Asian option is an option which payoff depends on the average underlying price over some some specific time period. Although there is no closed form solution of asian option, appropriate change of variable and Num¡¦eraire would reduce some terms of equation satisfies the Asian call price function. This thesis presents asian option¡¦s properties and process of reduction terms.

Shape Memory Alloy actuators can be used for morphing, or shape change, by controlling their temperature, which is effectively done by applying a voltage difference across their length. Control of these actuators requires determination of the relationship between voltage and strain so that an input-output map can be developed. In this research, a computer simulation uses a hyperbolic tangent curve to simulate the hysteresis behavior of a virtual Shape Memory Alloy wire in temperature-strain space, and uses a Reinforcement Learning algorithm called Sarsa to learn a near-optimal control policy and map the hysteretic region. The algorithm developed in simulation is then applied to an experimental apparatus where a Shape Memory Alloy wire is characterized in temperature-strain space. This algorithm is then modified so that the learning is done in voltage-strain space. This allows for the learning of a control policy that can provide a direct input-output mapping of voltage to position for a real wire. This research was successful in achieving its objectives. In the simulation phase, the Reinforcement Learning algorithm proved to be capable of controlling a virtual Shape Memory Alloy wire by determining an accurate input-output map of temperature to strain. The virtual model used was also shown to be accurate for characterizing Shape Memory Alloy hysteresis by validating it through comparison to the commonly used modified Preisach model. The validated algorithm was successfully applied to an experimental apparatus, in which both major and minor hysteresis loops were learned in temperature-strain space. Finally, the modified algorithm was able to learn the control policy in voltage-strain space with the capability of achieving all learned goal states within a tolerance of +-0.5% strain, or +-0.65mm. This policy provides the capability of achieving any learned goal when starting from any initial strain state. This research has validated that Reinforcement Learning is capable of determining a control policy for Shape Memory Alloy crystal phase transformations, and will open the door for research into the development of length controllable Shape Memory Alloy actuators.

In the thesis, we study the data collected from twelve-note music of Webern and Schoenberg, including opus 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30 and opus 31 of Webern and opus 25, 33a and opus 37 of Schoenberg. The data consists of the following two kinds. The data of the first kind consists of the four basic forms of the twelve-tone music. And the data of the second kind consists of the twelve-tone derived from the matrix of the twelve-note music. We will introduce the twelve-note music first and then study two main topics about twelve-note music in this thesis. In the first part, we consider the Markov properties of the first kind data. We compare the sample autocorrelation function and autocorrelation function of the fitted model to determine the fitness of the Markovian model. In the second part, we build the time series model for the second kind data. Sample autocorrelation function¡Bpartial autocorrelation function and extended autocorrelation function are used to determine the orders of the models. The best model is selected based on the AICC. Finally, we check the fitness of the models using sample autocorrelation function and partial autocorrelation function of the residuals.

The purpose of the thesis is to provide a useful concept in the framework of stochastic analysis applicable in finance. The thesis offers proof for Doob- Meyer theorem for boundend martingales which is then extended for local martingales. It also proves the strong Markov theorem for Wiener process and some of its significant consequences. The built framework is then used for creating a method for solution of different tasks in applied finance. 1

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Le thème de cette thèse est la théorie spatiale des valeurs extrêmes, et les objets principalement étudiés sont les processus max-stables à trajectoires continues. Nous commençons par déterminer la convergence des maximums de processus stochastiques indépendants, en utilisant la convergence de mesures empiriques vers des processus ponctuels de Poisson. Ensuite, nous déterminons les lois conditionnelles des processus max infiniment divisibles (max-i.d). La représentation des processus max-i.d par des processus ponctuels de Poisson permet l'introduction de notions telles que les fonctions extrémales et le hitting scénario qui permettent d'aboutir au résultat. Les processus max-stables étant des processus max-i.d, nous proposons un algorithme de simulation conditionnelle pour les champs max-stables puis nous l'utilisons pour des applications avec des données de précipitations autour de Zurich et de températures en Suisse. Nous trouvons aussi, une majoration du coefficient de β-mélange entre les restrictions d'un processus max-i.d sur deux sous-ensembles fermés et disjoints d'un espace métrique localement compact. Cette majoration permet d'obtenir de nouveaux critères pour le théorème de la limite central des processus stationnaires mélangeant. Enfin, nous terminons en démontrant qu'un processus stationnaire max-stable vérifiant la propriété de Markov est, quitte à renverser le temps, un processus max-autorégressif d’ordre 1. / The theme of this thesis is spatial extreme value theory and we focus on continuous max-stable processes. We begin with the convergence of the maximum of independent stochastic processes, by using the convergence of empirical measures to Poisson point processes. After that, we determine the regular conditional distributions of max infinitely divisible (max-i.d) processes. The representation of max-i.d. processes by Poisson point processes allows us to introduce the notions of extremal functions and hitting scenario. Our result relies on these new notions. Max-stable processes are max-i.d. processes, so we give an algorithm for conditional sampling and give an application to extreme precipitations around Zurich and extreme temperatures in Switzerland. We also find a upper bound for the β-mixing coefficient between the restrictions of a max-i.d. process on two disjoint closed subsets of a locally compact metric space. This entails a central limit theorem for stationary max-i.d processes. Finally, we prove that the class of stationary maxstable processes with the Markov property is equal, up to time reversal, to the class of stationary max-autoregressive processes of order 1.

This study is concerned with the numerical solution of certain stochastic models of chemical reactions. Such descriptions have been shown to be useful tools when studying biochemical processes inside living cells where classical deterministic rate equations fail to reproduce actual behavior. The main contribution of this thesis lies in its theoretical and practical investigation of different methods for obtaining numerical solutions to such descriptions. In a preliminary study, a simple but often quite effective approach to the moment closure problem is examined. A more advanced program is then developed for obtaining a consistent representation of the high dimensional probability density of the solution. The proposed method gains efficiency by utilizing a rapidly converging representation of certain functions defined over the semi-infinite integer lattice. Another contribution of this study, where the focus instead is on the spatially distributed case, is a suggestion for how to obtain a consistent stochastic reaction-diffusion model over an unstructured grid. Here it is also shown how to efficiently collect samples from the resulting model by making use of a hybrid method. In a final study, a time-parallel stochastic simulation algorithm is suggested and analyzed. Efficiency is here achieved by moving parts of the solution phase into the deterministic regime given that a parallel architecture is available. Necessary background material is developed in three chapters in this summary. An introductory chapter on an accessible level motivates the purpose of considering stochastic models in applied physics. In a second chapter the actual stochastic models considered are developed in a multi-faceted way. Finally, the current state-of-the-art in numerical solution methods is summarized and commented upon.

In this thesis we describe subclasses of a class of graphs with three types of edges, called loopless mixed graphs (LMGs). The class of LMGs contains almost all known classes of graphs used in the literature of graphical Markov models. We focus in particular on the subclass of ribbonless graphs (RGs), which as special cases include undirected graphs, bidirected graphs, and directed acyclic graphs, as well as ancestral graphs and summary graphs. We define a unifying interpretation of independence structure for LMGs and pairwise and global Markov properties for RGs, discuss their maximality, and, in particular, prove the equivalence of pairwise and global Markov properties for graphoids defined over the nodes of RGs. Three subclasses of LMGs (MC, summary, and ancestral graphs) capture the modified independence model after marginalisation over unobserved variables and conditioning on selection variables of variables satisfying independence restrictions represented by a directed acyclic graph (DAG). We derive algorithms to generate these graphs from a given DAG or from a graph of a specific subclass, and we study the relationships between these classes of graphs. Finally, a manual and codes are provided that explain methods and functions in R for implementing and generating various graphs studied in this thesis.

Cette thèse porte sur les relations entre les processus SLE, les ensembles CLE et le champ libre Gaussien. Dans le chapitre 2, nous donnons une construction des processus SLE(k,r) à partir des boucles des CLE(k) et d'échantillons de restriction chordale. Sheffield et Werner ont prouvé que les CLE(k) peuvent être construits à partir des processus d'exploration symétriques des SLE(k,r).Nous montrons dans le chapitre 3 que la configuration des boucles construites à partir du processus d'exploration asymétrique des SLE(k,k-6) donne la même loi CLE(k). Le processus SLE(4) peut être considéré comme les lignes de niveau du champ libre Gaussien et l'ensemble CLE(4) correspond à la collection des lignes de niveau de ce champ libre Gaussien. Dans la deuxième partie du chapitre 3, nous définissons un paramètre de temps invariant conforme pour chaque boucle appartenant à CLE(4) et nous donnons ensuite dans le chapitre 4 un couplage entre le champ libre Gaussien et l'ensemble CLE(4) à l'aide du paramètre de temps. Les processus SLE(k) peuvent être considérés comme les lignes de flot du champ libre Gaussien. Nous explicitons la dimension de Hausdorff de l'intersection de deux lignes de flot du champ libre Gaussien. Cela nous permet d'obtenir la dimension de l'ensemble des points de coupure et des points doubles de la courbe SLE, voir le chapitre 5. Dans le chapitre 6, nous définissons la mesure de restriction radiale, prouvons la caractérisation de ces mesures, et montrons la condition nécessaire et suffisante de l'existence des mesures de restriction radiale. / This thesis focuses on various relations between SLE, CLE and GFF. In Chapter 2, we give a construction of SLE(k,r) processes from CLE(k) loop configuration and chordal restriction samples. Sheffield and Werner has proved that CLE(k) can be constructed from symmetric SLE(k,k-6) exploration processes. We prove in Chapter 3 that the loop configuration constructed from the asymmetric SLE(k,k-6) exploration processes also give the same law CLE(k). SLE(4) can be viewed as level lines of GFF and CLE(4) can be viewed as the collection of level lines of GFF. We define a conformally invariant time parameter for each loop in CLE(4) in the second part of Chapter 3 and then give a coupling between GFF and CLE(4) with time parameter in Chapter 4. SLE(k,r) can be viewed as flow lines of GFF. We derive the Hausdorff dimension of the intersection of two flow lines in GFF. Then, from there, we obtain the dimension of the cut and double point set of SLE curve in Chapter 5. In Chapter 6, we define the radial restriction measure, prove the characterization of these measures, and show the if and only if condition for the existence of radial restriction measure.

資產配置乃是將資金分散投資到主要的資產類別中，諸如股票、債券、現金等。傳統的均數/變異數方法在資產配置上早已被廣泛的運用。但是，現今的金融情勢多變，多期配置的需求提高，傳統均數/變異數方法只處理單一期間的資產配置，且反應未來的能力不佳，顯然已經不適用。 本論文提供一種多期動態的資產配置，可以改良過去單點估計值的缺點，同時能夠將未來情境納入考量，使多期資產配置更富策略性。並實證在兩期的情況下，期中調整資產組合與不調整的差異性。從而瞭解持續的動態規劃，方能提升資產配置的效率性。 / Asset allocation is the process of dividing an investment fund among major asset classes such as equities, bonds, cash, etc. Traditional mean-variance portfolio selection is widely used for asset allocation. However, as time goes by, the financial condition changes rapidly. The method of mean-variance analysis has some limitations. It not only can’t deal with multiperiod asset allocation, but also cannot reflect future economic circumstances, especially for long-term investments. This research tries to use the method of multi-stage dynamic programming for asset allocation. This method can improve the pits of single estimate in using mean-variance analysis, and take future scenarios into account so that the model will become more useful in practice. The two-period empirical results have shown that using continuous dynamic programming to build strategic asset allocation decision can improve the efficiency of asset allocation.