In this thesis we use nonstandard methods for the valuation of derivatives on electricity. We model the dynamics of electricity spot price as mean reverting processes on the hyperfinite binomial tree and by switching to the risk-neutral world we derive analytical formulas for the price of forward contracts. Both of our models are fitted to the German electricity market and forward price predictions are compared with forward products traded on the exchange. We conclude that both the Ornstein-Uhlenbeck and the Schwartz one factor model fit long-term forward contracts well while our prediction results for short-term forward prod- ucts are not conclusive due to low liquidity and alternative approaches might be suitable. 1

Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.

The application of different unit root statistics is by now a standard practice in empirical work. Even when it is a practical issue, these statistics have complex nonstandard distributions depending on functionals of certain stochastic processes, and their derivations represent a barrier even for many theoretical econometricians. These derivations are based on rigorous and fundamental statistical tools which are not (very) well known by standard econometricians. This paper aims to fill this gap by explaining in a simple way one of these fundamental tools: namely, the Functional Central Limit Theorem. To this end, this paper analyzes the foundations and applicability of two versions of the Functional Central Limit Theorem within the framework of a unit root with a structural break. Initial attention is focused on the probabilistic structure of the time series to be considered. Thereafter, attention is focused on the asymptotic theory for nonstationary time series proposed by Phillips (1987a), which is applied by Perron (1989) to study the effects of an (assumed) exogenous structural break on the power of the augmented Dickey-Fuller test and by Zivot and Andrews (1992) to criticize the exogeneity assumption and propose a method for estimating an endogenous breakpoint. A systematic method for dealing with efficiency issues is introduced by Perron and Rodriguez (2003), which extends the Generalized Least Squares detrending approach due to Elliot et al. (1996). An empirical application is provided. / Hoy en día es una práctica estándar de trabajo empírico la aplicación de diferentes estadísticos de contraste de raíz unitaria. A pesar de ser un aspecto práctico, estos estadísticos poseen distribuciones complejas y no estándar que dependen de funcionales de ciertos procesos estocásticos y sus derivaciones representan una barrera incluso para varios econometristas teóricos. Estas derivaciones están basadas en herramientas estadísticas fundamentales y rigurosas que no son (muy) bien conocidas por econometristas estándar. El presente artículo completa esta brecha al explicar en una forma simple una de estas herramientas fundamentales la cual es el Teorema del Límite Central Funcional. Por lo tanto, este documento analiza los fundamentos y la aplicabilidad de dos versiones del Teorema del Límite Central Funcional dentro del marco de una raíz unitaria con un quiebre estructural. La atención inicial se centra en la estructura probabilística de las series de tiempo propuesta por Phillips (1987a), la cual es aplicada por Perron (1989) para estudiar los efectos de un quiebre estructural (asumido) exógeno sobre la potencia de las pruebas Dickey-Fuller aumentadas y por Zivot y Andrews (1992) para criticar el supuesto de exogeneidad y proponer un método para estimar un punto de quiebre endógeno. Un método sistemático para tratar con aspectos de eficiencia es introducido por Perron y Rodríguez (2003), el cual extiende el enfoque de Mínimos Cuadrados Generalizados para eliminar los componentes determinísticos de Elliot et al. (1996). Se presenta además una aplicación empírica.

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>

In this paper, we study mathematical properties of a generalized bivariate Ornstein-Uhlenbeck model for financial assets. Originally introduced by Lo and Wang, this model possesses a stochastic drift term which influences the statistical properties of the asset in the real (observable) world. Furthermore, we generali- ze the model with respect to a time-dependent (but still non-random) volatility function. Although it is well-known, that drift terms - under weak regularity conditions - do not affect the behaviour of the asset in the risk-neutral world and consequently the Black-Scholes option pricing formula holds true, it makes sense to point out that these regularity conditions are fulfilled in the present model and that option pricing can be treated in analogy to the Black-Scholes case.

In the Thesis the problem of estimating an unknown parameter in a stochastic dif- ferential equation is studied. Linear equations with Volterra process as the source of noise are considered. Firstly, the properties of Volterra processes and the properties of stochastic integral with respect to a Volterra process are presented. Secondly, the prop- erties of the solution to the equation under consideration are discussed. This includes the existence of the strictly stationary solution, the properties of such solution and ergodic results. These results are then generalized to equations with a mixed noise. Ergodic results are used to derive strongly consistent estimators of the unknown parameter. 1

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Prévoir et estimer le risque de canicule est un enjeu politique majeur. Évaluer la probabilité d'apparition des canicules et leurs sévérités serait possible en connaissant la température en temps continu. Cependant, les extrêmes journaliers (maxima et minima) sont parfois les seules données disponibles. Pour modéliser la dynamique des températures, il est courant d'utiliser un processus d'Ornstein-Uhlenbeck. Une estimation des paramètres de ce processus n'utilisant que les suprema journaliers observés est proposée. Cette nouvelle approche se base sur une minimisation des moindres carrés faisant intervenir la fonction de répartition du supremum. Les mesures de risque liées aux canicules sont ensuite obtenues numériquement. Pour exprimer explicitement ces mesures de risque, il peut être utile d'avoir la loi jointe du processus d'Ornstein-Uhlenbeck et de son supremum. L'étude se limite tout d'abord à la fonction de répartition / densité jointe du point final du processus et de son supremum. Cette probabilité admet une densité, solution de l'équation de Fokker-Planck, obtenue explicitement et utilisant les fonctions spéciales paraboliques cylindriques. La preuve de l'expression de la densité repose sur une décomposition sur une base hilbertienne de l'espace via une méthode spectrale. On étudie également le processus d'Ornstein-Uhlenbeck oscillant, dont le paramètre de drift est constant par morceaux selon le signe du processus. La transformée de Laplace du temps d'atteinte de ce processus est déterminée et la probabilité que le processus soit positif en un temps donné est calculée. / Forecasting and assessing the risk of heat waves is a crucial public policy stake. Evaluate the probability of heat waves and their severity can be possible by knowing the temperature in continuous time. However, daily extremes (maxima and minima) might be the only available data. The Ornstein-Uhlenbeck process is commonly used to model temperature dynamic. An estimation of the process parameters using only daily observed suprema of temperatures is proposed here. This new approach is based on a least square minimization using the cumulative distribution function of the supremum. Risk measures related to heat waves are then obtained numerically. In order to calculate explicitly those risk measures, it can be useful to have the joint law of the Ornstein-Uhlenbeck process and its supremum. The study is _rst limited to the joint density / distribution of the endpoint and supremum of the Ornstein-Uhlenbeck process. This probability admits a density, solution of the Fokker-Planck equation and explicitly obtained as an expansion involving parabolic cylinder functions. The proof of the density expression relies on a decomposition on a Hilbert basis of the space via a spectral method. We also study the oscillating Ornstein-Uhlenbeck process, which drift parameter is piecewise constant depending on the sign of the process. The Laplace transform of this process hitting time is determined and we also calculate the probability for the process to be positive on a fixed time.

Title: Stochastic Models in Financial Mathematics Author: Bc. Oliver Waczulík Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Jan Hurt, CSc., Department of Probability and Mathe- matical Statistics Abstract: This thesis looks into the problems of ordinary stochastic models used in financial mathematics, which are often influenced by unrealistic assumptions of Brownian motion. The thesis deals with and suggests more sophisticated alternatives to Brownian motion models. By applying the fractional Brownian motion we derive a modification of the Black-Scholes pricing formula for a mixed fractional Bro- wnian motion. We use Lévy processes to introduce subordinated stable process of Ornstein-Uhlenbeck type serving for modeling interest rates. We present the calibration procedures for these models along with a simulation study for estima- tion of Hurst parameter. To illustrate the practical use of the models introduced in the paper we have used real financial data and custom procedures program- med in the system Wolfram Mathematica. We have achieved almost 90% decline in the value of Kolmogorov-Smirnov statistics by the application of subordinated stable process of Ornstein-Uhlenbeck type for the historical values of the monthly PRIBOR (Prague Interbank Offered Rate) rates in...