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Estimation de processus de sauts / Estimation of the jump processesNguyen, Thi Thu Huong 06 December 2018 (has links)
Dans cette thèse, on considère une équation différentielle stochastique gouvernée par un processus de Lévy de saut pur dont l’indice d’activité des sauts α ∈ (0, 2) et on observe des données haute fréquence de ce processus sur un intervalle de temps fixé. Cette thèse est consacrée tout d’abord à l’étude du comportement de la densité du processus en temps petit. Ces résultats permettent ensuite de montrer la propriété LAMN (Local Asymptotic Mixed Normality) pour les paramètres de dérive et d’échelle. Enfin, on étudie des estimateurs de l’indice α du processus.La première partie traite du comportement asymptotique de la densité en temps petit du processus. Le processus est supposé dépendre d’un paramètre β = (θ,σ) et on étudie, dans cette partie, la sensibilité de la densité par rapport à ce paramètre. Cela étend les résultats de [17] qui étaient restreints à l’indice α ∈ (1,2) et ne considéraient que la sensibilité par rapport au paramètre de dérive. En utilisant le calcul de Malliavin, on obtient la représentation de la densité, de sa dérivée et de sa dérivée logarithmique comme une espérance et une espérance conditionnelle. Ces formules de représentation font apparaître des poids de Malliavin dont les expressions sont données explicitement, ce qui permet d’analyser le comportement asymptotique de la densité en temps petit, en utilisant la propriété d’autosimilarité du processus stable.La deuxième partie de cette thèse concerne la propriété LAMN (Local Asymptotic Mixed Normality) pour les paramètres. Le coefficient de dérive et le coefficient d’échelle dépendent tous les deux de paramètres inconnus et on étend les résultats de [17]. On identifie l’information de Fisher asymptotique ainsi que les vitesses optimales de convergence. Ces quantités dépendent de l’indice αLa troisième partie propose des estimateurs pour l’indice d’activité des sauts α ∈ (0,2) basés sur des méthodes de moments qui généralisent les résultats de Masuda [53]. On montre la consistence et la normalité asymptotique des estimateurs et on illustre les résultats par des simulations numériques / In this thesis, we consider a stochastic differential equation driven by a truncated pure jump Lévy process with index α ∈(0,2) and observe high frequency data of the process on a fixed observation time. We first study the behavior of the density of the process in small time. Next, we prove the Local Asymptotic Mixed Normality (LAMN) property for the drift and scaling parameters from high frequency observations. Finally, we propose some estimators of the index parameter of the process.The first part deals with the asymptotic behavior of the density in small time of the process. The process is assumed to depend on a parameter β = (θ,σ) and we study, in this part, the sensitivity of the density with respect to this parameter. This extends the results of [17] which were restricted to the index α ∈ (1,2) and considered only the sensitivity with respect to the drift coefficient. By using Malliavin calculus, we obtain the representation of the density, its derivative and its logarithm derivative as an expectation and a conditional expectation. These representation formulas involve some Malliavin weights whose expressions are given explicitly and this permits to analyze the asymptotic behavior in small time of the density, using the self-similarity property of the stable process.The second part of this thesis concerns the Local Asymptotic Mixed Normality property for the parameters. Both the drift coefficient and scale coefficient depend on the unknown parameters. Extending the results of [17], we compute the asymptotic Fisher information and find that the rate in the Local Asymptotic Mixed Normality property depends on the index α.The third part proposes some estimators of the jump activity index α ∈ (0,2) based on the method of moments as in Masuda [53]. We prove the consistency and asymptotic normality of the estimators and give some simulations to illustrate the finite-sample behaviors of the estimators
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On probability distributions of diffusions and financial models with non-globally smooth coefficientsDe Marco, Stefano 23 November 2010 (has links) (PDF)
Some recent works in the field of mathematical finance have brought new light on the importance of studying the regularity and the tail asymptotics of distributions for certain classes of diffusions with non-globally smooth coefficients. In this Ph.D. dissertation we deal with some issues in this framework. In a first part, we study the existence, smoothness and space asymptotics of densities for the solutions of stochastic differential equations assuming only local conditions on the coefficients of the equation. Our analysis is based on Malliavin calculus tools and on " tube estimates " for Ito processes, namely estimates for the probability that the trajectory of an Ito process remains close to a deterministic curve. We obtain significant estimates of densities and distribution functions in general classes of option pricing models, including generalisations of CIR and CEV processes and Local-Stochastic Volatility models. In the latter case, the estimates we derive have an impact on the moment explosion of the underlying price and, consequently, on the large-strike behaviour of the implied volatility. Parametric implied volatility modeling, in its turn, makes the object of the second part. In particular, we focus on J. Gatheral's SVI model, first proposing an effective quasi-explicit calibration procedure and displaying its performances on market data. Then, we analyse the capability of SVI to generate efficient approximations of symmetric smiles, building an explicit time-dependent parameterization. We provide and test the numerical application to the Heston model (without and with displacement), for which we generate semi-closed expressions of the smile
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Geometria dos caminhos em grupos de Lie / Path geometry in Lie groupsFélix, Luciano Vianna, 1986- 13 August 2018 (has links)
Orientador: Pedro Jose Catuogno / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T12:34:53Z (GMT). No. of bitstreams: 1
Felix_LucianoVianna_M.pdf: 566321 bytes, checksum: f717034fada0c65f1b886ba7bd821902 (MD5)
Previous issue date: 2009 / Resumo: Neste trabalho estudamos a geometria dos caminhos em grupos de Lie usando a exponencial estocástica e o logaritmo estocástico. Apresentamos as construções geométricas do espaço tangente, uma métrica e uma conexão natural as caminhos em grupos de Lie. Finalmente apresentamos uma situação em que essa conexão é Levi-Civita e outra que não é / Abstract: In this work, we study the path geometry in Lie groups using the stochastic exponential and the stochastic logarithm. We show the geometric constructions of tangent space, one metric and one natural conection of Lie groups valued path. Finelly we show one situation that this conection is Levi-Civita and another one that is not / Mestrado / Geometria / Mestre em Matemática
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Approximation et estimation de densité pour des équations d'évolution stochastique / No English title availableAboura, Omar 19 December 2013 (has links)
Dans la première partie de cette thèse, nous obtenons l’existence d’une densité et des estimées gaussiennes pour la solution d’une équation différentielle stochastique rétrograde. C’est une application du calcul de Malliavin et plus particulièrement d’une formule d’I. Nourdin et de F. Viens. La deuxième partie de cette thèse est consacrée à la simulation d’une équation aux dérivées partielles stochastique par une méthode probabiliste qui repose sur la représentation de l’équation aux dérivées partielles stochastique en terme d’équation différentielle doublement stochastique rétrograde, introduite par E. Pardoux et S. Peng. On étend dans ce cadre les idées de F. Zhang et E. Gobet et al. sur la simulation d’une équation différentielle stochastique rétrograde. Dans la dernière partie, nous étudions l’erreur faible du schéma d’Euler implicite pour les processus de diffusion et l’équation de la chaleur stochastique. Dans le premier cas, nous étendons les résultats de D. Talay et L. Tubaro. Dans le second cas, nous étendons les travaux de A. Debussche. / No English summary available.
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Malliavin-Stein Method in Stochastic GeometrySchulte, Matthias 19 March 2013 (has links)
In this thesis, abstract bounds for the normal approximation of Poisson functionals are computed by the Malliavin-Stein method and used to derive central limit theorems for problems from stochastic geometry. As a Poisson functional we denote a random variable depending on a Poisson point process. It is known from stochastic analysis that every square integrable Poisson functional has a representation as a (possibly infinite) sum of multiple Wiener-Ito integrals. This decomposition is called Wiener-Itô chaos expansion, and the integrands are denoted as kernels of the Wiener-Itô chaos expansion. An explicit formula for these kernels is known due to Last and Penrose.
Via their Wiener-Itô chaos expansions the so-called Malliavin operators are defined. By combining Malliavin calculus and Stein's method, a well-known technique to derive limit theorems in probability theory, bounds for the normal approximation of Poisson functionals in the Wasserstein distance and vectors of Poisson functionals in a similar distance were obtained by Peccati, Sole, Taqqu, and Utzet and Peccati and Zheng, respectively. An analogous bound for the univariate normal approximation in Kolmogorov distance is derived.
In order to evaluate these bounds, one has to compute the expectation of products of multiple Wiener-Itô integrals, which are complicated sums of deterministic integrals. Therefore, the bounds for the normal approximation of Poisson functionals reduce to sums of integrals depending on the kernels of the Wiener-Itô chaos expansion.
The strategy to derive central limit theorems for Poisson functionals is to compute the kernels of their Wiener-Itô chaos expansions, to put the kernels in the bounds for the normal approximation, and to show that the bounds vanish asymptotically.
By this approach, central limit theorems for some problems from stochastic geometry are derived. Univariate and multivariate central limit theorems for some functionals of the intersection process of Poisson k-flats and the number of vertices and the total edge length of a Gilbert graph are shown. These Poisson functionals are so-called Poisson U-statistics which have an easier structure since their Wiener-Itô chaos expansions are finite, i.e. their Wiener-Itô chaos expansions consist of finitely many multiple Wiener-Itô integrals. As examples for Poisson functionals with infinite Wiener-Itô chaos expansions, central limit theorems for the volume of the Poisson-Voronoi approximation of a convex set and the intrinsic volumes of Boolean models are proven.
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On probability distributions of diffusions and financial models with non-globally smooth coefficients / Sur les lois de diffusions et de modèles financiers avec coefficients non globalement réguliersDe Marco, Stefano 23 November 2010 (has links)
Des travaux récents dans le domaine des mathématiques financières ont fait émerger l'importance de l'étude de la régularité et du comportement fin des queues de distribution pour certaines classes de diffusions à coefficients non globalement réguliers. Dans cette thèse, nous traitons des problèmes issus de ce contexte. Nous étudions d'abord l'existence, la régularité et l'asymptotique en espace de densités pour les solutions d'équations différentielles stochastiques en n'imposant que des conditions locales sur les coefficients de l'équation. Notre analyse se base sur les outils du calcul de Malliavin et sur des estimations pour les processus d'Ito confinés dans un tube autour d'une courbe déterministe. Nous obtenons des estimations significatives de la fonction de répartition et de la densité dans des classes de modèles comprenant des généralisations du CIR et du CEV et des modèles à volatilité locale-stochastique : dans ce deuxième cas, les estimations entraînent l'explosion des moments du sous-jacent et ont ainsi un impact sur le comportement asymptotique en strike de la volatilité implicite. La modélisation paramétrique de la surface de volatilité, à son tour, fait l'objet de la deuxième partie. Nous considérons le modèle SVI de J. Gatheral, en proposant une nouvelle stratégie de calibration quasi-explicite, dont nous illustrons les performances sur des données de marché. Ensuite, nous analysons la capacité du SVI à générer des approximations pour les smiles symétriques, en le généralisant à un modèle dépendant du temps. Nous en testons l'application à un modèle de Heston (sans et avec déplacement), en générant des approximations semi-fermées pour le smile de volatilité / Some recent works in the field of mathematical finance have brought new light on the importance of studying the regularity and the tail asymptotics of distributions for certain classes of diffusions with non-globally smooth coefficients. In this Ph.D. dissertation we deal with some issues in this framework. In a first part, we study the existence, smoothness and space asymptotics of densities for the solutions of stochastic differential equations assuming only local conditions on the coefficients of the equation. Our analysis is based on Malliavin calculus tools and on « tube estimates » for Ito processes, namely estimates for the probability that the trajectory of an Ito process remains close to a deterministic curve. We obtain significant estimates of densities and distribution functions in general classes of option pricing models, including generalisations of CIR and CEV processes and Local-Stochastic Volatility models. In the latter case, the estimates we derive have an impact on the moment explosion of the underlying price and, consequently, on the large-strike behaviour of the implied volatility. Parametric implied volatility modeling, in its turn, makes the object of the second part. In particular, we focus on J. Gatheral's SVI model, first proposing an effective quasi-explicit calibration procedure and displaying its performances on market data. Then, we analyse the capability of SVI to generate efficient approximations of symmetric smiles, building an explicit time-dependent parameterization. We provide and test the numerical application to the Heston model (without and with displacement), for which we generate semi-closed expressions of the smile
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Three essays on valuation and investment in incomplete marketsRinger, Nathanael David 01 June 2011 (has links)
Incomplete markets provide many challenges for both investment decisions and valuation
problems. While both problems have received extensive attention in complete markets,
there remain many open areas in the theory of incomplete markets. We present the results
in three parts. In the first essay we consider the Merton investment problem of optimal portfolio
choice when the traded instruments are the set of zero-coupon bonds. Working within
a Markovian Heath-Jarrow-Morton framework of the interest rate term structure driven by
an infinite dimensional Wiener process, we give sufficient conditions for the existence and
uniqueness of an optimal investment strategy. When there is uniqueness, we provide a characterization
of the optimal portfolio. Furthermore, we show that a specific Gauss-Markov random field model can be treated within this framework, and explicitly calculate the optimal
portfolio. We show that the optimal portfolio in this case can be identified with the discontinuities
of a certain function of the market parameters. In the second essay we price a claim,
using the indifference valuation methodology, in the model presented in the first section. We
appeal to the indifference pricing framework instead of the classic Black-Scholes method due
to the natural incompleteness in such a market model. Because we price time-sensitive interest
rate claims, the units in which we price are very important. This will require us to take
care in formulating the investor’s utility function in terms of the units in which we express the
wealth function. This leads to new results, namely a general change-of-numeraire theorem in
incomplete markets via indifference pricing. Lastly, in the third essay, we propose a method
to price credit derivatives, namely collateralized debt obligations (CDOs) using indifference.
We develop a numerical algorithm for pricing such CDOs. The high illiquidity of the CDO
market coupled with the allowance of default in the underlying traded assets creates a very
incomplete market. We explain the market-observed prices of such credit derivatives via the
risk aversion of investors. In addition to a general algorithm, several approximation schemes
are proposed. / text
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Frakcionální Brownův pohyb ve financích / Fractional Brownian Motion in FinanceKratochvíl, Matěj January 2016 (has links)
This thesis deals with the stochastic integral with respect to Gaussian processes, which can be expressed in the form Bt = t 0 K(t, s)dWs. Here W stands for a Brownian motion and K for a square integrable Volterra kernel. Such processes generalize fractional Brownian motion. Since these processes are not semimartin- gales, Itô calculus cannot be used and other methods must be employed to define the stochastic integral with respect to these proceses. Two ways are considered in this thesis. If both the integrand and the process B are regular enough, it is possible to define the integral in the pathwise sense as a generalization of Lebesgue-Stieltjes integral. The other method uses the methods of Malliavin cal- culus and defines the integral as an adjoint operator to the Malliavin derivative. As an application, the stochastic differential equation dSt = µStdt + σStdBt, which is used to model price of a stock, is solved. Implications of such a model are briefly discussed. 1
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Some Extensions of Fractional Ornstein-Uhlenbeck Model : Arbitrage and Other ApplicationsMorlanes, José Igor January 2017 (has links)
This doctoral thesis endeavors to extend probability and statistical models using stochastic differential equations. The described models capture essential features from data that are not explained by classical diffusion models driven by Brownian motion. New results obtained by the author are presented in five articles. These are divided into two parts. The first part involves three articles on statistical inference and simulation of a family of processes related to fractional Brownian motion and Ornstein-Uhlenbeck process, the so-called fractional Ornstein-Uhlenbeck process of the second kind (fOU2). In two of the articles, we show how to simulate fOU2 by means of circulant embedding method and memoryless transformations. In the other one, we construct a least squares consistent estimator of the drift parameter and prove the central limit theorem using techniques from Stochastic Calculus for Gaussian processes and Malliavin Calculus. The second phase of my research consists of two articles about jump market models and arbitrage portfolio strategies for an insider trader. One of the articles describes two arbitrage free markets according to their risk neutral valuation formula and an arbitrage strategy by switching the markets. The key aspect is the difference in volatility between the markets. Statistical evidence of this situation is shown from a sequential data set. In the other one, we analyze the arbitrage strategies of an strong insider in a pure jump Markov chain financial market by means of a likelihood process. This is constructed in an enlarged filtration using Itô calculus and general theory of stochastic processes. / Föreliggande doktorsavhandling strävar efter att utöka sannolikhetsbaserade och statistiska modeller med stokastiska differentialekvationer. De beskrivna modellerna fångar väsentliga egenskaper i data som inte förklaras av klassiska diffusionsmodeller för brownsk rörelse. Nya resultat, som författaren har härlett, presenteras i fem uppsatser. De är ordnade i två delar. Del 1 innehåller tre uppsatser om statistisk inferens och simulering av en familj av stokastiska processer som är relaterade till fraktionell brownsk rörelse och Ornstein-Uhlenbeckprocessen, så kallade andra ordningens fraktionella Ornstein-Uhlenbeckprocesser (fOU2). I två av uppsatserna visar vi hur vi kan simulera fOU2-processer med hjälp av cyklisk inbäddning och minneslös transformering. I den tredje uppsatsen konstruerar vi en minsta-kvadratestimator som ger konsistent skattning av driftparametern och bevisar centrala gränsvärdessatsen med tekniker från statistisk analys för gaussiska processer och malliavinsk analys. Del 2 av min forskning består av två uppsatser om marknadsmodeller med plötsliga hopp och portföljstrategier med arbitrage för en insiderhandlare. En av uppsatserna beskriver två arbitragefria marknader med riskneutrala värderingsformeln och en arbitragestrategi som består i växla mellan marknaderna. Den väsentliga komponenten är skillnaden mellan marknadernas volatilitet. Statistisk evidens i den här situationen visas utifrån ett sekventiellt datamaterial. I den andra uppsatsen analyserar vi arbitragestrategier hos en insiderhandlare i en finansiell marknad som förändrar sig enligt en Markovkedja där alla förändringar i tillstånd består av plötsliga hopp. Det gör vi med en likelihoodprocess. Vi konstruerar detta med utökad filtrering med hjälp av Itôanalys och allmän teori för stokastiska processer. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Manuscript. Paper 5: Manuscript.</p>
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Étude et modélisation des équations différentielles stochastiques / High weak order discretization schemes for stochastic differential equationRey, Clément 04 December 2015 (has links)
Durant les dernières décennies, l'essor des moyens technologiques et particulièrement informatiques a permis l'émergence de la mise en œuvre de méthodes numériques pour l'approximation d'Equations Différentielles Stochastiques (EDS) ainsi que pour l'estimation de leurs paramètres. Cette thèse aborde ces deux aspects et s'intéresse plus spécifiquement à l'efficacité de ces méthodes. La première partie sera consacrée à l'approximation d'EDS par schéma numérique tandis que la deuxième partie traite l'estimation de paramètres. Dans un premier temps, nous étudions des schémas d'approximation pour les EDSs. On suppose que ces schémas sont définis sur une grille de temps de taille $n$. On dira que le schéma $X^n$ converge faiblement vers la diffusion $X$ avec ordre $h in mathbb{N}$ si pour tout $T>0$, $vert mathbb{E}[f(X_T)-f(X_T^n)] vertleqslant C_f /n^h$. Jusqu'à maintenant, sauf dans certains cas particulier (schémas d'Euler et de Ninomiya Victoir), les recherches sur le sujet imposent que $C_f$ dépende de la norme infini de $f$ mais aussi de ses dérivées. En d'autres termes $C_f =C sum_{vert alpha vert leqslant q} Vert partial_{alpha} f Vert_{ infty}$. Notre objectif est de montrer que si le schéma converge faiblement avec ordre $h$ pour un tel $C_f$, alors, sous des hypothèses de non dégénérescence et de régularité des coefficients, on peut obtenir le même résultat avec $C_f=C Vert f Vert_{infty}$. Ainsi, on prouve qu'il est possible d'estimer $mathbb{E}[f(X_T)]$ pour $f$ mesurable et bornée. On dit alors que le schéma converge en variation totale vers la diffusion avec ordre $h$. On prouve aussi qu'il est possible d'approximer la densité de $X_T$ et ses dérivées par celle $X_T^n$. Afin d'obtenir ce résultat, nous emploierons une méthode de calcul de Malliavin adaptatif basée sur les variables aléatoires utilisées dans le schéma. L'intérêt de notre approche repose sur le fait que l'on ne traite pas le cas d'un schéma particulier. Ainsi notre résultat s'applique aussi bien aux schémas d'Euler ($h=1$) que de Ninomiya Victoir ($h=2$) mais aussi à un ensemble générique de schémas. De plus les variables aléatoires utilisées dans le schéma n'ont pas de lois de probabilité imposées mais appartiennent à un ensemble de lois ce qui conduit à considérer notre résultat comme un principe d'invariance. On illustrera également ce résultat dans le cas d'un schéma d'ordre 3 pour les EDSs unidimensionnelles. La deuxième partie de cette thèse traite le sujet de l'estimation des paramètres d'une EDS. Ici, on va se placer dans le cas particulier de l'Estimateur du Maximum de Vraisemblance (EMV) des paramètres qui apparaissent dans le modèle matriciel de Wishart. Ce processus est la version multi-dimensionnelle du processus de Cox Ingersoll Ross (CIR) et a pour particularité la présence de la fonction racine carrée dans le coefficient de diffusion. Ainsi ce modèle permet de généraliser le modèle d'Heston au cas d'une covariance locale. Dans cette thèse nous construisons l'EMV des paramètres du Wishart. On donne également la vitesse de convergence et la loi limite pour le cas ergodique ainsi que pour certains cas non ergodiques. Afin de prouver ces convergences, nous emploierons diverses méthodes, en l'occurrence : les théorèmes ergodiques, des méthodes de changement de temps, ou l'étude de la transformée de Laplace jointe du Wishart et de sa moyenne. De plus, dans dernière cette étude, on étend le domaine de définition de cette transformée jointe / The development of technology and computer science in the last decades, has led the emergence of numerical methods for the approximation of Stochastic Differential Equations (SDE) and for the estimation of their parameters. This thesis treats both of these two aspects. In particular, we study the effectiveness of those methods. The first part will be devoted to SDE's approximation by numerical schemes while the second part will deal with the estimation of the parameters of the Wishart process. First, we focus on approximation schemes for SDE's. We will treat schemes which are defined on a time grid with size $n$. We say that the scheme $ X^n $ converges weakly to the diffusion $ X $, with order $ h in mathbb{N} $, if for every $ T> 0 $, $ vert mathbb{E} [f (X_T) -f (X_T^n)]vert leqslant C_f / h^n $. Until now, except in some particular cases (Euler and Victoir Ninomiya schemes), researches on this topic require that $ C_f$ depends on the supremum norm of $ f $ as well as its derivatives. In other words $C_f =C sum_{vert alpha vert leqslant q} Vert partial_{alpha} f Vert_{ infty}$. Our goal is to show that, if the scheme converges weakly with order $ h $ for such $C_f$, then, under non degeneracy and regularity assumptions, we can obtain the same result with $ C_f=C Vert f Vert_{infty}$. We are thus able to estimate $mathbb{E} [f (X_T)]$ for a bounded and measurable function $f$. We will say that the scheme converges for the total variation distance, with rate $h$. We will also prove that the density of $X^n_T$ and its derivatives converge toward the ones of $X_T$. The proof of those results relies on a variant of the Malliavin calculus based on the noise of the random variable involved in the scheme. The great benefit of our approach is that it does not treat the case of a particular scheme and it can be used for many schemes. For instance, our result applies to both Euler $(h = 1)$ and Ninomiya Victoir $(h = 2)$ schemes. Furthermore, the random variables used in this set of schemes do not have a particular distribution law but belong to a set of laws. This leads to consider our result as an invariance principle as well. Finally, we will also illustrate this result for a third weak order scheme for one dimensional SDE's. The second part of this thesis deals with the topic of SDE's parameter estimation. More particularly, we will study the Maximum Likelihood Estimator (MLE) of the parameters that appear in the matrix model of Wishart. This process is the multi-dimensional version of the Cox Ingersoll Ross (CIR) process. Its specificity relies on the square root term which appears in the diffusion coefficient. Using those processes, it is possible to generalize the Heston model for the case of a local covariance. This thesis provides the calculation of the EMV of the parameters of the Wishart process. It also gives the speed of convergence and the limit laws for the ergodic cases and for some non-ergodic case. In order to obtain those results, we will use various methods, namely: the ergodic theorems, time change methods or the study of the joint Laplace transform of the Wishart process together with its average process. Moreover, in this latter study, we extend the domain of definition of this joint Laplace transform
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