Dissertations / Theses on the topic 'McKean stochastic differential equation'
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1
McMurray, Eamon Finnian Valentine. "Regularity of McKean-Vlasov stochastic differential equations and applications." Thesis, Imperial College London, 2015. http://hdl.handle.net/10044/1/28918.
Abstract:
In this thesis, we study time-inhomogeneous and McKean-Vlasov type stochastic differential equations (SDEs), along with related partial differential equations (PDEs). We are particularly interested in regularity estimates and their applications to numerical methods. In the first part of the thesis, we build on the work of Kusuoka \& Stroock to develop sharp estimates on the derivatives of solutions to time-inhomogeneous parabolic PDEs. The basis of these estimates is an integration by parts formula for derivatives of the solution under the UFG condition, which is weaker than the uniform Hoermander condition. This integration by parts formula is obtained using Malliavin Calculus. The formula allows us to extend the notion of classical solution to a framework where differentiability does not necessarily hold in all directions. As an application, we extend the error analysis for the cubature on Wiener space method to time-inhomogeneous stochastic differential equations. We then present two cubature on Wiener space algorithms for the numerical solution of McKean-Vlasov SDEs with smooth scalar interaction. The analysis involves the regularity estimates proved previously and takes place under a uniform strong Hoermander condition. Finally, we develop integration by parts formulas on Wiener space for solutions of SDEs with general McKean-Vlasov interaction and uniformly elliptic coefficients. These formulas hold both for derivatives with respect to a real variable and derivatives with respect to a measure in the sense of Lions. This allows us to develop estimates on the density of solutions of the McKean-Vlasov SDEs. We also prove the existence of a classical solution to a related PDE with irregular terminal condition.
2
Mezerdi, Mohamed Amine. "Equations différentielles stochastiques de type McKean-Vlasov et leur contrôle optimal." Electronic Thesis or Diss., Toulon, 2020. http://www.theses.fr/2020TOUL0014.
Abstract:
Nous considérons les équations différentielles stochastiques (EDS) de Mc Kean-Vlasov, qui sont des EDS dont les coefficients de dérive et de diffusion dépendent non seulement de l'état du processus inconnu, mais également de sa loi de probabilité. Ces EDS, également appelées EDS à champ moyen, ont d'abord été étudiées en physique statistique et représentent en quelque sorte le comportement moyen d'un nombre infini de particules. Récemment, ce type d'équations a suscité un regain d'intérêt dans le contexte de la théorie des jeux à champ moyen. Cette théorie a été inventée par P.L. Lions et J.M. Lasry en 2006, pour résoudre le problème de l'existence d'un équilibre de Nash approximatif pour les jeux différentiels, avec un grand nombre de joueurs. Ces équations ont trouvé des applications dans divers domaines tels que la théorie des jeux, la finance mathématique, les réseaux de communication et la gestion des ressources pétrolières. Dans cette thèse, nous avons étudié les questions de stabilité par rapport aux données initiales, aux coefficients et aux processus directeurs des équations de McKean-Vlasov. Les propriétés génériques de ce type d'équations stochastiques, telles que l'existence et l'unicité, la stabilité par rapport aux paramètres, ont été examinées. En théorie du contrôle, notre attention s'est portée sur l'existence et l'approximation de contrôles relaxés pour les systèmes gouvernés par des EDS de Mc Kean-VlasovWe consider Mc Kean-Vlasov stochastic differential equations (SDEs), which are SDEs where the drift and diffusion coefficients depend not only on the state of the unknown process but also on its probability distribution. These SDEs called also mean- field SDEs were first studied in statistical physics and represent in some sense the average behavior of an infinite number of particles. Recently there has been a renewed interest for this kind of equations in the context of mean-field game theory. Since the pioneering papers by P.L. Lions and J.M. Lasry, mean-field games and mean-field control theory has raised a lot of interest, motivated by applications to various fields such as game theory, mathematical finance, communications networks and management of oil resources. In this thesis, we studied questions of stability with respect to initial data, coefficients and driving processes of Mc Kean-Vlasov equations. Generic properties for this type of SDEs, such as existence and uniqueness, stability with respect to parameters, have been investigated. In control theory, our attention were focused on existence, approximation of relaxed controls for controlled Mc Kean-Vlasov SDEs
3
Izydorczyk, Lucas. "Probabilistic backward McKean numerical methods for PDEs and one application to energy management." Electronic Thesis or Diss., Institut polytechnique de Paris, 2021. http://www.theses.fr/2021IPPAE008.
Abstract:
Cette thèse s'intéresse aux équations différentielles stochastiques de type McKean(EDS) et à leur utilisation pour représenter des équations aux dérivées partielles (EDP) non linéaires. Ces équations ne dépendent pas seulement du temps et de la position d'une certaine particule mais également de sa loi. En particulier nous traitons le cas inhabituel de la représentation d'EDP de type Fokker-Planck avec condition terminale fixée. Nous discutons existence et unicité pour ces EDP et de leur représentation sous la forme d'une EDS de type McKean, dont l'unique solutioncorrespond à la dynamique du retourné dans le temps d'un processus de diffusion.Nous introduisons la notion de représentation complètement non-linéaire d'une EDP semilinéaire. Celle-ci consiste dans le couplage d'une EDS rétrograde et d'un processus solution d'une EDS évoluant de manière rétrograde dans le temps. Nous discutons également une application à la représentation d'une équation d'Hamilton-Jacobi-Bellman (HJB) en contrôle stochastique. Sur cette base, nous proposonsun algorithme de Monte-Carlo pour résoudre des problèmes de contrôle. Celui ciest avantageux en termes d'efficience calculatoire et de mémoire, en comparaisonavec les approches traditionnelles progressive rétrograde. Nous appliquons cette méthode dans le contexte de la gestion de la demande dans les réseaux électriques. Pour finir, nous faisons le point sur l'utilisation d'EDS de type McKean généralisées pour représenter des EDP non-linéaires et non-conservatives plus générales que Fokker-PlanckThis thesis concerns McKean Stochastic Differential Equations (SDEs) to representpossibly non-linear Partial Differential Equations (PDEs). Those depend not onlyon the time and position of a given particle, but also on its probability law. In particular, we treat the unusual case of Fokker-Planck type PDEs with prescribed final data. We discuss existence and uniqueness for those equations and provide a probabilistic representation in the form of McKean type equation, whose unique solution corresponds to the time-reversal dynamics of a diffusion process.We introduce the notion of fully backward representation of a semilinear PDE: thatconsists in fact in the coupling of a classical Backward SDE with an underlying processevolving backwardly in time. We also discuss an application to the representationof Hamilton-Jacobi-Bellman Equation (HJB) in stochastic control. Based on this, we propose a Monte-Carlo algorithm to solve some control problems which has advantages in terms of computational efficiency and memory whencompared to traditional forward-backward approaches. We apply this method in the context of demand side management problems occurring in power systems. Finally, we survey the use of generalized McKean SDEs to represent non-linear and non-conservative extensions of Fokker-Planck type PDEs
4
Le, cavil Anthony. "Représentation probabiliste de type progressif d'EDP nonlinéaires nonconservatives et algorithmes particulaires." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLY023.
Abstract:
Dans cette thèse, nous proposons une approche progressive (forward) pour la représentation probabiliste d'Equations aux Dérivées Partielles (EDP) nonlinéaires et nonconservatives, permettant ainsi de développer un algorithme particulaire afin d'en estimer numériquement les solutions. Les Equations Différentielles Stochastiques Nonlinéaires de type McKean (NLSDE) étudiées dans la littérature constituent une formulation microscopique d'un phénomène modélisé macroscopiquement par une EDP conservative. Une solution d'une telle NLSDE est la donnée d'un couple $(Y,u)$ où $Y$ est une solution d' équation différentielle stochastique (EDS) dont les coefficients dépendent de $u$ et de $t$ telle que $u(t,cdot)$ est la densité de $Y_t$. La principale contribution de cette thèse est de considérer des EDP nonconservatives, c'est-à- dire des EDP conservatives perturbées par un terme nonlinéaire de la forme $Lambda(u,nabla u)u$. Ceci implique qu'un couple $(Y,u)$ sera solution de la représentation probabiliste associée si $Y$ est un encore un processus stochastique et la relation entre $Y$ et la fonction $u$ sera alors plus complexe. Etant donnée la loi de $Y$, l'existence et l'unicité de $u$ sont démontrées par un argument de type point fixe via une formulation originale de type Feynmann-KacThis thesis performs forward probabilistic representations of nonlinear and nonconservative Partial Differential Equations (PDEs), which allowto numerically estimate the corresponding solutions via an interacting particle system algorithm, mixing Monte-Carlo methods and non-parametric density estimates.In the literature, McKean typeNonlinear Stochastic Differential Equations (NLSDEs) constitute the microscopic modelof a class of PDEs which are conservative. The solution of a NLSDEis generally a couple $(Y,u)$ where $Y$ is a stochastic process solving a stochastic differential equation whose coefficients depend on $u$ and at each time $t$, $u(t,cdot)$ is the law density of the random variable $Y_t$.The main idea of this thesis is to consider this time a non-conservative PDE which is the result of a conservative PDE perturbed by a term of the type $Lambda(u, nabla u) u$. In this case, the solution of the corresponding NLSDE is again a couple $(Y,u)$, where again $Y$ is a stochastic processbut where the link between the function $u$ and $Y$ is more complicated and once fixed the law of $Y$, $u$ is determined by a fixed pointargument via an innovating Feynmann-Kac type formula
5
Le, cavil Anthony. "Représentation probabiliste de type progressif d'EDP nonlinéaires nonconservatives et algorithmes particulaires." Electronic Thesis or Diss., Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLY023.
Abstract:
Dans cette thèse, nous proposons une approche progressive (forward) pour la représentation probabiliste d'Equations aux Dérivées Partielles (EDP) nonlinéaires et nonconservatives, permettant ainsi de développer un algorithme particulaire afin d'en estimer numériquement les solutions. Les Equations Différentielles Stochastiques Nonlinéaires de type McKean (NLSDE) étudiées dans la littérature constituent une formulation microscopique d'un phénomène modélisé macroscopiquement par une EDP conservative. Une solution d'une telle NLSDE est la donnée d'un couple (Y,u) où Y est une solution d' équation différentielle stochastique (EDS) dont les coefficients dépendent de u et de t telle que u(t,.) est la densité de Yt. La principale contribution de cette thèse est de considérer des EDP nonconservatives, c'est-à- dire des EDP conservatives perturbées par un terme nonlinéaire de la forme Lambda(u,nabla u)u. Ceci implique qu'un couple (Y,u) sera solution de la représentation probabiliste associée si Y est un encore un processus stochastique et la relation entre Y et la fonction u sera alors plus complexe. Etant donnée la loi de Y, l'existence et l'unicité de u sont démontrées par un argument de type point fixe via une formulation originale de type Feynmann-KacThis thesis performs forward probabilistic representations of nonlinear and nonconservative Partial Differential Equations (PDEs), which allowto numerically estimate the corresponding solutions via an interacting particle system algorithm, mixing Monte-Carlo methods and non-parametric density estimates.In the literature, McKean typeNonlinear Stochastic Differential Equations (NLSDEs) constitute the microscopic modelof a class of PDEs which are conservative. The solution of a NLSDEis generally a couple (Y,u) where Y is a stochastic process solving a stochastic differential equation whose coefficients depend on u and at each time t, u(t,.) is the law density of the random variable Yt.The main idea of this thesis is to consider this time a non-conservative PDE which is the result of a conservative PDE perturbed by a term of the type Lambda(u, nabla u) u. In this case, the solution of the corresponding NLSDE is again a couple (Y,u), where again Y is a stochastic processbut where the link between the function u and Y is more complicated and once fixed the law of Y, u is determined by a fixed pointargument via an innovating Feynmann-Kac type formula
6
Treacy, Brian. "A stochastic differential equation derived from evolutionary game theory." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-377554.
7
Al-Saadony, Muhannad. "Bayesian stochastic differential equation modelling with application to finance." Thesis, University of Plymouth, 2013. http://hdl.handle.net/10026.1/1530.
Abstract:
In this thesis, we consider some popular stochastic differential equation models used in finance, such as the Vasicek Interest Rate model, the Heston model and a new fractional Heston model. We discuss how to perform inference about unknown quantities associated with these models in the Bayesian framework. We describe sequential importance sampling, the particle filter and the auxiliary particle filter. We apply these inference methods to the Vasicek Interest Rate model and the standard stochastic volatility model, both to sample from the posterior distribution of the underlying processes and to update the posterior distribution of the parameters sequentially, as data arrive over time. We discuss the sensitivity of our results to prior assumptions. We then consider the use of Markov chain Monte Carlo (MCMC) methodology to sample from the posterior distribution of the underlying volatility process and of the unknown model parameters in the Heston model. The particle filter and the auxiliary particle filter are also employed to perform sequential inference. Next we extend the Heston model to the fractional Heston model, by replacing the Brownian motions that drive the underlying stochastic differential equations by fractional Brownian motions, so allowing a richer dependence structure across time. Again, we use a variety of methods to perform inference. We apply our methodology to simulated and real financial data with success. We then discuss how to make forecasts using both the Heston and the fractional Heston model. We make comparisons between the models and show that using our new fractional Heston model can lead to improve forecasts for real financial data.
8
Li, Shuang. "Study of Various Stochastic Differential Equation Models for Finance." Thesis, Curtin University, 2017. http://hdl.handle.net/20.500.11937/56545.
Abstract:
The first part of the study focuses on European and American option pricing. We explore the jump diffusion models with stochastic volatility within the general equilibrium framework and use the minimal martingale measure as martingale measure. The second part of the thesis is on portfolio optimization. We formulate optimal asset allocation problem with multiple-periods under mean variance utility in the game theoretic framework, develop and solve a series of extended HJB equations for the problem.
9
Botha, Imke. "Bayesian inference for stochastic differential equation mixed effects models." Thesis, Queensland University of Technology, 2020. https://eprints.qut.edu.au/198039/1/Imke_Botha_Thesis.pdf.
Abstract:
Stochastic differential equation mixed effects models (SDEMEMs) are increasingly used in biomedical and pharmacokinetic/pharmacodynamic research. However, the complexity of these models means that previous research has focussed on approximate parameter estimation methods. This thesis develops three novel Bayesian parameter estimation methods for SDEMEMs. The new methods can produce parameter estimates that are more accurate and provide more reliable uncertainty quantification. The new methods are applied to both real and simulated data from a tumour xenography study on mice.
10
Zararsiz, Zarife. "On an epidemic model given by a stochastic differential equation." Thesis, Växjö University, School of Mathematics and Systems Engineering, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:vxu:diva-5747.
11
Ahmad, Ferhana. "A stochastic partial differential equation approach to mortgage backed securities." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:ee33aa2d-b9fa-4cc4-a399-5f681966bc77.
Abstract:
The market for mortgage backed securities (MBS) was active and fast growing from the issuance of the first MBS in 1981. This enabled financial firms to transform risky individual mortgages into liquid and tradable market instruments. The subprime mortgage crisis of 2007 shows the need for a better understanding and development of mathematical models for these securities. The aim of this thesis is to develop a model for MBS that is flexible enough to capture both regular and subprime MBS. The thesis considers two models, one for a single mortgage in an intensity based framework and the second for mortgage backed securities using a stochastic partial differential equation approach. In the model for a single mortgage, we capture the prepayment and default incentives of the borrower using intensity processes. Using the minimum of the two intensity processes, we develop a nonlinear equation for the mortgage rate and solve it numerically and present some case studies. In modelling of an MBS in a structural framework using stochastic PDEs (SPDEs), we consider a large number of individuals in a mortgage pool and assume that the wealth of each individual follows a stochastic process, driven by two Brownian mo- tions, one capturing the idiosyncratic noise of each individual and the second a common market factor. By defining the empirical measure of a large pool of these individuals we study the evolution of the limit empirical measure and derive an SPDE for the evolution of the density of the limit empirical measure. We numerically solve the SPDE to demonstrate its flexibility in different market environments. The calibration of the model to financial data is the focus of the final part of thesis. We discuss the different parameters and demonstrate how many can be fitted to observed data. Finally, for the key model parameters, we present a strategy to estimate them given observations of the loss function and use this to determine implied model parameters of ABX.HE.
12
Brites, Nuno M. "Stochastic differential equation harvesting models: sustainable policies and profit optimization." Doctoral thesis, Universidade de Évora, 2017. http://hdl.handle.net/10174/21965.
Abstract:
We describe the growth dynamics of a fish or some other harvested population in a random environment
using a stochastic differential equation ;
general model, where the harvest term depends on a constant or on
a variable fishing effort. We compare the profit obtained by the fishing activity with two types of harvesting
policies, one based on variable effort, which is inapplicable, and the other based on a constant effort, which
is applicable, sustainable and is socially advantageous. We use real data and consider a logistic and a
Gompertz growth models to perform such comparisons. For both optimal policies, profitwise comparisons
are also made when considering a logistic-type growth model with weak Allee effects. The mean and
variance of the first passage times by a lower and by an upper thresholds are studied and, for a particular
threshold value, we estimate the probability density function of the first passage time using the inversion
of the Laplace transform; Resumo:
MODELOS DE PESCA USANDO EQUAÇÕES
DIFERENCIAIS ESTOCÁSTICAS: POLÍTICAS
SUSTENTÁVEIS E OTIMIZAÇÃO DO LUCRO
A dinâmica de crescimento de uma população sujeita a pesca em ambiente aleatório é descrita através de
modelos de equações diferenciais estocásticas, onde o termo de captura depende de um esforço de pesca
constante ou variável. Comparamos o lucro obtido pela atividade de pesca usando dois tipos de políticas de
pesca, uma inaplicável e baseada em esforço variável e a outra aplicável, sustentável e socialmente vantajosa,
baseada em esforço constante. As comparações são realizadas recorrendo a dados reais e considerando dois
modelos de crescimento, o modelo logístico e o modelo de Gompertz. Para ambas as políticas ótimas, as
comparações do lucro também são feitas quando se considera um modelo de crescimento do tipo logístico
com efeitos de Allee fracos. A média e a variância dos tempos de primeira passagem por um limite inferior e
por um limite superior são estudados e, para um determinado valor limite, estimamos a função de densidade
do tempo de primeira passagem usando a inversa da transformada de Laplace.
13
Wang, Yong Tian. "Stochastic differential delay equation with jumps and application to finance." Thesis, Swansea University, 2007. https://cronfa.swan.ac.uk/Record/cronfa43121.
14
Zhou, Yanli. "Computational methods for various stochastic differential equation models in finance." Thesis, Curtin University, 2014. http://hdl.handle.net/20.500.11937/247.
Abstract:
This study develops efficient numerical methods for solving jumpdiffusion stochastic delay differential equations and stochastic differential equations with fractional order. In addition, two novel algorithms are developed for the estimation of parameters in the stochastic models. One of the algorithms is based on the implementation of the Bayesian inference and the Markov Chain Monte Carlo method, while the other one is developed by using an implicit numerical scheme integrated with the particle swarm optimization.
15
Shedlock, Andrew James. "A Numerical Method for solving the Periodic Burgers' Equation through a Stochastic Differential Equation." Thesis, Virginia Tech, 2021. http://hdl.handle.net/10919/103947.
Abstract:
The Burgers equation, and related partial differential equations (PDEs), can be numerically challenging for small values of the viscosity parameter. For example, these equations can develop discontinuous solutions (or solutions with large gradients) from smooth initial data. Aside from numerical stability issues, standard numerical methods can also give rise to spurious oscillations near these discontinuities. In this study, we consider an equivalent form of the Burgers equation given by Constantin and Iyer, whose solution can be written as the expected value of a stochastic differential equation. This equivalence is used to develop a numerical method for approximating solutions to Burgers equation. Our preliminary analysis of the algorithm reveals that it is a natural generalization of the method of characteristics and that it produces approximate solutions that actually improve as the viscosity parameter vanishes. We present three examples that compare our algorithm to a recently published reference method as well as the vanishing viscosity/entropy solution for decreasing values of the viscosity.Master of Science
Burgers equation is a Partial Differential Equation (PDE) used to model how fluids evolve in time based on some initial condition and viscosity parameter. This viscosity parameter helps describe how the energy in a fluid dissipates. When studying partial differential equations, it is often hard to find a closed form solution to the problem, so we often approximate the solution with numerical methods. As our viscosity parameter approaches 0, many numerical methods develop problems and may no longer accurately compute the solution. Using random variables, we develop an approximation algorithm and test our numerical method on various types of initial conditions with small viscosity coefficients.
16
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.
Abstract:
MathematicsPh.D.
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
17
Pereira, Lo Bernardino. "Parameter estimation for a stochastic differential equation model of polymer rheology." Thesis, Imperial College London, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.441346.
18
Zhao, Lin. "Portfolio selection of stochastic differential equation with jumps under regime switching." Thesis, Swansea University, 2010. https://cronfa.swan.ac.uk/Record/cronfa42401.
Abstract:
In this thesis, we are interested in the stochastic differential equation with jumps under regime switching. Firstly, we investigate a continuous-time version of the mean-variance portfolio selection model with jumps under regime switching. The portfolio selection proposed and analyzed for a market consisting of one bank account an d multiple stocks. The random regime switching is assumed to be independent of the underlying Brownian motion and jump processes. Secondly, we consider the problem of pricing contigent claims on a stock whose price process is modeled by a Levy process. Since the market is incomplete and there is not a unique equivalent martingale measure. We study approaches to pricing options. Finally, we investigate a continuous-time version Markowitz's mean-variance portfolio selection problem which is studied in a market with one bank account, one stock and proportional transaction costs. This is a singular stochastic control problem. Via a series of transformations, the problem is turned into a double obstacle problem.
19
Messerschmidt, Reinhardt. "Hattendorff’s theorem and Thiele’s differential equation generalized." Diss., University of Pretoria, 2005. http://hdl.handle.net/2263/30476.
Abstract:
Hattendorff's theorem on the zero means and uncorrelatedness of losses in disjoint time periods on a life insurance policy is derived for payment streams, discount functions and time periods that are all stochastic. Thiele's differential equation, describing the development of life insurance policy reserves over the contract period, is derived for stochastic payment streams generated by point processes with intensities. The development follows that by Norberg. In pursuit of these aims, the basic properties of Lebesgue-Stieltjes integration are spelled out in detail. An axiomatic approach to the discounting of payment streams is presented, and a characterization in terms of the integral of a discount function is derived, again following the development by Norberg. The required concepts and tools from the theory of continuous time stochastic processes, in particular point processes, are surveyed.Dissertation (MSc (Actuarial Science))--University of Pretoria, 2007.
Insurance and Actuarial Science
unrestricted
20
Lemos, Alice Loureiro Leocádio Botelho de. "A study on Thiele's Differential Equation." Master's thesis, Instituto Superior de Economia e Gestão, 2014. http://hdl.handle.net/10400.5/7975.
Abstract:
Mestrado em Ciências ActuariaisThorvald Nicolai Thiele foi um importante investigador dinamarquês. Entre os seus contributos, destaca-se em particular o facto de ter provado que para um seguro de vida inteira com benefício de valor 1, emitido sobre uma pessoa e pago imediatamente após a morte, as reservas prospetivas satisfazem uma equação diferencial linear: a chamada equação diferencial de Thiele. De um modo mais geral, as equações diferenciais de Thiele são um sistema diferencial linear de equações que descrevem a dinâmica das reservas nos seguros de vida e pensões em tempo contínuo. Este texto tem como principal objetivo rever de forma tão completa quanto possível as contribuições relacionadas com a equação de Thiele que foram surgindo ao longo do tempo, dando assim o presente estado de arte deste relevante tópico. Começando por fazer uma revisão breve do essencial da matemática atuarial avança depois para a derivação da equação de Thiele, considerando os dois modelos de mortalidade, o clássico e o de múltiplos estados, sobre uma pessoa e sobre várias pessoas. Algumas ilustrações, para vários tipos de contrato, são seguidamente introduzidas. Dos desenvolvimentos conhecidos, dá-se especial destaque às generalizações da equação diferencial que incluem um processo estocástico de pagamentos e um processo de difusão para a taxa de juro. Apresenta-se também o uso da equação como ferramenta para o desenvolvimento de produtos de seguro de vida e descreve-se uma generalização da equação diferencial para uma carteira fechada de seguros. A última parte do trabalho faz um resumo de outros contributos relacionados com a equação.
Thiele's differential equation has a long history, dating back to an unpublished note of Thiele, 1875. Thorvald Nicolai Thiele was a Danish researcher who worked as an actuary, astronomer, mathematician and statistician. He proved that for a whole life assurance of a single individual with benefit of amount 1, payable immediately on death, the prospective reserve satisfies a certain linear differential equation, which is extremely useful for the understanding of reality: Thiele's differential equation. In a more general framework, Thiele's differential equations for the prospective reserve are a linear system of differential equations describing the dynamics of reserves in life and pension insurance in continuous time. This text has the main purpose of reviewing in a comprehensive way the contributions related to Thiele's equation that appeared over time, presenting the status of the art on this important topic. A revision of life insurance mathematics is first and then Thiele?s differential equation is derived under the classical and multiple state model of human mortality for one life and for multiple lives After this, some illustrations are presented under different types of contracts. Following the developments in the literature, more general differential equations are obtained, including a stochastic payment process and a diffusion process for interest rate. The technique of using Thiele's differential equation as a tool for life insurance product development and the generalization of the equation for a closed insurance portfolio are also discussed. Finally, other developments are summarised.
21
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.
22
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.
Abstract:
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.
23
Grecksch, Wilfried, and Christian Roth. "Approximation of a Quasilinear Stochastic Partial Differential Equation driven by Fractional White Noise." Universitätsbibliothek Chemnitz, 2008. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200800521.
Abstract:
We approximate the solution of a quasilinear stochastic partial differential equa-
tion driven by fractional Brownian motion B_H(t); H in (0,1), which was calculated
via fractional White Noise calculus, see [5].
24
Luo, Ye. "Random periodic solutions of stochastic functional differential equations." Thesis, Loughborough University, 2014. https://dspace.lboro.ac.uk/2134/16112.
Abstract:
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.
25
Yevik, Andrei. "Numerical approximations to the stationary solutions of stochastic differential equations." Thesis, Loughborough University, 2011. https://dspace.lboro.ac.uk/2134/7777.
Abstract:
This thesis investigates the possibility of approximating stationary solutions of stochastic differential equations using numerical methods. We consider a particular class of stochastic differential equations, which are known to generate random dynamical systems. The existence of stochastic stationary solution is proved using global attractor approach. Euler's numerical method, applied to the stochastic differential equation, is proved to generate a discrete random dynamical system. The existence of stationary solution is proved again using global attractor approach. At last we prove that the approximate stationary point converges in mean-square sense to the exact one as the time step of the numerical scheme diminishes.
26
Fantozzi, Marco. "Large deviations for differential stochastic equations with additive noise." Doctoral thesis, Scuola Normale Superiore, 2007. http://hdl.handle.net/11384/85673.
27
Ahlip, Rehez Ajmal. "Stability & filtering of stochastic systems." Thesis, Queensland University of Technology, 1997.
28
Wei, Fajin. "Stochastic Infinity-Laplacian equation and One-Laplacian equation in image processing and mean curvature flows : finite and large time behaviours." Thesis, Loughborough University, 2010. https://dspace.lboro.ac.uk/2134/7345.
Abstract:
The existence of pathwise stationary solutions of this stochastic partial differential equation (SPDE, for abbreviation) is demonstrated. In Part II, a connection between certain kind of state constrained controlled Forward-Backward Stochastic Differential Equations (FBSDEs) and Hamilton-Jacobi-Bellman equations (HJB equations) are demonstrated. The special case provides a probabilistic representation of some geometric flows, including the mean curvature flows. Part II includes also a probabilistic proof of the finite time existence of the mean curvature flows.
29
Nguyen, Cu Ngoc. "Stochastic differential equations with long-memory input." Thesis, Queensland University of Technology, 2001.
30
Knani, Habiba. "Backward stochastic differential equations driven by Gaussian Volterra processes." Electronic Thesis or Diss., Université de Lorraine, 2020. http://www.theses.fr/2020LORR0014.
Abstract:
Cette thèse porte sur les équations différentielles stochastiques rétrogrades (EDSR) dirigées par une classe de processus de Volterra qui contient le mouvement brownien multifractionnaire et le processus Ornstein-Uhlenbeck multifractionnaire. Dans la première partie, nous étudions la solution des EDSRs multidimensionnelles avec des générateurs linéaires. Par la formule d’Itô pour les processus de Volterra nous réduisons l’EDSR à une équation aux dérivées partielles (EDP) de second ordre linéaire avec la condition terminale. Sous une condition d’intégrabilité dans un voisinage du temps terminal de la variance du processus de Volterra, nous résolvons l’EDP associée explicitement et en déduisons la solution des EDSR linéaire. Puis, nous discutons une application dans le contexte des stratégies autofinancées. La seconde partie de la thèse traite des EDSRs non linéaires dirigées par la même classe de processus de Volterra. Les résultats principaux sont l’existence et l’unicité de la solution de l’EDSR dans un espace de fonctionnelles régulières du processus de Volterra et un théorème de comparaison qui porte sur les générateurs et les conditions terminales. Nous donnons deux preuves de l’existence et de l’unicité de la solution de l’EDSR, l’une basée sur l’EDP associée et l’autre sans référence à l’EDP, mais avec des méthodes probabilistes. Cette seconde preuve est techniquement difficile et, en raison de l’absence de propriétés de martingale dans le contexte des processus de Volterra, la preuve nécessite différentes normes sur l’espace de Hilbert sous-jacent défini par le noyau du processus de Volterra. Pour la construction de la solution, nous avons besoin de la notion de l’espérance quasi-conditionnelle, d’une formule de type Clark-Ocone et d’une autre formule d’Itô pour les processus de Volterra. Contrairement au cas classique des EDSR dirigées par le mouvement brownien ou brownien fractionnaire, une hypothèse sur le comportement du noyau est nécessaire pour l’existence et l’unicité de la solution de l’EDSR. Pour le mouvement brownien multifractionnaire, cette hypothèse est liée à la fonction de HurstThis thesis treats of backward stochastic differential equations (BSDE) driven by a class of Gaussian Volterra processes that includes multifractional Brownian motion and multifractional Ornstein-Uhlenbeck processes. In the first part we study multidimensional BSDE with generators that are linear functions of the solution. By means of an Itoˆ formula for Volterra processes, a linear second order partial differential equation (PDE) with terminal condition is associated to the BSDE. Under an integrability condition on a functional of the second moment of the Volterra process in a neighbourhood of the terminal time, we solve the associated PDE explicitely and deduce the solution of the linear BSDE. We discuss an application in the context of self-financing trading stategies. The second part of the thesis treats of non-linear BSDE driven by the same class of Gaussian Volterra processes. The main results are the existence and uniqueness of the solution in a space of regular functionals of the Volterra process, and a comparison theorem for the solutions of BSDE. We give two proofs for the existence and uniqueness of the solution, one is based on the associated PDE and a second one without making reference to this PDE, but with probabilistic and functional theoretic methods. Especially this second proof is technically quite complex, and, due to the absence of mar- tingale properties in the context of Volterra processes, requires to work with different norms on the underlying Hilbert space that is defined by the kernel of the Volterra process. For the construction of the solution we need the notion of quasi-conditional expectation, a Clark-Ocone type formula and another Itoˆ formula for Volterra processes. Contrary to the more classical cases of BSDE driven by Brownian or fractional Brownian motion, an assumption on the behaviour of the kernel of the driv- ing Volterra process is in general necessary for the wellposedness of the BSDE. For multifractional Brownian motion this assumption is closely related to the behaviour of the Hurst function
31
Miserocchi, Andrea. "The Fokker-Planck equation as model for the stochastic gradient descent in deep learning." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/18290/.
Abstract:
La discesa stocastica del gradiente (SGD) è alla base degli algoritmi di
ottimizzazione di reti di Deep Learning più usati in AI, dal riconoscimento
delle immagini all’elaborazione del linguaggio naturale. Questa tesi si propone di descrivere un modello basato sull’equazione di Fokker-Planck della dinamica del SGD. Si introduce la teoria dei processi stocastici, con particolare enfasi sulle equazioni di Langevin e sull’equazione di Fokker-Planck. Si mostra come il SGD minimizzi un funzionale sulla densità di probabilità dei pesi, non dipendente direttamente dalla funzione di costo. Infine si discutono le implicazioni di questa inferenza variazionale ottenuta dal SGD.
32
Thomas, Philipp. "Systematic approximation methods for stochastic biochemical kinetics." Thesis, University of Edinburgh, 2015. http://hdl.handle.net/1842/16197.
Abstract:
Experimental studies have shown that the protein abundance in living cells varies from few tens to several thousands molecules per species. Molecular fluctuations roughly scale as the inverse square root of the number of molecules due to the random timing of reactions. It is hence expected that intrinsic noise plays an important role in the dynamics of biochemical networks. The Chemical Master Equation is the accepted description of these systems under well-mixed conditions. Because analytical solutions to this equation are available only for simple systems, one often has to resort to approximation methods. A popular technique is an expansion in the inverse volume to which the reactants are confined, called van Kampen's system size expansion. Its leading order terms are given by the phenomenological rate equations and the linear noise approximation that quantify the mean concentrations and the Gaussian fluctuations about them, respectively. While these approximations are valid in the limit of large molecule numbers, it is known that physiological conditions often imply low molecule numbers. We here develop systematic approximation methods based on higher terms in the system size expansion for general biochemical networks. We present an asymptotic series for the moments of the Chemical Master Equation that can be computed to arbitrary precision in the system size expansion. We then derive an analytical approximation of the corresponding time-dependent probability distribution. Finally, we devise a diagrammatic technique based on the path-integral method that allows to compute time-correlation functions. We show through the use of biological examples that the first few terms of the expansion yield accurate approximations even for low number of molecules. The theory is hence expected to closely resemble the outcomes of single cell experiments.
33
Mayo, Nardone Pablo Sabino. "Modeling the Heat Flow Dynamics of a Houses Using Stochastic Differential Equations." Thesis, KTH, Skolan för industriell teknik och management (ITM), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-302557.
Abstract:
This research aims to explore new ways of assessing energy performance within housing units. The mainobjective of this work is to propose a heat dynamics model based on monitoring data, to contribute towardsan energy-efficient transition in the building sector. An extensive study on the available mathematical and statistical tools is described in order to determine aholistic solution, found in grey-box models. This model approach offers the possibility of understandingmultivariate systems, which can be applied to a housing-unit heat flow dynamics. Through the iterative process of testing each possible model, this work determines the one with bestexplanatory power, defining the thermal characteristics of the studied housing unit. This method allows thedetection of underperforming dwellings among constructions with high energy-efficiency standards. This investigation reflects the feasibility of employing grey-box models to predict the dynamics of heatrelated systems. Moreover, it sets the basis for new ways of employing the monitoring data of dwellings.Denna forskning syftar till att utforska nya sätt att bedöma energiprestanda inom bostäder. Huvudsyftetmed detta arbete är att föreslå en värmedynamikmodell baserad på övervakningsdata för att bidra till enenergieffektiv övergång inom byggsektorn. En omfattande studie av tillgängliga matematiska och statistiska verktyg beskrivs för att bestämma enhelhetslösning, som finns i gråboxmodeller. Denna modellstrategi ger möjlighet att förstå multivariatasystem, som kan tillämpas på en hushålls värmedynamik. Genom den iterativa processen att testa varje möjlig modell bestämmer detta arbete den med bästförklarande kraft, och definierar de studerade husenhetens termiska egenskaper. Denna metod gör detmöjligt att upptäcka underpresterande bostäder bland anläggningar med hög energieffektivitetsstandard. Denna undersökning återspeglar möjligheten att använda gråboxmodeller för att förutsäga dynamiken ivärmerelaterade system. Dessutom lägger den grunden för nya sätt att använda övervakningsdata förbostäder.
34
Tempone, Olariaga Raul. "Numerical Complexity Analysis of Weak Approximation of Stochastic Differential Equations." Doctoral thesis, KTH, Numerisk analys och datalogi, NADA, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3413.
Abstract:
The thesis consists of four papers on numerical complexityanalysis of weak approximation of ordinary and partialstochastic differential equations, including illustrativenumerical examples. Here by numerical complexity we mean thecomputational work needed by a numerical method to solve aproblem with a given accuracy. This notion offers a way tounderstand the efficiency of different numerical methods. The first paper develops new expansions of the weakcomputational error for Ito stochastic differentialequations using Malliavin calculus. These expansions have acomputable leading order term in a posteriori form, and arebased on stochastic flows and discrete dual backward problems.Beside this, these expansions lead to efficient and accuratecomputation of error estimates and give the basis for adaptivealgorithms with either deterministic or stochastic time steps.The second paper proves convergence rates of adaptivealgorithms for Ito stochastic differential equations. Twoalgorithms based either on stochastic or deterministic timesteps are studied. The analysis of their numerical complexitycombines the error expansions from the first paper and anextension of the convergence results for adaptive algorithmsapproximating deterministic ordinary differential equations.Both adaptive algorithms are proven to stop with an optimalnumber of time steps up to a problem independent factor definedin the algorithm. The third paper extends the techniques to theframework of Ito stochastic differential equations ininfinite dimensional spaces, arising in the Heath Jarrow Mortonterm structure model for financial applications in bondmarkets. Error expansions are derived to identify differenterror contributions arising from time and maturitydiscretization, as well as the classical statistical error dueto finite sampling. The last paper studies the approximation of linear ellipticstochastic partial differential equations, describing andanalyzing two numerical methods. The first method generates iidMonte Carlo approximations of the solution by sampling thecoefficients of the equation and using a standard Galerkinfinite elements variational formulation. The second method isbased on a finite dimensional Karhunen- Lo`eve approximation ofthe stochastic coefficients, turning the original stochasticproblem into a high dimensional deterministic parametricelliptic problem. Then, adeterministic Galerkin finite elementmethod, of either h or p version, approximates the stochasticpartial differential equation. The paper concludes by comparingthe numerical complexity of the Monte Carlo method with theparametric finite element method, suggesting intuitiveconditions for an optimal selection of these methods. 2000Mathematics Subject Classification. Primary 65C05, 60H10,60H35, 65C30, 65C20; Secondary 91B28, 91B70.QC 20100825
35
Zhou, Bo. "The existence of bistable stationary solutions of random dynamical systems generated by stochastic differential equations and random difference equations." Thesis, Loughborough University, 2009. https://dspace.lboro.ac.uk/2134/14255.
Abstract:
In this thesis, we study the existence of stationary solutions for two cases. One is for random difference equations. For this, we prove the existence and uniqueness of the stationary solutions in a finite-dimensional Euclidean space Rd by applying the coupling method. The other one is for semi linear stochastic evolution equations. For this case, we follows Mohammed, Zhang and Zhao [25]'s work. In an infinite-dimensional Hilbert space H, we release the Lipschitz constant restriction by using Arzela-Ascoli compactness argument. And we also weaken the globally bounded condition for F by applying forward and backward Gronwall inequality and coupling method.
36
Melichov, Dmitrij. "On estimation of the Hurst index of solutions of stochastic differential equations." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2011. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2011~D_20111228_165042-00002.
Abstract:
The main topic of this dissertation is the estimation of the Hurst index H of the solutions of stochastic differential equations (SDEs) driven by the fractional Brownian motion (fBm).
Firstly, the limit behavior of the first and second order quadratic variations of the solutions of SDEs driven by the fBm is analyzed. This yields several strongly consistent estimators of the Hurst index H. Secondly, it is proved that in case the solution of the SDE is replaced by its Milstein approximation, the estimators remain strongly consistent. Additionally, the possibilities of applying the increment ratios (IR) statistic based estimator of H originally obtained by J. M. Bardet and D. Surgailis in 2010 to the fractional geometric Brownian motion are examined.
Furthermore, this dissertation derives the convergence rate of the modified Gladyshev’s estimator of the Hurst index to its real value.
The estimators obtained in the dissertation were compared with several other known estimators of the Hurst index H, namely the naive and ordinary least squares Gladyshev and eta-summing oscillation estimators, the variogram estimator and the IR estimator. The models chosen for comparison of these estimators were the fractional Ornstein-Uhlenbeck (O-U) process and the fractional geometric Brownian motion (gBm). The initial inference about the behavior of these estimators was drawn for the O-U process which is Gaussian, while the gBm process was used to check how the estimators behave in a... [to full text]Pagrindinė šios disertacijos tema – stochastinių diferencialinių lygčių (SDL), valdomų trupmeninio Brauno judesio (tBj), sprendinių Hursto indekso H vertinimas. Pirmiausia disertacijoje išnagrinėta SDL, valdomų tBj, sprendinių pirmos ir antros eilės kvadratinių variacijų ribinė elgsena. Iš šių rezultatų seka keli stipriai pagrįsti Hursto indekso H įvertiniai. Įrodyta, kad šie įvertiniai išlieka stipriai pagrįsti, jei tikra sprendinio trajektorija keičiama jos Milšteino aproksimacija. Taip pat išnagrinėtos pokyčių santykio (increment ratios) statistikos H įvertinio, gauto J. M. Bardeto ir D. Surgailio 2010 m., taikymo trupmeninio geometrinio Brauno judesio Hursto indekso vertinimui galimybės bei nustatytas modifikuoto Gladyševo H įvertinio konvergavimo i tikrąją parametro reikšme greitis. Gauti įvertiniai palyginti su kai kuriais kitais žinomais Hursto indekso H įvertiniais: naiviais bei mažiausių kvadratų Gladyševo ir eta-sumavimo osciliacijos įvertiniais, variogramos įvertiniu ir pokyčių santykio statistikos įvertiniu. Įvertinių elgsena buvo palyginta trupmeniniam Ornšteino-Ulenbeko (OU) procesui bei trupmeniniam geometriniam Brauno judesiui (gBj). Pradinės išvados buvo padarytos O-U procesui, kuris yra Gauso, o gBj procesas buvo naudojamas patikrinti, kaip šie įvertiniai elgiasi, kai procesas yra ne Gauso. Disertaciją sudaro įvadas, 3 pagrindiniai skyriai, išvados, literatūros sąrašas, autoriaus publikacijų disertacijos tema sąrašas ir du priedai.
37
Redmon, Jessica. "Stochastic Bubble Formation and Behavior in Non-Newtonian Fluids." Case Western Reserve University School of Graduate Studies / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=case15602738261697.
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Masike, Kakanyo Knowledge. "The required ansatz to construct Lie point transformations and the symmetries of a first-order stochastic differential equation." Master's thesis, University of Cape Town, 2011. http://hdl.handle.net/11427/14145.
Abstract:
In this thesis we demonstrate how to obtain the required ansatz to determine Lie point transformations of evolution-type equations from the contact transformation approach. We indicate that the Lie point transformations of the Fokker-Planck equation (FPE), which is a second-order linear parabolic partial differential equation (PDE), are projectable by using the ansatz. We further obtain the symmetries of a stochastic ordinary differential equation (SODE) which corresponds to those of the FPE. This is possible because there exists a relationship between an SODE and the associated (deterministic) FPE. The study of SODEs is an interesting and applicable concept in the real world and one of the building factors to this study is an Ito integral. These Ito integrals are of much use, for instance, in the field of mathematical finance whereby its use has shown the relationship between call options and their non-deterministic underlying stock prices. Wiener processes must be considered in finding an approximation of these integrals. Acclimatization of Sophus Lie's work to SODEs has been done by (Gaeta and Quintero [2]; Wafo Soh and Mahomed [41]; Unal [42]; Fredericks and Mahomed [43]). The determining equations for the first-order SODEs are derived in an Ito calculus context and are non-stochastic. Consequently, symmetries of an SODE are obtained without the consultation of its corresponding FPE.
39
McKinley, Scott Alister. "An existence result from the theory of fluctuating hydrodynamics of polymers in dilute solution." Columbus, Ohio : Ohio State University, 2006. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1149020682.
40
Wang, Shuo. "Analysis and Application of Haseltine and Rawlings's Hybrid Stochastic Simulation Algorithm." Diss., Virginia Tech, 2016. http://hdl.handle.net/10919/82717.
Abstract:
Stochastic effects in cellular systems are usually modeled and simulated with Gillespie's stochastic simulation algorithm (SSA), which follows the same theoretical derivation as the chemical master equation (CME), but the low efficiency of SSA limits its application to large chemical networks.
To improve efficiency of stochastic simulations, Haseltine and Rawlings proposed a hybrid of ODE and SSA algorithm, which combines ordinary differential equations (ODEs) for traditional deterministic models and SSA for stochastic models. In this dissertation, accuracy analysis, efficient implementation strategies, and application of of Haseltine and Rawlings's hybrid method (HR) to a budding yeast cell cycle model are discussed.
Accuracy of the hybrid method HR is studied based on a linear chain reaction system, motivated from the modeling practice used for the budding yeast cell cycle control mechanism. Mathematical analysis and numerical results both show that the hybrid method HR is accurate if either numbers of molecules of reactants in fast reactions are above certain thresholds, or rate constants of fast reactions are much larger than rate constants of slow reactions. Our analysis also shows that the hybrid method HR allows for a much greater region in system parameter space than those for the slow scale SSA (ssSSA) and the stochastic quasi steady state assumption (SQSSA) method.
Implementation of the hybrid method HR requires a stiff ODE solver for numerical integration and an efficient event-handling strategy for slow reaction firings. In this dissertation, an event-handling strategy is developed based on inverse interpolation. Performances of five wildly used stiff ODE solvers are measured in three numerical experiments.
Furthermore, inspired by the strategy of the hybrid method HR, a hybrid of ODE and SSA stochastic models for the budding yeast cell cycle is developed, based on a deterministic model in the literature. Simulation results of this hybrid model match very well with biological experimental data, and this model is the first to do so with these recently available experimental data. This study demonstrates that the hybrid method HR has great potential for stochastic modeling and simulation of large biochemical networks.Ph. D.
41
Cormier, Quentin. "Comportement en temps long d'un modèle champ moyen de neurones à décharge en interactions." Thesis, Université Côte d'Azur, 2021. http://www.theses.fr/2021COAZ4008.
Abstract:
Nous étudions le comportement en temps long d'une équation différentielle stochastique (EDS) de type McKean-Vlasov, dirigée par une mesure de Poisson. En neurosciences, cette EDS modélise la dynamique du potentiel de membrane d'un neurone typique dans un grand réseau. Le modèle peut-être obtenu en considérant un réseau fini de neurones de type Intègre-Et-Tire généralisé et en prenant la limite où le nombre de neurones tend vers l'infini. Cette EDS est donc un modèle champ moyen de neurones à décharge.Nous étudions l'existence et l'unicité de la solution de cette EDS McKean-Vlasov et nous donnons ses mesures de probabilité invariantes. Si le paramètre d'interaction J est suffisamment petit, nous prouvons l'unicité et la stabilité globale de la mesure invariante. Pour un J quelconque cependant, il peut y avoir plusieurs mesures de probabilité invariantes. Nous donnons une condition suffisante assurant la stabilité locale d'une telle mesure invariante. Notre critère fait intervenir les zéros d'une fonction holomorphe associée à la solution stationnaire considérée. Lorsque tous les zéros sont de partie réelle négative, nous prouvons la stabilité. Nous donnons finalement des conditions générales suffisantes assurant l'existence de solutions périodiques par le biais d'une bifurcation de Hopf : pour un certain paramètre d'interaction critique J0, la probabilité invariante perd sa stabilité et des solutions périodiques apparaissent pour J suffisamment proche de J0. Pour obtenir ces résultats, nous combinons des méthodes probabilistes et déterministes. En particulier, dans cette analyse, un outil clé est l'équation intégrale de Volterra non linéaire satisfaite par le courant synaptique. Enfin, nous illustrons ces résultats par des exemples que l'on peut traiter de manière analytique. En outre, nous donnons des méthodes numériques pour approximer la solution de l'équation champ moyen et pour prédire numériquement les bifurcationsWe study the long time behavior of a McKean-Vlasov stochastic differential equation (SDE), driven by a Poisson measure. In neuroscience, this SDE models the dynamics of the membrane potential of a typical neuron in a large network. The model can be derived by considering a finite network of generalized Integrate-And-Fire neurons and by taking the limit where the number of neurons goes to infinity. Hence the McKean-Vlasov SDE is a mean-field model of spiking neurons.We study existence and uniqueness of the solution this McKean-Vlasov SDE and describe its invariant probability measures. For small enough interaction parameter J, we prove uniqueness and global stability of the invariant measure. For J arbitrary large however, the invariant measures may not be unique. We give a sufficient condition ensuring the local stability of such a given invariant probability measure. Our criterion involves the location of the zeros of an explicit holomorphic function associated to the considered stationary solution. When all the zeros have negative real part, we prove that stability holds. We then give sufficient general conditions ensuring the existence of periodic solutions through a Hopf bifurcation: at some critical interaction parameter J0, the invariant probability losses its stability and periodic solutions appear for J close to J0. To obtain these results, we combine probabilistic and deterministic methods. In particular, a key tool in this analysis is a nonlinear Volterra Integral equation satisfied by the synaptic current.Finally, we illustrate these results with examples which are tractable analytically. Additionally, we give numerical methods to approximate the solution of the mean-field equation and to predict numerically the bifurcations
42
Kateregga, Michael. "Perturbation methods in derivatives pricing under stochastic volatility." Thesis, Stellenbosch : Stellenbosch University, 2012. http://hdl.handle.net/10019.1/71708.
Abstract:
Thesis (MSc)--Stellenbosch University, 2012.ENGLISH ABSTRACT: This work employs perturbation techniques to price and hedge financial derivatives in a stochastic volatility framework. Fouque et al. [44] model volatility as a function of two processes operating on different time-scales. One process is responsible for the fast-fluctuating feature of volatility and corresponds to the slow time-scale and the second is for slowfluctuations or fast time-scale. The former is an Ergodic Markov process and the latter is a strong solution to a Lipschitz stochastic differential equation. This work mainly involves modelling, analysis and estimation techniques, exploiting the concept of mean reversion of volatility. The approach used is robust in the sense that it does not assume a specific volatility model. Using singular and regular perturbation techniques on the resulting PDE a first-order price correction to Black-Scholes option pricing model is derived. Vital groupings of market parameters are identified and their estimation from market data is extremely efficient and stable. The implied volatility is expressed as a linear (affine) function of log-moneyness-tomaturity ratio, and can be easily calibrated by estimating the grouped market parameters from the observed implied volatility surface. Importantly, the same grouped parameters can be used to price other complex derivatives beyond the European and American options, which include Barrier, Asian, Basket and Forward options. However, this semi-analytic perturbative approach is effective for longer maturities and unstable when pricing is done close to maturity. As a result a more accurate technique, the decomposition pricing approach that gives explicit analytic first- and second-order pricing and implied volatility formulae is discussed as one of the current alternatives. Here, the method is only employed for European options but an extension to other options could be an idea for further research. The only requirements for this method are integrability and regularity of the stochastic volatility process. Corrections to [3] remarkable work are discussed here.
AFRIKAANSE OPSOMMING: Hierdie werk gebruik steuringstegnieke om finansiële afgeleide instrumente in ’n stogastiese wisselvalligheid raamwerk te prys en te verskans. Fouque et al. [44] gemodelleer wisselvalligheid as ’n funksie van twee prosesse wat op verskillende tyd-skale werk. Een proses is verantwoordelik vir die vinnig-wisselende eienskap van die wisselvalligheid en stem ooreen met die stadiger tyd-skaal en die tweede is vir stadig-wisselende fluktuasies of ’n vinniger tyd-skaal. Die voormalige is ’n Ergodiese-Markov-proses en die laasgenoemde is ’n sterk oplossing vir ’n Lipschitz stogastiese differensiaalvergelyking. Hierdie werk behels hoofsaaklik modellering, analise en skattingstegnieke, wat die konsep van terugkeer to die gemiddelde van die wisseling gebruik. Die benadering wat gebruik word is rubuust in die sin dat dit nie ’n aanname van ’n spesifieke wisselvalligheid model maak nie. Deur singulêre en reëlmatige steuringstegnieke te gebruik op die PDV kan ’n eerste-orde pryskorreksie aan die Black-Scholes opsie-waardasiemodel afgelei word. Belangrike groeperings van mark parameters is geïdentifiseer en hul geskatte waardes van mark data is uiters doeltreffend en stabiel. Die geïmpliseerde onbestendigheid word uitgedruk as ’n lineêre (affiene) funksie van die log-geldkarakter-tot-verval verhouding, en kan maklik gekalibreer word deur gegroepeerde mark parameters te beraam van die waargenome geïmpliseerde wisselvalligheids vlak. Wat belangrik is, is dat dieselfde gegroepeerde parameters gebruik kan word om ander komplekse afgeleide instrumente buite die Europese en Amerikaanse opsies te prys, dié sluit in Barrier, Asiatiese, Basket en Stuur opsies. Hierdie semi-analitiese steurings benadering is effektief vir langer termyne en onstabiel wanneer pryse naby aan die vervaldatum beraam word. As gevolg hiervan is ’n meer akkurate tegniek, die ontbinding prys benadering wat eksplisiete analitiese eerste- en tweede-orde pryse en geïmpliseerde wisselvalligheid formules gee as een van die huidige alternatiewe bespreek. Hier word slegs die metode vir Europese opsies gebruik, maar ’n uitbreiding na ander opsies kan’n idee vir verdere navorsing wees. Die enigste vereistes vir hierdie metode is integreerbaarheid en reëlmatigheid van die stogastiese wisselvalligheid proses. Korreksies tot [3] se noemenswaardige werk word ook hier bespreek.
43
Li, Yao. "Stochastic perturbation theory and its application to complex biological networks -- a quantification of systematic features of biological networks." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/49013.
Abstract:
The primary objective of this thesis is to make a quantitative study of complex biological networks. Our fundamental motivation is to obtain the statistical dependency between modules by injecting external noise. To accomplish this, a deep study of stochastic dynamical systems would be essential. The first chapter is about the stochastic dynamical system theory. The classical estimation of invariant measures of Fokker-Planck equations is improved by the level set method. Further, we develop a discrete Fokker-Planck-type equation to study the discrete stochastic dynamical systems. In the second part, we quantify systematic measures including degeneracy, complexity and robustness. We also provide a series of results on their properties and the connection between them. Then we apply our theory to the JAK-STAT signaling pathway network.
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Salhi, Rym. "Contributions to quadratic backward stochastic differential equations with jumps and applications." Thesis, Le Mans, 2019. http://www.theses.fr/2019LEMA1023.
Abstract:
Cette thèse porte sur l'étude des équations différentielles stochastiques rétrogrades (EDSR) avec sauts et leurs applications.Dans le chapitre 1, nous étudions une classe d'EDSR lorsque le bruit provient d'un mouvement Brownien et d'une mesure aléatoire de saut indépendante à activité infinie. Plus précisément, nous traitons le cas où le générateur est à croissance quadratique et la condition terminale est non bornée. L'existence et l'unicité de la solution sont prouvées en combinant à la fois la procédure d'approximation monotone et une approche progressive. Cette méthode permet de résoudre le cas où la condition terminale est non bornée.Le chapitre 2 est consacré aux EDSR avec sauts généralisées doublement réfléchies sous des hypothèses d’intégrabilités faibles. Plus précisément, on montre l'existence d'une solution pour un générateur à croissance quadratique stochastique et une condition terminale non bornée. Nous montrons également, dans un cadre approprié, la connexion entre notre classe d’équations différentielles stochastiques rétrogrades et les jeu à somme nuls.Dans le chapitre 3, nous considérons une classe générale d'EDSR progressive-rétrograde couplée avec sauts de type Mackean Vlasov sous une condition faible de monotonicité. Les résultats d'existence et d'unicité sont établis sous deux classes d'hypothèses en se basant sur des schémas de perturbations soit de l’équation différentielle stochastique progressive, soit de l’équation différentielle stochastique rétrograde. On conclut le chapitre par un problème de stockage optimal d’énergie dans un parc électrique de type champs moyenThis thesis focuses on backward stochastic differential equation with jumps and their applications. In the first chapter, we study a backward stochastic differential equation (BSDE for short) driven jointly by a Brownian motion and an integer valued random measure that may have infinite activity with compensator being possibly time inhomogeneous. In particular, we are concerned with the case where the driver has quadratic growth and unbounded terminal condition. The existence and uniqueness of the solution are proven by combining a monotone approximation technics and a forward approach. Chapter 2 is devoted to the well-posedness of generalized doubly reflected BSDEs (GDRBSDE for short) with jumps under weaker assumptions on the data. In particular, we study the existence of a solution for a one-dimensional GDRBSDE with jumps when the terminal condition is only measurable with respect to the related filtration and when the coefficient has general stochastic quadratic growth. We also show, in a suitable framework, the connection between our class of backward stochastic differential equations and risk sensitive zero-sum game. In chapter 3, we investigate a general class of fully coupled mean field forward-backward under weak monotonicity conditions without assuming any non-degeneracy assumption on the forward equation. We derive existence and uniqueness results under two different sets of conditions based on proximation schema weither on the forward or the backward equation. Later, we give an application for storage in smart grids
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Schwarz, Daniel Christopher. "Price modelling and asset valuation in carbon emission and electricity markets." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:7de118d2-a61b-4125-a615-29ff82ac7316.
Abstract:
This thesis is concerned with the mathematical analysis of electricity and carbon emission markets. We introduce a novel, versatile and tractable stochastic framework for the joint price formation of electricity spot prices and allowance certificates. In the proposed framework electricity and allowance prices are explained as functions of specific fundamental factors, such as the demand for electricity and the prices of the fuels used for its production. As a result, the proposed model very clearly captures the complex dependency of the modelled prices on the aforementioned fundamental factors. The allowance price is obtained as the solution to a coupled forward-backward stochastic differential equation. We provide a rigorous proof of the existence and uniqueness of a solution to this equation and analyse its behaviour using asymptotic techniques. The essence of the model for the electricity price is a carefully chosen and explicitly constructed function representing the supply curve in the electricity market. The model we propose accommodates most regulatory features that are commonly found in implementations of emissions trading systems and we analyse in detail the impact these features have on the prices of allowance certificates. Thereby we reveal a weakness in existing regulatory frameworks, which, in rare cases, can lead to allowance prices that do not conform with the conditions imposed by the regulator. We illustrate the applicability of our model to the pricing of derivative contracts, in particular clean spread options and numerically illustrate its ability to "see" relationships between the fundamental variables and the option contract, which are usually unobserved by other commonly used models in the literature. The results we obtain constitute flexible tools that help to efficiently evaluate the financial impact current or future implementations of emissions trading systems have on participants in these markets.
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Roelly, Sylvie, and Myriam Fradon. "Infinite system of Brownian balls : equilibrium measures are canonical Gibbs." Universität Potsdam, 2006. http://opus.kobv.de/ubp/volltexte/2006/672/.
Abstract:
We consider a system of infinitely many hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with a local time term. We prove that the set of all equilibrium measures, solution of a detailed balance equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.
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Fradon, Myriam, and Sylvie Roelly. "Infinite system of Brownian Balls: Equilibrium measures are canonical Gibbs." Universität Potsdam, 2005. http://opus.kobv.de/ubp/volltexte/2011/5159/.
Abstract:
We consider a system of infinitely many hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional Stochastic Differential Equation with a local time term. We prove that the set of all equilibrium measures, solution of a Detailed Balance Equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.
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Kleinen, Thomas Christopher. "Stochastic information in the assessment of climate change." Phd thesis, [S.l. : s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=975745441.
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Noubiagain, Chomchie Fanny Larissa. "Contributions to second order reflected backward stochastic differentials equations." Thesis, Le Mans, 2017. http://www.theses.fr/2017LEMA1016/document.
Abstract:
Cette thèse traite des équations différentielles stochastiques rétrogrades réfléchies du second ordre dans une filtration générale . Nous avons traité tout d'abord la réflexion à une barrière inférieure puis nous avons étendu le résultat dans le cas d'une barrière supérieure. Notre contribution consiste à démontrer l'existence et l'unicité de la solution de ces équations dans le cadre d'une filtration générale sous des hypothèses faibles. Nous remplaçons la régularité uniforme par la régularité de type Borel. Le principe de programmation dynamique pour le problème de contrôle stochastique robuste est donc démontré sous les hypothèses faibles c'est à dire sans régularité sur le générateur, la condition terminal et la barrière. Dans le cadre des Équations Différentielles Stochastiques Rétrogrades (EDSRs ) standard, les problèmes de réflexions à barrières inférieures et supérieures sont symétriques. Par contre dans le cadre des EDSRs de second ordre, cette symétrie n'est plus valable à cause des la non linéarité de l'espérance sous laquelle est définie notre problème de contrôle stochastique robuste non dominé. Ensuite nous un schéma d'approximation numérique d'une classe d'EDSR de second ordre réfléchies. En particulier nous montrons la convergence de schéma et nous testons numériquement les résultats obtenusThis thesis deals with the second-order reflected backward stochastic differential equations (2RBSDEs) in general filtration. In the first part , we consider the reflection with a lower obstacle and then extended the result in the case of an upper obstacle . Our main contribution consists in demonstrating the existence and the uniqueness of the solution of these equations defined in the general filtration under weak assumptions. We replace the uniform regularity by the Borel regularity(through analytic measurability). The dynamic programming principle for the robust stochastic control problem is thus demonstrated under weak assumptions, that is to say without regularity on the generator, the terminal condition and the obstacle. In the standard Backward Stochastic Differential Equations (BSDEs) framework, there is a symmetry between lower and upper obstacles reflection problem. On the contrary, in the context of second order BSDEs, this symmetry is no longer satisfy because of the nonlinearity of the expectation under which our robust stochastic non-dominated stochastic control problem is defined. In the second part , we get a numerical approximation scheme of a class of second-order reflected BSDEs. In particular we show the convergence of our scheme and we test numerically the results
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Celep, Saziye Betul. "Stochastic Volatility And Stochastic Interest Rate Model With Jump And Its Application On General Electric Data." Master's thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12613192/index.pdf.
Abstract:
In this thesis, we present two different approaches for the stochastic volatility and stochastic interest rate model with jump and analyze the performance of four alternative models. In the first approach, suggested by Scott, the closed form solution for prices on European call stock options are developed by deriving characteristic functions with the help of martingale methods. Here, we study the asset price process and give in detail the derivation of the European call option price process. The second approach, suggested by Bashki-Cao-Chen, describes the closed form solution of European call option by deriving the partial integro-differential equation. In this one we g ive the derivations of both asset price dynamics and the European call option price process. Finally, in the application part of the thesis, we examine the performance of four alternative models using General Electric Stock Option Data. These models are constructed by using the theoretical results of the second approach.