Dans ce travail de thèse, nous nous intéressons à certaines propriétés d'une classe de processus stochastiques à accroissements stationnaires et autosimilaires. Ces processus sont représentés par des intégrales multiples de Wiener-Itô. Dans le premier chapitre, nous étudions les propriétés géométriques des trajectoires de ce type de processus. En particulier, nous obtenons un développement en ondelettes presque-sûr. Celui-ci permet alors de trouver une borne supérieure pour le module de continuité uniforme, une borne supérieure pour le comportement asymptotique du processus et un résultat presque-sûr concernant les coefficients ponctuel et local de Hölder. De plus, nous obtenons des bornes inférieures et supérieures pour les dimensions de Hausdorff du graphe et de l'image des versions multidimensionnelles anisotropes de la classe de processus considérée. Dans le deuxième et le troisième chapitre de cette thèse, nous nous intéressons au calcul différentiel stochastique relatif au processus de Rosenblatt. A l'aide de la théorie des distributions de Hida, nous définissons une intégrale stochastique par rapport au processus de Rosenblatt. Nous obtenons une formule d'Itô pour certaines fonctionnelles du processus de Rosenblatt. Nous calculons explicitement la variance de l'intégrale stochastique par rapport au processus de Rosenblatt pour une classe spécifique d'intégrandes aléatoires. Enfin, nous comparons l'intégrale introduite avec d'autres définitions utilisées dans la littérature et procédons à une étude fine des termes résiduels faisant le lien entre ces différentes définitions. / In this PhD thesis, we are concerned with some properties of a class of self-similar stochastic processes with stationary increments. These processes are represented by multiple Wiener-Itô integrals. In the first chapter, we study geometric properties of the sample path of this type of processes. Specifically, we obtain an almost sure wavelet expansion which, in turn, allows us to compute an upper bound for the uniform modulus of continuity, an upper bound for the asymptotic growth at infinity of the processes and the almost sure values of the pointwise and local Hölder exponents at any points. Moreover, we obtain lower and upper bounds for the Hausdorff dimensions of the graph and the image of multidimensional anisotropic versions of the class of processes previously considered. In the second and in the third chapters, we are interested in the stochastic calculus with respect to the Rosenblatt process. Using Hida distributions theory, we define a stochastic integral with respect to the Rosenblatt process. We obtain an Itô formula for some functional of the Rosenblatt process. We compute explicitly the variance of the stochastic integral with respect to the Rosenblatt process for a specific class of stochastic integrands. At last, we compare the considered integral with other definitions used in the literature and provide a careful analysis of the residual terms linking the different definitions of integrals.

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Dans cette thèse, on s'intéresse à la notion de temps fort de stationnarité et à celle, étroitement liée, de dual de stationnarité forte. Ces outils permettent d'étu- dier la convergence de processus ergodiques, en déterminant un instant aléatoire où l'équilibre est atteint. Les espaces d'état des processus considérés ici sont des graphes continus ou discrets. Dans la première partie, on considère le cas discret, et on dégage une condition nécessaire et suffisante à l'existence, pour n'importe quelle loi initiale, d'un temps fort de stationnarité fini. Pour cela, on construit explicitement un dual de station- narité forte, à valeurs dans l'ensemble des parties connexes du graphe, qui évolue à chaque étape en ajoutant ou en enlevant des points de sa frontière. Lorsque cette opération sépare l'ensemble dual en plusieurs parties, afin de ne pas le déconnecter, une de ces parties est choisie au hasard, avec une probabilité proportionnelle à son poids par la mesure invariante. On s'intéresse également au comportement général d'un processus dual, et on donne quelques exemples différents de celui construit précédemment. Dans la deuxième partie, on traite le cas continu, et le processus étudié est alors une diffusion. On caractérise notamment sa mesure invariante, et on explicite un générateur infinitésimal qui devrait être celui d'un processus dual. Néanmoins, ce cas s'avère plus compliqué que le cas discret. Le processus dual n'est donc construit que pour un mouvement brownien sur un graphe particulier, comme l'unique so- lution d'un problème de martingale. Des pistes sont présentées pour traiter des diffusions sur des graphes plus généraux, notamment en utilisant la convergence d'une suite de processus de saut tels que ceux présentés dans la première partie. / In this thesis, we are interested in the notion of strong stationary time, and in that, strongly connected, of strong stationary dual. These tools allow to study the convergence of ergodic processes, by determining a random time when the equilibrium is reached. The state space of the considered processes are discrete or continuous graphs. In the first part, we consider the discrete case, and we explicit a necessary and sufficient condition to the existence, for any initial distribution, of a finite strong stationary time. To do so, we construct explicitly a strong stationary dual, with values in the set of connected subsets of the graph, which evolves at each step by adding or removing some points at its border. Whenever this operation separates the dual set in several parts, in order not to disconnect it, one of these parts is chosen randomly, with a probability proportionnal to its weight relative to the invariant distribution. We also study the general behaviour of any dual process,2 and we give some other examples. In the second part, we deal with the continuous case, and the studied process is then a diffuion. We caracterize its invariant distribution, and we explicit an infinitesimal generator, which is expected to be that of a dual process. Nevertheless, this case turns out to be a little more involved that the discrete one. The dual process is thus constructed only for a brownian motion on a particular graph, as the unique solution of a martingale problem. Some leads are given to solve the case of diffusions on more general graphs, especially by using the convergence of a sequence of jump processes such as those presented in the first part.

The concept of dynamic programming was originally used in late 1949, mostly during the 1950s, by Richard Bellman to describe decision making problems. By 1952, he refined this to the modern meaning, referring specifically to nesting smaller decision problems inside larger decisions. Also, the Bellman equation, one of the basic concepts in dynamic programming, is named after him. Dynamic programming has become an important argument which was used in various fields; such as, economics, finance, bioinformatics, aerospace, information theory, etc. Since Richard Bellman's invention of dynamic programming, economists and mathematicians have formulated and solved a huge variety of sequential decision making problems both in deterministic and stochastic cases; either finite or infinite time horizon. This thesis is comprised of five chapters where the major objective is to study both deterministic and stochastic dynamic programming models in finance. In the first chapter, we give a brief history of dynamic programming and we introduce the essentials of theory. Unlike economists, who have analyzed the dynamic programming on discrete, that is, periodic and continuous time domains, we claim that trading is not a reasonably periodic or continuous act. Therefore, it is more accurate to demonstrate the dynamic programming on non-periodic time domains. In the second chapter we introduce time scales calculus. Moreover, since it is more realistic to analyze a decision maker’s behavior without risk aversion, we give basics of Stochastic Calculus in this chapter. After we introduce the necessary background, in the third chapter we construct the deterministic dynamic sequence problem on isolated time scales. Then we derive the corresponding Bellman equation for the sequence problem. We analyze the relation between solutions of the sequence problem and the Bellman equation through the principle of optimality. We give an example of the deterministic model in finance with all details of calculations by using guessing method, and we prove uniqueness and existence of the solution by using the Contraction Mapping Theorem. In the fourth chapter, we define the stochastic dynamic sequence problem on isolated time scales. Then we derive the corresponding stochastic Bellman equation. As in the deterministic case, we give an example in finance with the distributions of solutions.

It is well established that stock market volatility has a memory of the past, moreover it is found that volatility correlations are long ranged. As a consequence, volatility cannot be characterized by a single correlation time in general. Recent empirical work suggests that the volatility correlation functions of various assets actually decay as a power law. Moreover it is well established that the distribution functions for the returns do not obey a Gaussian distribution, but follow more the type of distributions that incorporate what are commonly known as fat–tailed distributions. As a result, if one is to model the evolution of the stock price, stock market or any financial derivative, then standard Brownian motion models are inaccurate. One must take into account the results obtained from empirical studies and work with models that include realistic features observed on the market. In this thesis we show that it is possible to derive the path integral for a non-Gaussian option pricing model that can capture fat–tails. However we find that the path integral technique can only be used on a very small set of problems, as a number of situations of interest are shown to be intractable. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1378473 / Thesis (Ph.D.) -- University of Adelaide, School of Electrical and Electronic Engineering, 2010

It is well established that stock market volatility has a memory of the past, moreover it is found that volatility correlations are long ranged. As a consequence, volatility cannot be characterized by a single correlation time in general. Recent empirical work suggests that the volatility correlation functions of various assets actually decay as a power law. Moreover it is well established that the distribution functions for the returns do not obey a Gaussian distribution, but follow more the type of distributions that incorporate what are commonly known as fat–tailed distributions. As a result, if one is to model the evolution of the stock price, stock market or any financial derivative, then standard Brownian motion models are inaccurate. One must take into account the results obtained from empirical studies and work with models that include realistic features observed on the market. In this thesis we show that it is possible to derive the path integral for a non-Gaussian option pricing model that can capture fat–tails. However we find that the path integral technique can only be used on a very small set of problems, as a number of situations of interest are shown to be intractable. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1378473 / Thesis (Ph.D.) -- University of Adelaide, School of Electrical and Electronic Engineering, 2010

De nombreux phénomènes de physique quantique ne peuvent être compris que par l'analyse des systèmes ouverts. Un appareil de mesure, par exemple, est un système macroscopique en contact avec un système quantique. Ainsi, tout modèle d'expérience doit prendre en compte les dynamiques propres aux systèmes ouverts. Ces dynamiques peuvent être complexes : l'interaction du système avec son environnement peut modifier ses propriétés, l'interaction peu créer des effets de mémoire dans l'évolution du système, . . . Ces dynamiques sont particulièrement importantes dans l'étude des expériences d'optique quantique. Nous sommes aujourd'hui capables de manipuler individuellement des particules. Pour cela la compréhension et le contrôle de l'influence de l'environnement est crucial. Dans cette thèse nous étudions d'un point de vue théorique quelques procédures communément utilisées en optique quantique. Avant la présentation de nos résultats, nous introduisons et motivons l'utilisation de la description markovienne des systèmes quantiques ouverts. Nous présentons a la fois les équations maîtresses et le calcul stochastique quantique. Nous introduisons ensuite la notion de trajectoire quantique pour la description des mesures indirectes continues. C'est dans ce contexte que l'on présente les résultats obtenus au cours de cette thèse. Dans un premier temps, nous étudions la convergence des mesures non destructives. Nous montrons qu'elles reproduisent la réduction du paquet d'onde du système mesuré. Nous montrons que cette convergence est exponentielle avec un taux fixe. Nous bornons le temps moyen de convergence. Dans ce cadre, en utilisant les techniques de changement de mesure par martingale, nous obtenons la limite continue des trajectoires quantiques discrètes. Dans un second temps, nous étudions l'influence de l'enregistrement des résultats de mesure sur la préparation d'état par ingénierie de réservoir. Nous montrons que l'enregistrement des résultats de mesure n'a pas d'influence sur la convergence proprement dite. Cependant, nous trouvons que l'enregistrement des résultats de mesure modifie le comportement du système avant la convergence. Nous retrouvons une convergence exponentielle avec un taux équivalent au taux sans enregistrement. Mais nous trouvons aussi un nouveau taux de convergence correspondant a une stabilité asymptotique. Ce dernier taux est interprété comme une mesure non destructive ajoutée. Ainsi l'état du système ne converge qu'après un temps aléatoire. A partir de ce temps la convergence peut être bien plus rapide. Nous obtenons aussi une borne sur le temps moyen de convergence. / Many quantum physics phenomena can only be understood in the context of open system analysis. For example a measurement apparatus is a macroscopic system in contact with a quantum system. Therefore any experiment model needs to take into account open system behaviors. These behaviors can be complex: the interaction of the system with its environment might modify its properties, the interaction may induce memory effects in the system evolution, ... These dynamics are particularly important when studying quantum optic experiments. We are now able to manipulate individual particles. Understanding and controlling the environment influence is therefore crucial. In this thesis we investigate at a theoretical level some commonly used quantum optic procedures. Before the presentation of our results, we introduce and motivate the Markovian approach to open quantum systems. We present both the usual master equation and quantum stochastic calculus. We then introduce the notion of quantum trajectory for the description of continuous indirect measurements. It is in this context that we present the results obtained during this thesis. First, we study the convergence of non demolition measurements. We show that they reproduce the system wave function collapse. We show that this convergence is exponential with a fixed rate. We bound the mean convergence time. In this context, we obtain the continuous time limit of discrete quantum trajectories using martingale change of measure techniques. Second, we investigate the influence of measurement outcome recording on state preparation using reservoir engineering techniques. We show that measurement outcome recording does not influence the convergence itself. Nevertheless, we find that measurement outcome recording modifies the system behavior before the convergence. We recover an exponential convergence with a rate equivalent to the rate without measurement outcome recording. But we also find a new convergence rate corresponding to an asymptotic stability. This last rate is interpreted as an added non demolition measurement. Hence, the system state converges only after a random time. At this time the convergence can be much faster. We also find a bound on the mean convergence time.

Dans la première partie de cette thèse, on présente des inégalités de concentration convexe pour des intégrales stochastiques. Ces résultats sont obtenus par calcul stochastique e tpar calcul de Malliavin forward/backward. On présente également des inégalités de déviation pour les exponentielles martingales à saut.Dans une deuxième partie on présente des théorèmes limites pour le conditionnement du mouvement brownien. / In the first part of this thesis, we present some convex concentration inequalities for stochastic integrals. These results are obtained by forward/backward stochastic calculus combined with Malliavin calculus. We also present deviation inequalities for exponentialjump-diffusion.In the second part, we present some limit theorems for the conditionning of Brownian motion.

This thesis considers the filtering and prediction problems of nonlinear noisy econometric systems. As a filter/predictor, the standard tool Extended Kalman Filter and new approaches Discrete Quantization Filter and Sequential Importance Resampling Filter are used. The algorithms are compared by using Monte Carlo Simulation technique. The advantages of the new algorithms over Extended Kalman Filter are shown.

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