Stochastic Evolution Equations Petr Čoupek Doctoral Thesis Abstract Linear stochastic evolution equations with additive regular Volterra noise are studied in the thesis. Regular Volterra processes need not be Gaussian, Markov or semimartingales, but they admit a certain covariance structure instead. Particular examples cover the fractional Brownian motion of H > 1/2 and, in the non-Gaussian case, the Rosenblatt process. The solution is considered in the mild form, which is given by the variation of constants formula, and takes values either in a separable Hilbert space or the space Lp(D, µ) for large p. In the Hilbert-space setting, existence, space-time regularity and large-time behaviour of the solutions are studied. In the Lp setting, existence and regularity is studied, and in concrete cases of stochastic partial differential equations, the solution is shown to be a space-time continuous random field.

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

We partially solve the adaptive ergodic stochastic optimal control problem where the driving process is a fractional Brownian motion with Hurst parameter H > 1/2. A formula is provided for an optimal feedback control given a strongly consistent estimator of the parameters of the controlled system is avail- able. There are some special conditions imposed on the estimator which means the results are not completely general. They apply, for example, in the case where the estimator is independent of the driving fractional Brownian motion. In the course of the thesis, construction of stochastic integrals of suitable determinis- tic functions with respect to fractional Brownian motion with Hurst parameter H > 1/2 over the unbounded positive real half-line is presented as well. 1

We approximate the solution of a quasilinear stochastic partial differential equa- tion driven by fractional Brownian motion B_H(t); H in (0,1), which was calculated via fractional White Noise calculus, see [5].

In the Thesis we study the problem of linear filtration of Gaussian signals in finite-dimensional space. We use the Kalman-type equations for the filter to show that the filter depends continuously on the signal. Secondly, we show the same continuity property for the covariance of the error and verify existence and uniqueness of a solution to an integral equation that is satisfied by the filter even under more general assumptions. We present several examples of application of the continuity property that are based on the theory of stochastic differential equations driven by fractional Brownian motion. 1

>Magister Scientiae - MSc / Warrant pricing has become very crucial in the present market scenario. See, for example, M. Hanke and K. Potzelberger, Consistent pricing of warrants and traded options, Review Financial Economics 11(1) (2002) 63-77 where the authors indicate that warrants issuance affects the stock price process of the issuing company. This change in the stock price process leads to subsequent changes in the prices of options written on the issuing company's stocks. Another notable work is W.G. Zhang, W.L. Xiao and C.X. He, Equity warrant pricing model under Fractional Brownian motion and an empirical study, Expert System with Applications 36(2) (2009) 3056-3065 where the authors construct equity warrants pricing model under Fractional Brownian motion and deduce the European options pricing formula with a simple method. We study this paper in details in this mini-thesis. We also study some of the mathematical models on warrant pricing using the Black-Scholes framework. The relationship between the price of the warrants and the price of the call accounts for the dilution effect is also studied mathematically. Finally we do some numerical simulations to derive the value of warrants.

In this thesis, we study various notions of variation of certain stochastic processes, namely $p$-variation, pathwise $p$-th variation along sequence of partitions and $p$-th variation along sequence of partitions. We study these concepts for fractional Brownian motions and Rosenblatt processes. A fractional Brownian motion is a Gaussian process and it has been intensively developed and studied over the last two decades because of its importance in modeling various phenomena. On the other hand, a Rosenblatt process, which is a non- Gaussian process that can be used for modeling non-Gaussian fluctuations, has not been getting as much attention as fractional Brownian motion. For that reason, we concentrate in this thesis on this process and we present some original results that deal with ergodicity, $p$-variation, pathwise $p$-th variation along sequence of partitions and $p$-th variation along sequence of partitions. Boris Kiška

The aim of this thesis is to simulate stochastic models that are driven by a fractional Brownian motion process and to apply these methods to financial applications related to yield rate and asset price modelling. Several rough volatility processes are used to model the asset price and yield dynamics. Firstly fractional processes of Cox-Ingersoll-Ross, CEV and Vasicek types are introduced as models for volatility and yield data. In this framework it holds that the Hurst parameter that determines the covariance structure of the fBM process can be directly estimated from observed data series using a least squares log-periodogram approach. The remaining parameters in the model are estimated using a combination of Maximum Likelihood estimates and expectation estimations. In the modelling and pricing of assets one model that is studied is the fractional Heston model, that is used to model an asset price process using both observed asset and volatility data. Similarly two other similar rough volatility models are also studied, which are constructed so as to have log-Normal returns. These processes which in the thesis are called the exponential models 1 and 2 have rough volatility that are characterized by the CEV and Vasicek processes. Additionally the first order Wiener Chaos Expansion is implemented and explored in two ways. Firstly the Chaos Expansion is applied to a parametric fractional stochastic model which is used to generate a Wick product process, which is found to resemble the underlying process. It is also used to generate an approximate expansion of real yield rate data using a bootstrap sampling approach. / Den här uppsatsen syftar till att simulera stokastiska modeller som drivs av fraktionell Brownsk rörelse och att använda dessa modeller i finansiella tillämpningar relaterade till räntor och finansiella tillgångar. Flera volatilitetsprocesser som är rough används för att modellera ränte- och aktiedynamiken. Först introduceras de fraktionella varianterna av Cox-Ingersoll-Ross, CEV och Vasicek processer, vilka används för att modellera volatilitet och ränteprocesser. Med detta tillvägagångssätt gäller det att Hurstparametern, vilken bestämmer covariansstrukturen för den fraktionella Brownska rörelsen, kan uppskattas direkt från observerad data med en minsta kvadrat log-periodogram-metod. Samtliga andra parametrar i modellen uppskattas med en kombination av Maximum Likelihood och uppskattning av väntevärden. I modelleringen och prissättningen av finansiella tillgångar är en model som studeras den fraktionella Hestonmodellen, som används för att modellera en tillgång baserat på både volatilitets- och aktiedata. Ytterligare två liknande modeller studeras, vilka också har volatilitet som är rough och är konstruerade så att deras avkastning är log-Normal. Dessa processer, vilka i uppsatsen är benämnda som de exponentiella modellerna 1 och 2 har volatilitet som karaktäriseras av CEV- och Vasicekprocesser. Ytterligare är Wiener's Kaosexpansion av första ordningen också implementerad och undersöks från två håll. Först används den på en parameterbestämd fraktionell stokastisk modell, vilken används för att generera en Wickproduktprocess. Expansionen används även med hjälp av en bootstrap-metod för att generera en process från observerad data.

In the Thesis, linear stochastic differential equations in a Hilbert space driven by a cylindrical fractional Brownian motion with the Hurst parameter in the interval H < 1/2 are considered. Under the conditions on the range of the diffusion coefficient, existence of the mild solution is proved together with measurability and continuity. Existence of a limiting distribution is shown for exponentially stable semigroups. The theory is modified for the case of analytical semigroups. In this case, the conditions for the diffusion coefficient are weakened. The scope of the theory is illustrated on the Heath-Jarrow-Morton model, the wave equation, and the heat equation. 1

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...