Les caractéristiques des rayonnements solaires dépendent fortement de certains événements météorologiques non observés comme fréquence, taille et type des nuages et leurs propriétés optiques (aérosols atmosphériques, al- bédo du sol, vapeur d’eau, poussière et turbidité atmosphérique) tandis qu’une séquence du rayonnement solaire peut être observée et mesurée à une station donnée. Ceci nous a suggéré de modéliser les processus de rayonnement solaire (ou d’indice de clarté) en utilisant un modèle Markovien caché (HMM), paire corrélée de processus stochastiques. Notre modèle principal est un HMM à temps continu (Xt, yt)t_0 est tel que (yt), le processus observé de rayonnement, soit une solution de l’équation différentielle stochastique (EDS) : dyt = [g(Xt)It − yt]dt + _(Xt)ytdWt, où It est le rayonnement extraterrestre à l’instant t, (Wt) est un mouvement Brownien standard et g(Xt), _(Xt) sont des fonctions de la chaîne de Markov non observée (Xt) modélisant la dynamique des régimes environnementaux. Pour ajuster nos modèles aux données réelles observées, les procédures d’estimation utilisent l’algorithme EM et la méthode du changement de mesures par le théorème de Girsanov. Des équations de filtrage sont établies et les équations à temps continu sont approchées par des versions robustes. Les modèles ajustés sont appliqués à des fins de comparaison et classification de distributions et de prédiction. / Characteristics of solar radiation highly depend on some unobserved meteorological events such as frequency, height and type of the clouds and their optical properties (atmospheric aerosols, ground albedo, water vapor, dust and atmospheric turbidity) while a sequence of solar radiation can be observed and measured at a given station. This has suggested us to model solar radiation (or clearness index) processes using a hidden Markov model (HMM), a pair of correlated stochastic processes. Our main model is a continuous-time HMM (Xt, yt)t_0 is such that the solar radiation process (yt)t_0 is a solution of the stochastic differential equation (SDE) : dyt = [g(Xt)It − yt]dt + _(Xt)ytdWt, where It is the extraterrestrial radiation received at time t, (Wt) is a standard Brownian motion and g(Xt), _(Xt) are functions of the unobserved Markov chain (Xt) modelling environmental regimes. To fit our models to observed real data, the estimation procedures combine the Expectation Maximization (EM) algorithm and the measure change method due to Girsanov theorem. Filtering equations are derived and continuous-time equations are approximated by robust versions. The models are applied to pdf comparison and classification and prediction purposes.

We consider infinite-dimensional diffusions where the interaction between the coordinates has a finite extent both in space and time. In particular, it is not supposed to be smooth or Markov. The initial state of the system is Gibbs, given by a strong summable interaction. If the strongness of this initial interaction is lower than a suitable level, and if the dynamical interaction is bounded from above in a right way, we prove that the law of the diffusion at any time t is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion in space uniformly in time of the Girsanov factor coming from the dynamics and exponential ergodicity of the free dynamics to an equilibrium product measure.

A num¡¦eraire is the unit of account in which other assets are denominated. One usually takes the num¡¦eraire to be the currency of a country. In some applications one must change the num¡¦eraire due to the finance considerations. And sometimes it is convenient to change the num¡¦eraire because of modeling considerations. A model can be complicated or simple, depending on the choice of thenum¡¦eraire for the method. When change the num¡¦eraire, denominating the asset in some other unit of account, it is no longer a martingale under ˜P . When we change the num¡¦eraire, we need to also change the risk-neutral measure in order to maintain risk neutrality. The details and some applications of this idea developed in this thesis.

A dynamic semiparametric pricing method is proposed for financial derivatives including European and American type options and convertible bonds. The proposed method is an iterative procedure which uses nonparametric regression to approximate derivative values and parametric asset models to derive the continuation values. Extension to higher dimensional option pricing is also developed, in which the dependence structure of financial time series is modeled by copula functions. In the simulation study, we valuate one dimensional American options, convertible bonds and multi-dimensional American geometric average options and max options. The considered one-dimensional underlying asset models include the Black-Scholes, jump-diffusion, and nonlinear asymmetric GARCH models and for multivariate case we study copula models such as the Gaussian, Clayton and Gumbel copulae. Convergence of the method is proved under continuity assumption on the transition densities of the underlying asset models. And the orders of the supnorm errors are derived. Both the theoretical findings and the simulation results show the proposed approach to be tractable for numerical implementation and provides a unified and accurate technique for financial derivative pricing. The second part of this thesis studies the option pricing and hedging problems for conditional leptokurtic returns which is an important feature in financial data. The risk-neutral models for log and simple return models with heavy-tailed innovations are derived by an extended Girsanov change of measure, respectively. The result is applicable to the option pricing of the GARCH model with t innovations (GARCH-t) for simple eturn series. The dynamic semiparametric approach is extended to compute the option prices of conditional leptokurtic returns. The hedging strategy consistent with the extended Girsanov change of measure is constructed and is shown to have smaller cost variation than the commonly used delta hedging under the risk neutral measure. Simulation studies are also performed to show the effect of using GARCH-normal models to compute the option prices and delta hedging of GARCH-t model for plain vanilla and exotic options. The results indicate that there are little pricing and hedging differences between the normal and t innovations for plain vanilla and Asian options, yet significant disparities arise for barrier and lookback options due to improper distribution setting of the GARCH innovations.

Dans la première partie de ma thèse, en adaptant l'idée de Jien et Ma (2010), l'objectif principal est étudier les équations différentielles doublement stochastiques rétrogrades, semi-linéaires ou nonlinéaires, régies par un mouvement brownien standard et un mouvement brownien fractionnaire indépendant, ainsi que les équations différentielles partielles stochastiques associées régies par le mouvement brownien fractionnaire. Pour le cas semi-linéaire, dans un papier en collaboration avec Jorge A. Leόn (CINVESTAV, Mexique), nous utilisons le calcul de Malliavin dans le cadre du mouvement brownien fractionnaire et la transformation de Girsanov anticipative. Pour le cas nonlinéaire, nous appliquons la transformation de Doss-Sussmann. Dans la deuxième partie nous étudions la régularité, à savoir la continuité de Lipschitz conjointe et la semiconcavité conjointe, de la solution de viscosité pour une classe générale d'équations aux dérivées partielles-intégrales non locales de type Hamilton-Jacobi-Bellman. Pour cette fin nous employons l'interprétation stochastique par une équation différentielle stochastique rétrograde contrôlée avec sauts, en appliquant du changement de temps pour le mouvement brownien et la transformation de Kulik pour la mesure aléatoire de Poisson. Notre travail est une généralisation des travaux de Buckdahn, Cannarsa et Quincampoix (2010) et Buckdahn, Huang et Li (2011).

Les caractéristiques des rayonnements solaires dépendent fortement de certains événements météorologiques non observés comme fréquence, taille et type des nuages et leurs propriétés optiques (aérosols atmosphériques, al- bédo du sol, vapeur d'eau, poussière et turbidité atmosphérique) tandis qu'une séquence du rayonnement solaire peut être observée et mesurée à une station donnée. Ceci nous a suggéré de modéliser les processus de rayonnement solaire (ou d'indice de clarté) en utilisant un modèle Markovien caché (HMM), paire corrélée de processus stochastiques. Notre modèle principal est un HMM à temps continu (Xt, yt)t_0 est tel que (yt), le processus observé de rayonnement, soit une solution de l'équation différentielle stochastique (EDS) : dyt = [g(Xt)It − yt]dt + _(Xt)ytdWt, où It est le rayonnement extraterrestre à l'instant t, (Wt) est un mouvement Brownien standard et g(Xt), _(Xt) sont des fonctions de la chaîne de Markov non observée (Xt) modélisant la dynamique des régimes environnementaux. Pour ajuster nos modèles aux données réelles observées, les procédures d'estimation utilisent l'algorithme EM et la méthode du changement de mesures par le théorème de Girsanov. Des équations de filtrage sont établies et les équations à temps continu sont approchées par des versions robustes. Les modèles ajustés sont appliqués à des fins de comparaison et classification de distributions et de prédiction.

>Magister Scientiae - MSc / We present various methods of pricing Asian options. The methods include Monte Carlo simulations designed using control and antithetic variates, numerical solution of partial differential equation and using lower bounds.The price of the Asian option is known to be a certain risk-neutral expectation. Using the Feynman-Kac theorem, we deduce that the problem of determining the expectation implies solving a linear parabolic partial differential equation. This partial differential equation does not admit explicit solutions due to the fact that the distribution of a sum of lognormal variables is not explicit. We then solve the partial differential equation numerically using finite difference and Monte Carlo methods.Our Monte Carlo approach is based on the pseudo random numbers and not deterministic sequence of numbers on which Quasi-Monte Carlo methods are designed. To make the Monte Carlo method more effective, two variance reduction techniques are discussed.Under the finite difference method, we consider explicit and the Crank-Nicholson’s schemes. We demonstrate that the explicit method gives rise to extraneous solutions because the stability conditions are difficult to satisfy. On the other hand, the Crank-Nicholson method is unconditionally stable and provides correct solutions. Finally, we apply the pricing methods to a similar problem of determining the price of a European-style arithmetic basket option under the Black-Scholes framework. We find the optimal lower bound, calculate it numerically and compare this with those obtained by the Monte Carlo and Moment Matching methods.Our presentation here includes some of the most recent advances on Asian options, and we contribute in particular by adding detail to the proofs and explanations. We also contribute some novel numerical methods. Most significantly, we include an original contribution on the use of very sharp lower bounds towards pricing European basket options.

Stochastic partial differential equations have proven useful in many applied areas of mathematics, such as physics or mathematical finance. A major part of such equations consists of linear equations with additive noise. In certain cases, however, the drift part of the differential equation additionally contains a possibly problematic non-linear term, which makes it unsolvable by the standard methods and even a solution in the mild sense may be out of reach. In such situations, we may still find a solution in the weak sense by employing a suitable transformation of the probability space. This thesis deals with semilinear stochastic evolution equations in a separable Hilbert space, where the driving process is an element of a large class of processes - so called Volterra processes, which can be understood as a generalisation of the Wiener process and may be of use to model a wide range of phenomena. The weak solutions, however, have been studied so far only for equations with the cylindrical fractional Brownian motion as the driving process. In this thesis, we introduce a generalisation of the Girsanov theorem for cylindrical Gaussian Volterra processes and give, in full generality, sufficient conditions for the existence of a weak solution and the uniqueness of the equation in law. Further, we introduce...

到期還本的指數連動型證券為一種連結權益(equity)的債權證券，所連結的權益部分通常以隱含選擇權的方式建立。指數連動證券具有自動資產配置調整的特性，當股票市場表現不錯時，此契約給予投資人較高的股票市場風險暴露（因為股票上漲時，Delta值增加）。若股票市場表現不佳，則契約收益特徵接近債券的型式。所以是保守型投資得以參與部分股票市場表現之設計。 本論文所研究之中短期連動型票券，係以零息債券持有至到期（其面額等於到期還本金額），期間不可贖回或申購，並以期初零息債券貼現的部分來購買不同的請求權，以做為連動股票市場表現的機制。在推導多重標的資產請求權評價模型上，係採Martingale方式，其中並證明在Gisanov轉換機率測度下，多重標的之隨機項轉換的規則。 本文主要研究Rainbow Call與Spread Call的評價模型與避險參數；進一步研究標的資產間相關係數對選擇權價值之影響與避險上的財務經濟意義。另一方面，運用Martingale此一有力的工具，佐以現金流量分析，來推導跨國標的之評價模型，並提出跨國之避險操作方法，與說明標的資產與匯率間相關係數在避險上的財務經濟意義。 本文最後就兩套請求權設計之指數連動票券，模擬比較在不同相關係數下，與其他選擇權設計之指數連動票券的表現。並嘗試提出該設計票券之較佳表現時機。

A main thrust of this thesis is to develop and explore linearization-based numeric-analytic integration techniques in the context of stochastically driven nonlinear oscillators of relevance in structural dynamics. Unfortunately, unlike the case of deterministic oscillators, available numerical or numeric-analytic integration schemes for stochastically driven oscillators, often modelled through stochastic differential equations (SDE-s), have significantly poorer numerical accuracy. These schemes are generally derived through stochastic Taylor expansions and the limited accuracy results from difficulties in evaluating the multiple stochastic integrals. We propose a few higher-order methods based on the stochastic version of transversal linearization and another method of linearizing the nonlinear drift field based on a Girsanov change of measures. When these schemes are implemented within a Monte Carlo framework for computing the response statistics, one typically needs repeated simulations over a large ensemble. The statistical error due to the finiteness of the ensemble (of size N, say)is of order 1/√N, which implies a rather slow convergence as N→∞. Given the prohibitively large computational cost as N increases, a variance reduction strategy that enables computing accurate response statistics for small N is considered useful. This leads us to propose a weak variance reduction strategy. Finally, we use the explicit derivative-free linearization techniques for state and parameter estimations for structural systems using the extended Kalman filter (EKF). A two-stage version of the EKF (2-EKF) is also proposed so as to account for errors due to linearization and unmodelled dynamics. In Chapter 2, we develop higher order locally transversal linearization (LTL) techniques for strong and weak solutions of stochastically driven nonlinear oscillators. For developing the higher-order methods, we expand the non-linear drift and multiplicative diffusion fields based on backward Euler and Newmark expansions while simultaneously satisfying the original vector field at the forward time instant where we intend to find the discretized solution. Since the non-linear vector fields are conditioned on the solution we wish to determine, the methods are implicit. We also report explicit versions of such linearization schemes via simple modifications. Local error estimates are provided for weak solutions. Weak linearized solutions enable faster computation vis-à-vis their strong counterparts. In Chapter 3, we propose another weak linearization method for non-linear oscillators under stochastic excitations based on Girsanov transformation of measures. Here, the non-linear drift vector is appropriately linearized such that the resulting SDE is analytically solvable. In order to account for the error in replacing of non-linear drift terms, the linearized solutions are multiplied by scalar weighting function. The weighting function is the solution of a scalar SDE(i.e.,Radon-Nikodym derivative). Apart from numerically illustrating the method through applications to non-linear oscillators, we also use the Girsanov transformation of measures to correct the truncation errors in lower order discretizations. In order to achieve efficiency in the computation of response statistics via Monte Carlo simulation, we propose in Chapter 4 a weak variance reduction strategy such that the ensemble size is significantly reduced without seriously affecting the accuracy of the predicted expectations of any smooth function of the response vector. The basis of the variance reduction strategy is to appropriately augment the governing system equations and then weakly replace the associated stochastic forcing functions through variance-reduced functions. In the process, the additional computational cost due to system augmentation is generally far less besides the accrued advantages due to a drastically reduced ensemble size. The variance reduction scheme is illustrated through applications to several non-linear oscillators, including a 3-DOF system. Finally, in Chapter 5, we exploit the explicit forms of the LTL techniques for state and parameters estimations of non-linear oscillators of engineering interest using a novel derivative-free EKF and a 2-EKF. In the derivative-free EKF, we use one-term, Euler and Newmark replacements for linearizations of the non-linear drift terms. In the 2-EKF, we use bias terms to account for errors due to lower order linearization and unmodelled dynamics in the mathematical model. Numerical studies establish the relative advantages of EKF-DLL as well as 2-EKF over the conventional forms of EKF. The thesis is concluded in Chapter 6 with an overall summary of the contributions made and suggestions for future research.