Ciencias de la Ingeniería, Mención Modelación MatemáticaDesde muy temprano en el desarrollo de la teoría de procesos estocásticos ha existido un creciente interés por aplicar sus herramientas en diferentes contextos; ya en el año 1900 Bachelier creó un modelo de movimiento Browniano para describir el mercado de acciones en París [7], y desde entonces el rango de aplicaciones del modelamiento estocástico ha seguido creciendo y hoy en día incluye desde economía hasta biología. Esta tesis tiene dos partes, cada una de ellas dedicada al estudio de un modelo estocástico diferente. En la primera se estudia la aproximación numérica de la solución de una ecuación diferencial estocástica con aplicaciones en finanzas. Mientras que en la segunda se estudia un modelo estocástico para neuronas con énfasis en los comportamientos asintóticos . La primera parte de esta tesis se organiza como sigue. En el Capítulo 2 se presenta una breve introducción a los métodos clásicos de aproximación de soluciones de ecuaciones difer- enciales estocásticas y se recuerdan sus propiedades de convergencia. Luego, en el Capítulo 3 se estudia un esquema numérico para aproximar las soluciones de dX_t =b(X_s)ds + σ|Xs|^αdWs. Esta ecuación se puede ver como la generalización del modelo CIR para tasas de interés y tiene un gran rango de aplicaciones en finanzas. El principal resultado de este capítulo es la convergencia fuerte con tasa 1 del esquema numérico estudiado a la solución exacta de la ecuación. Este capítulo está basado en un trabajo conjunto con Mireille Bossy [16], el cual ha sido aceptado para su publicación en la revista Bernoulli. En la segunda parte de esta tesis se estudia el modelo de Morris-Lecar para una red de neuronas. El principal objetivo es estudiar el comportamiento del sistema cuando el tiempo o el número de neuronas se va a infinito. Sin embargo, antes de abordar esas temáticas, se discuten dos versiones estocásticas para el modelo de Morris-Lecar, y la relación entre ellas. Los resultados principales de esta parte de la tesis son la caracterización del comportamiento límite, en intervalos de tiempo finito, para una red de neuronas cuando el número de neuronas diverge a infinito y un resultado de sincronización para una red finita de neuronas cuando el tiempo diverge a infinito.
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Angoshtari, Bahman. "Stochastic modeling and methods for portfolio management in cointegrated markets." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:1ae9236c-4bf0-4d9b-a694-f08e1b8713c0.
Full textAbstract:
In this thesis we study the utility maximization problem for assets whose prices are cointegrated, which arises from the investment practice of convergence trading and its special forms, pairs trading and spread trading. The major theme in the first two chapters of the thesis, is to investigate the assumption of market-neutrality of the optimal convergence trading strategies, which is a ubiquitous assumption taken by practitioners and academics alike. This assumption lacks a theoretical justification and, to the best of our knowledge, the only relevant study is Liu and Timmermann (2013) which implies that the optimal convergence strategies are, in general, not market-neutral. We start by considering a minimalistic pairs-trading scenario with two cointegrated stocks and solve the Merton investment problem with power and logarithmic utilities. We pay special attention to when/if the stochastic control problem is well-posed, which is overlooked in the study done by Liu and Timmermann (2013). In particular, we show that the problem is ill-posed if and only if the agent’s risk-aversion is less than a constant which is an explicit function of the market parameters. This condition, in turn, yields the necessary and sufficient condition for well-posedness of the Merton problem for all possible values of agent’s risk-aversion. The resulting well-posedness condition is surprisingly strict and, in particular, is equivalent to assuming the optimal investment strategy in the stocks to be market-neutral. Furthermore, it is shown that the well-posedness condition is equivalent to applying Novikov’s condition to the market-price of risk, which is a ubiquitous sufficient condition for imposing absence of arbitrage. To the best of our knowledge, these are the only theoretical results for supporting the assumption of market-neutrality of convergence trading strategies. We then generalise the results to the more realistic setting of multiple cointegrated assets, assuming risk factors that effects the asset returns, and general utility functions for investor’s preference. In the process of generalising the bivariate results, we also obtained some well-posedness conditions for matrix Riccati differential equations which are, to the best of our knowledge, new. In the last chapter, we set up and justify a Merton problem that is related to spread-trading with two futures assets and assuming proportional transaction costs. The model possesses three characteristics whose combination makes it different from the existing literature on proportional transaction costs: 1) finite time horizon, 2) Multiple risky assets 3) stochastic opportunity set. We introduce the HJB equation and provide rigorous arguments showing that the corresponding value function is the viscosity solution of the HJB equation. We end the chapter by devising a numerical scheme, based on the penalty method of Forsyth and Vetzal (2002), to approximate the viscosity solution of the HJB equation.
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Roininen, L. (Lassi). "Discretisation-invariant and computationally efficient correlation priors for Bayesian inversion." Doctoral thesis, University of Oulu, 2015. http://urn.fi/urn:isbn:9789526207544.
Full textAbstract:
Abstract
We are interested in studying Gaussian Markov random fields as correlation priors for Bayesian inversion. We construct the correlation priors to be discretisation-invariant, which means, loosely speaking, that the discrete priors converge to continuous priors at the discretisation limit. We construct the priors with stochastic partial differential equations, which guarantees computational efficiency via sparse matrix approximations. The stationary correlation priors have a clear statistical interpretation through the autocorrelation function.
We also consider how to make structural model of an unknown object with anisotropic and inhomogeneous Gaussian Markov random fields. Finally we consider these fields on unstructured meshes, which are needed on complex domains.
The publications in this thesis contain fundamental mathematical and computational results of correlation priors. We have considered one application in this thesis, the electrical impedance tomography. These fundamental results and application provide a platform for engineers and researchers to use correlation priors in other inverse problem applications.
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Norgren, Axel, and Martin Olsson. "Institutional Dynamics in the Global FDI Network : Examining The Co-evolution of Institutions and FDI with Stochastic Actor-Oriented Modelling." Thesis, Linköpings universitet, Nationalekonomi, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-176549.
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This thesis addresses the relationship between institutions and foreign direct investments (FDI). While the issue of how institutions attract FDI (selection) is quite well-researched, the empirical evidence for institutions spreading through FDI (influence) is more ambiguous. We argue that past studies have neglected issues of endogeneity and interdependence in their modelling. We amend these issues by using a Stochastic Actor-Oriented network model which allows for interdependent and endogenous processes. The thesis also addresses the mechanisms governing the general relation between FDI and institutions and what these can tell us about institutional change and the process of globalisation. The model provides no evidence that FDI helps to spread institutions from home to host countries, but it does provide evidence that the selection effect can be an important dynamic between FDI and a certain set of institutions. Finally, we argue that FDI does not seem to be a contributory factor to institutional convergence.
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Messaci, Fatiha. "Estimation de la densité spectrale d'un processus en temps continu par échantillonage poissonnien." Rouen, 1986. http://www.theses.fr/1986ROUES036.
Full textAbstract:
Ce travail est consacré à l'estimation de la densité spectrale d'un processus réel, par échantillonnage poissonnien. Après l'étude théorique, le calcul des estimateurs a été effectué sur des données simulées d'un processus de Gauss Markov, puis d'un processus gaussien non markovien
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BRITO, MARGARIDA. "Encadrement presque sur des statistiques d'ordre." Paris 6, 1987. http://www.theses.fr/1987PA066284.
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Soit k(n) une suite non-decroissante d'entiers positifs. Sous certaines hypotheses pour la suite k(n), on determine des encadrements presque surement optimaux de la k(n)eme statistique d'ordre d'un echantillon de taille n. On commence par aborder le cas ou k(n) est inferieur ou egal a log(log(n)). En utilisant des approximations des queues de la loi binomiale, obtenues a partir des techniques usuelles de la theorie des grandes deviations, on determine d'abord des suites qui majorent ou minorent de facon optimale la k(n)eme statistique d'ordre d'un echantillon uniforme. On applique ensuite les resultats obtenus aux lois de probabilite actuelles
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Honore, Valentin. "Convergence HPC - Big Data : Gestion de différentes catégories d'applications sur des infrastructures HPC." Thesis, Bordeaux, 2020. http://www.theses.fr/2020BORD0145.
Full textAbstract:
Le calcul haute performance est un domaine scientifique dans lequel de très complexes et intensifs calculs sont réalisés sur des infrastructures de calcul à très large échelle appelées supercalculateurs. Leur puissance calculatoire phénoménale permet aux supercalculateurs de générer un flot de données gigantesque qu'il est aujourd'hui difficile d'appréhender, que ce soit d'un point de vue du stockage en mémoire que de l'extraction des résultats les plus importants pour les applications.Nous assistons depuis quelques années à une convergence entre le calcul haute performance et des domaines tels que le BigData ou l'intelligence artificielle qui voient leurs besoins en terme de capacité de calcul exploser. Dans le cadre de cette convergence, une grande diversité d'applications doit être traitée par les ordonnanceurs des supercalculateurs, provenant d'utilisateurs de différents horizons pour qui il n'est pas toujours aisé de comprendre le fonctionnement de ces infrastructures pour le calcul distribué.Dans cette thèse, nous exposons des solutions d'ordonnancement et de partitionnement de ressources pour résoudre ces problématiques. Pour ce faire, nous proposons une approche basée sur des modèles mathématiques qui permet d'obtenir des solutions avec de fortes garanties théoriques de leu performance. Dans ce manuscrit, nous nous focalisons sur deux catégories d'applications qui s'inscrivent en droite ligne avec la convergence entre le calcul haute performance et le BigData:les applications intensives en données et les applications à temps d'exécution stochastique.Les applications intensives en données représentent les applications typiques du domaine du calcul haute performance. Dans cette thèse, nous proposons d'optimiser cette catégorie d'applications exécutées sur des supercalculateurs en exposant des méthodes automatiques de partitionnement de ressources ainsi que des algorithmes d'ordonnancement pour les différentes phases de ces applications. Pour ce faire, nous utilisons le paradigme in situ, devenu à ce jour une référence pour ces applications. De nombreux travaux se sont attachés à proposer des solutions logicielles pour mettre en pratique ce paradigme pour les applications. Néanmoins, peu de travaux ont étudié comment efficacement partager les ressources de calcul les différentes phases des applications afin d'optimiser leur temps d'exécution.Les applications stochastiques constituent la deuxième catégorie d'applications que nous étudions dans cette thèse. Ces applications ont un profil différent de celles de la première partie de ce manuscrit. En effet, contrairement aux applications de simulation numérique, ces applications présentent de fortes variations de leur temps d'exécution en fonction des caractéristiques du jeu de données fourni en entrée. Cela est dû à leur structure interne composée d'une succession de fonctions, qui diffère des blocs de code massifs composant les applications intensive en données.L'incertitude autour de leur temps d'exécution est une contrainte très forte pour lancer ces applications sur les supercalculateurs. En effet, l'utilisateur doit réserver des ressources de calcul pour une durée qu'il ne connait pas. Dans cette thèse, nous proposons une approche novatrice pour aider les utilisateurs à déterminer une séquence de réservations optimale qui minimise l'espérance du coût total de toutes les réservations. Ces solutions sont par la suite étendues à un modèle d'applications avec points de sauvegarde à la fin de (certaines) réservations afin d'éviter de perdre le travail réalisé lors des réservations trop courtes. Enfin, nous proposons un profiling d'une application stochastique issue du domaine des neurosciences afin de mieux comprendre les propriétés de sa stochasticité. A travers cette étude, nous montrons qu'il est fondamental de bien connaître les caractéristiques des applications pour qui souhaite élaborer des stratégies efficaces du point de vue de l'utilisateurNumerical simulations are complex programs that allow scientists to solve, simulate and model complex phenomena. High Performance Computing (HPC) is the domain in which these complex and heavy computations are performed on large-scale computers, also called supercomputers.Nowadays, most scientific fields need supercomputers to undertake their research. It is the case of cosmology, physics, biology or chemistry. Recently, we observe a convergence between Big Data/Machine Learning and HPC. Applications coming from these emerging fields (for example, using Deep Learning framework) are becoming highly compute-intensive. Hence, HPC facilities have emerged as an appropriate solution to run such applications. From the large variety of existing applications has risen a necessity for all supercomputers: they mustbe generic and compatible with all kinds of applications. Actually, computing nodes also have a wide range of variety, going from CPU to GPU with specific nodes designed to perform dedicated computations. Each category of node is designed to perform very fast operations of a given type (for example vector or matrix computation).Supercomputers are used in a competitive environment. Indeed, multiple users simultaneously connect and request a set of computing resources to run their applications. This competition for resources is managed by the machine itself via a specific program called scheduler. This program reviews, assigns andmaps the different user requests. Each user asks for (that is, pay for the use of) access to the resources ofthe supercomputer in order to run his application. The user is granted access to some resources for a limited amount of time. This means that the users need to estimate how many compute nodes they want to request and for how long, which is often difficult to decide.In this thesis, we provide solutions and strategies to tackle these issues. We propose mathematical models, scheduling algorithms, and resource partitioning strategies in order to optimize high-throughput applications running on supercomputers. In this work, we focus on two types of applications in the context of the convergence HPC/Big Data: data-intensive and irregular (orstochastic) applications.Data-intensive applications represent typical HPC frameworks. These applications are made up oftwo main components. The first one is called simulation, a very compute-intensive code that generates a tremendous amount of data by simulating a physical or biological phenomenon. The second component is called analytics, during which sub-routines post-process the simulation output to extract,generate and save the final result of the application. We propose to optimize these applications by designing automatic resource partitioning and scheduling strategies for both of its components.To do so, we use the well-known in situ paradigm that consists in scheduling both components together in order to reduce the huge cost of saving all simulation data on disks. We propose automatic resource partitioning models and scheduling heuristics to improve overall performance of in situ applications.Stochastic applications are applications for which the execution time depends on its input, while inusual data-intensive applications the makespan of simulation and analytics are not affected by such parameters. Stochastic jobs originate from Big Data or Machine Learning workloads, whose performanceis highly dependent on the characteristics of input data. These applications have recently appeared onHPC platforms. However, the uncertainty of their execution time remains a strong limitation when using supercomputers. Indeed, the user needs to estimate how long his job will have to be executed by the machine, and enters this estimation as his first reservation value. But if the job does not complete successfully within this first reservation, the user will have to resubmit the job, this time requiring a longer reservation