Es sei ein Heath-Jarrow-Morton Zinsstrukturmodell df(t,T) = alpha(t,T)dt + sigma(t,T)dX_t gegeben, angetrieben von einem mehrdimensionalen Semimartingal X. Das Ziel dieser Arbeit besteht darin, die Existenz endlich dimensionaler Realisierungen für solche Modelle zu untersuchen, wobei wir als treibende Prozesse die Klasse der Grigelionis Prozesse wählen, die insbesondere Levy Prozesse enthält. Zur Bearbeitung der Fragestellung werden zwei veschiedene Ansätze verfolgt. Wir dehnen die Ideen aus der Differenzialgeometrie von Björk und Svensson (2001) auf die vorliegende Situation aus und zeigen, dass das in der zitierten Arbeit bewiesene Kriterium für die Existenz endlich dimensionaler Realisierungen in unserem Fall als notwendiges Kriterium dienlich ist. Dieses Resultat wird auf konkrete Volatilitätsstrukturen angewandt. Im Kontext von sogenannten Benchmark Realisierungen, die eine natürliche Verallgemeinerung von Short Rate Realisierungen darstellen, leiten wir Integro-Differenzialgleichungen her, die für die Untersuchung der Existenz endlich dimensionaler Realisierungen hilfreich sind. Als Verallgemeinerung eines Resultats von Jeffrey (1995) beweisen wir außerdem, dass Zinsstrukturmodelle, die eine generische Benchmark Realisierung besitzen, notwendigerweise eine singuläre Hessesche Matrix haben. Beide Ansätze zeigen, dass neue Phänomene auftreten, sobald der treibende Prozess X Sprünge macht. Es gibt dann auf einmal nur noch sehr wenige Zinsstrukturmodelle, die endlich dimensionale Realisierungen zulassen, was ein beträchtlicher Unterschied zu solchen Modellen ist, die von einer Brownschen Bewegung angetrieben werden. Aus diesem Grund zeigen wir, dass für die in der Literatur oft behandelten Modelle mit deterministischer Richtungsvolatilität eine Folge von endlich dimensionalen Systemen existiert, die gegen das Zinsmodell konvergieren. / Let f(t,T) be a term structure model of Heath-Jarrow-Morton type df(t,T) = alpha(t,T)dt + sigma(t,T)dX_t, driven by a multidimensional semimartingale X. Our objective is to study the existence of finite dimensional realizations for equations of this kind. Choosing the class of Grigelionis processes (including in particular Levy processes) as driving processes, we approach this problem from two different directions. Extending the ideas from differential geometry in Björk and Svensson (2001), we show that the criterion for the existence of finite dimensional realizations, proven in the aforementioned paper, still serves as a necessary condition in our setup. This result is applied to concrete volatility structures. In the context of benchmark realizations, which are a natural generalization of short rate realizations, we derive integro-differential equations, suitable for the analysis of the realization problem. Generalizing Jeffrey (1995), we also prove a result stating that forward rate models, which generically possess a benchmark realization, must have a singular Hessian matrix. Both approaches reveal that, with regard to the results known for driving Wiener processes, new phenomena emerge, as soon as the driving process X has jumps. In particular, the occurrence of jumps severely limits the range of models that admit finite dimensional realizations. For this reason we prove, for the often considered case of deterministic direction volatility structures, the existence of finite dimensional systems converging to the forward rate model.

Two optimisation problems were considered, in which market exposure is indirectly controlled. The first one models the capital of a company and an independent portfolio of new businesses, each one represented by a Cram\'r-Lundberg process. The company can choose the proportion of new business it wants to take on and can alter this proportion over time. Here the objective is to find a strategy that maximises the survival probability. We use a point processes framework to deal with the impact of an adapted strategy in the intensity of the new business. We prove that when Cram\'{e}r-Lundberg processes with exponentially distributed claims, it is optimal to choose a threshold type strategy, where the company switches between owning all new businesses or none depending on the capital level. For this type of processes that change both drift and jump measure when crossing the constant threshold, we solve the one and two-sided exit problems. This optimisation problem is also solved when the capital of the company and the new business are modelled by spectrally positive L\'vy processes of bounded variation. Here the one-sided exit problem is solved and we prove optimality of the same type of threshold strategy for any jump distribution. The second problem is a stochastic variation of the work done by Taylor about underwriting in a competitive market. Taylor maximised discounted future cash flows over a finite time horizon in a discrete time setting when the change of exposure from one period to the next has a multiplicative form involving the company's premium and the market average premium. The control is the company's premium strategy over a the mentioned finite time horizon. Taylor's work opened a rich line of research, and we discuss some of it. In contrast with Taylor's model, we consider the market average premium to be a Markov chain instead of a deterministic vector. This allows to model uncertainty in future conditions of the market. We also consider an infinite time horizon instead of finite. This solves the time dependency in Taylor's optimal strategies that were giving unrealistic results. Our main result is a formula to calculate explicitly the value function of a specific class of pricing strategies. Further we explore concrete examples numerically. We find a mix of optimal strategies where in some examples the company should follow the market while in other cases should go against it.

Aufgrund des großen Verhältnisses von Oberfläche zu Volumen zeigen Nanopartikel interessante, größenabhängige Eigenschaften, die man im ausgedehnten Festkörper nicht beobachtet. Sie sind daher von großem wissenschaftlichem und technologischem Interesse. Die Herstellung kleinster Partikel ist aus diesem Grund überaus wünschenswert. Dieses Ziel kann mit Hilfe von Mikroemulsionen als Templatphasen bei der Herstellung von Nanopartikeln erreicht werden. Mikroemulsionen sind thermodynamisch stabile, transparente und isotrope Mischungen von Wasser und Öl, die durch einen Emulgator stabilisiert sind. Sie können eine Vielzahl verschiedener Mikrostrukturen bilden. Die Kenntnis der einer Mikroemulsion zugrunde liegenden Struktur und Dynamik ist daher von außerordentlicher Bedeutung, um ein gewähltes System potentiell als Templatphase zur Nanopartikelherstellung einsetzen zu können.<br><br> In der vorliegenden Arbeit wurden komplexe Mehrkomponentensysteme auf der Basis einer natürlich vorkommenden Sojabohnenlecithin-Mischung, eines gereinigten Lecithins und eines Sulfobetains als Emulgatoren mit Hilfe der diffusionsgewichteten 1H-NMR-Spektroskopie unter Verwendung gepulster Feldgradienten (PFG) in Abhängigkeit des Zusatzes des Polykations Poly-(diallyl-dimethyl-ammoniumchlorid) (PDADMAC) untersucht. Der zentrale Gegenstand dieser Untersuchungen war die strukturelle und dynamische Charakterisierung der verwendeten Mikroemulsionen hinsichtlich ihrer potentiellen Anwendbarkeit als Templatphasen für die Herstellung möglichst kleiner Nanopartikel.<br><br> Die konzentrations- und zeit-abhängige NMR-Diffusionsmessung stellte sich dabei als hervorragend geeignete und genaue Methode zur Untersuchung der Mikrostruktur und Dynamik in den vorliegenden Systemen heraus. Die beobachtete geschlossene Wasser-in-Öl- (W/O-) Mikrostruktur der Mikroemulsionen zeigt deutlich deren potentielle Anwendbarkeit in der Nanopartikelsynthese. Das Gesamtdiffusionsverhalten des Tensides wird durch variierende Anteile aus der Verschiebung gesamter Aggregate, der Monomerdiffusion im Medium bzw. der medium-vermittelten Oberflächendiffusion bestimmt. Dies resultierte in einigen Fällen in einer anormalen Diffusionscharakteristik. In allen Systemen liegen hydrodynamische und direkte Wechselwirkungen zwischen den Tensidaggregaten vor.<br><br> Der Zusatz von PDADMAC zu den Mikroemulsionen resultiert in einer Stabilisierung der flüssigen Grenzfläche der Tensidaggregate aufgrund der Adsorption des Polykations auf den entgegengesetzt geladenen Tensidfilm und kann potentiell zu Nanopartikeln mit kleineren Dimensionen und schmaleren Größenverteilungen führen. / Owing to their large surface-to-volume ratio nanoparticles show interesting size-dependent properties that are not observable in bulk materials. Thus, they are of great scientific and technological interest. Thereby, the highly desirable preparation of as small particles as possible might be easily achieved using microemulsions as template phases. Microemulsions are thermodynamically stable, transparent and isotropic mixtures of water and oil stabilized by an emulsifying agent. However, microemulsions may form a great variety of different microstructures. Thus, it is of utmost importance to know the underlying microstructure and microdynamics of a chosen microemulsion system in order to use it as a template phase for nanoparticle formation.<br><br> In the present study complex multi-component microemulsion systems based on a naturally occurring soybean lecithin mixture, purified lecithin and sulfobetaine as emulsifiers were investigated by diffusion-weighted pulsed field gradient (PFG) 1H NMR spectroscopy in the presence and absence of the polycation poly-(diallyldimethylammonium chloride) (PDADMAC). The central topic of this study was to structurally and dynamically characterize the present microemulsions with respect to their potential use in nanoparticle formation.<br><br> The concentration- and time-dependent NMR diffusion measurements turned out to be a suitable and accurate tool to investigate the microstructure and microdynamics of the systems under investigation. They reveal closed water-in-oil (W/O) microemulsion microstructures which prove the potential suitability of the respective systems as template phases for the preparation of nano-sized particles. The overall diffusion behavior of surfactants were found to be governed by varying contributions from displacements of entire aggregates, monomer diffusion in the medium and bulk-mediated surface diffusion, respectively. In some cases this led to a marked anomalous diffusion characteristics. In all systems interactions between aggregates are dominated by hydrodynamic and direct forces.<br><br> The addition of PDADMAC to the microemulsion systems results in a stabilization of the liquid interface of surfactant aggregates due to the adsorption of the polycation at the oppositely charged surfactant film and may potentially lead to nanoparticles of smaller dimensions and narrower size distributions.

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本文係採公共政策理論方法，兼顧土地課稅政策目標與政治行政可行性兩方面分析比 較何類土地稅基較適合我國目前政治體制，俾政府所面臨之地價稅基偏低無法達成政 策目標，而地價高漲民眾又可能抗稅；及公告土地現值偏低，漲價歸私，但土地徵收 補償時，民眾又抱怨地價過低影響其權益等之兩難問題能獲得解決，全文共分七章廿 二節，茲扼要說明其內容如下： 第一章 緒論：說明本文之研究動機與目的，研究範圍與限制，研究方法與流程，並 就相關文獻加以探討。 第二章 土地課稅理論與立法沿革：說明土地稅之理論根據，平均地權與漲價歸公， 我國土地稅稅基立法演進，並就土地稅之功能加以分析。 第三章 土地稅稅基問題與影響：說明作為地價稅稅基之公告地價與土地增值稅計徵 標準之公告土地現值所產生的問題及其影響。 第四章 地價稅稅基分析：探討地價稅性質與課稅地價查估方式，並依公共政策理方 法就地價永以為定，每年規定一次與每隔三年規定一次作為地價稅稅基方式加以比較 分析，再以問卷調查社會各界對地價稅稅基之看法，以供分析結果參考。 第五章 土地增值稅稅基分析：說明土地增值稅特性，並依公共政策理論方法就公告 土地現值，實際移轉地價，標準宗地地價三種土地增值稅計徵標準方式加以比較分析 ，再以問卷調查社會各界對土地增值稅計徵標準之看法，以供分析結果參考。 第六章 土地稅稅基改進方向與配合措施：論述土地稅稅基改進方向與掌握課稅地價 之配合措施。 第七章 結論與建議：綜合各章所述作一結論分析，並提出建議。

The objective of the diploma thesis "Changes in Agricultural Enterpriser's Levy Charge" is to describe main legislative changes, which influence levy burden of a real farmer. Every chapter in its first part describes main changes that were made in tax acts. The other part of each chapter analyzes how the levy charge of the farmer is influenced by these changes and if there are some other impacts that affect his levy charge.

From both theoretical and applied perspectives, first passage<br />time problems for random processes are challenging and of great<br />interest. In this thesis, our contribution consists on providing<br />explicit or quasi-explicit solutions for these problems in two<br />different settings.<br /><br />In the first one, we deal with problems related to the<br />distribution of the first passage time (FPT) of a Brownian motion<br />over a continuous curve. We provide several representations for<br />the density of the FPT of a fixed level by an Ornstein-Uhlenbeck<br />process. This problem is known to be closely connected to the one<br />of the FPT of a Brownian motion over the square root boundary.<br />Then, we compute the joint Laplace transform of the $L^1$ and<br />$L^2$ norms of the $3$-dimensional Bessel bridges. This result is<br />used to illustrate a relationship which we establish between the<br />laws of the FPT of a Brownian motion over a twice continuously<br />differentiable curve and the quadratic and linear ones. Finally,<br />we introduce a transformation which maps a continuous function<br />into a family of continuous functions and we establish its<br />analytical and algebraic properties. We deduce a simple and<br />explicit relationship between the densities of the FPT over each<br />element of this family by a selfsimilar diffusion.<br /><br /> In the second setting, we are concerned with the study of<br />exit problems associated to Generalized Ornstein-Uhlenbeck<br />processes. These are constructed from the classical<br />Ornstein-Uhlenbeck process by simply replacing the driving<br />Brownian motion by a Lévy process. They are diffusions with<br />possible jumps. We consider two cases: The spectrally negative<br />case, that is when the process has only downward jumps and the<br />case when the Lévy process is a compound Poisson process with<br />exponentially distributed jumps. We derive an expression, in terms<br />of new special functions, for the joint Laplace transform of the<br />FPT of a fixed level and the primitives of theses processes taken<br />at this stopping time. This result allows to compute the Laplace<br />transform of the price of a European call option on the maximum on<br />the yield in the generalized Vasicek model. Finally, we study the<br />resolvent density of these processes when the Lévy process is<br />$\alpha$-stable ($1 < \alpha \leq 2$). In particular, we<br />construct their $q$-scale function which generalizes the<br />Mittag-Leffler function.

This thesis is concerned with the small-time asymptotics and expansions of call option prices, when the log-return processes of the underlying stock prices follow several Levy-based models. To be specific, we derive the time-to-maturity asymptotic behavior for both at-the-money (ATM), out-of-the-money (OTM) and in-the-money (ITM) call-option prices under several jump-diffusion models and stochastic volatility models with Levy jumps. In the OTM and ITM cases, we consider a general stochastic volatility model with independent Levy jumps, while in the ATM case, we consider the pure-jump CGMY model with or without an independent Brownian component. An accurate modeling of the option market and asset prices requires a mixture of a continuous diffusive component and a jump component. In this thesis, we first model the log-return process of a risk asset with a jump diffusion model by combining a stochastic volatility model with an independent pure-jump Levy process. By assuming smoothness conditions on the Levy density away from the origin and a small-time large deviation principle on the stochastic volatility model, we derive the small-time expansions, of arbitrary polynomial order, in time-t, for the tail distribution of the log-return process, and for the call-option price which is not at-the-money. Moreover, our approach allows for a unified treatment of more general payoff functions. As a consequence of our tail expansions, the polynomial expansion in t of the transition density is also obtained under mild conditions. The asymptotic behavior of the ATM call-option prices is more complicated to obtain, and, in general, is given by fractional powers of t, which depends on different choices of the underlying log-return models. Here, we focus on the CGMY model, one of the most popular tempered stable models used in financial modeling. A novel second-order approximation for ATM option prices under the pure-jump CGMY Levy model is derived, and then extended to a model with an additional independent Brownian component. The third-order asymptotic behavior of the ATM option prices as well as the asymptotic behavior of the corresponding Black-Scholes implied volatilities are also addressed.

In insurance risk theory, the surplus of an insurance company is modelled to monitor and quantify its risks. With the outgo of claims and inflow of premiums, the insurer needs to determine what financial portfolio ensures the soundness of the company’s future while satisfying the shareholders’ interests. It is usually assumed that the net profit condition (i.e. the expectation of the process is positive) is satisfied, which then implies that this process would drift towards infinity. To correct this unrealistic behaviour, the surplus process was modified to include the payout of dividends until the time of ruin. Under this more realistic surplus process, a topic of growing interest is determining which dividend strategy is optimal, where optimality is in the sense of maximizing the expected present value of dividend payments. This problem dates back to the work of Bruno De Finetti (1957) where it was shown that if the surplus process is modelled as a random walk with ± 1 step sizes, the optimal dividend payment strategy is a barrier strategy. Such a strategy pays as dividends any excess of the surplus above some threshold. Since then, other examples where a barrier strategy is optimal include the Brownian motion model (Gerber and Shiu (2004)) and the compound Poisson process model with exponential claims (Gerber and Shiu (2006)). In this thesis, we focus on the optimality of a barrier strategy in the more general Lévy risk models. The risk process will be formulated as a spectrally negative Lévy process, a continuous-time stochastic process with stationary increments which provides an extension of the classical Cramér-Lundberg model. This includes the Brownian and the compound Poisson risk processes as special cases. In this setting, results are expressed in terms of “scale functions”, a family of functions known only through their Laplace transform. In Loeffen (2008), we can find a sufficient condition on the jump distribution of the process for a barrier strategy to be optimal. This condition was then improved upon by Loeffen and Renaud (2010) while considering a more general control problem. The first chapter provides a brief review of theory of spectrally negative Lévy processes and scale functions. In chapter 2, we define the optimal dividends problem and provide existing results in the literature. When the surplus process is given by the Cramér-Lundberg process with a Brownian motion component, we provide a sufficient condition on the parameters of this process for the optimality of a dividend barrier strategy. Chapter 3 focuses on the case when the claims distribution is given by a univariate mixture of Erlang distributions with a common scale parameter. Analytical results for the Value-at-Risk and Tail-Value-at-Risk, and the Euler risk contribution to the Conditional Tail Expectation are provided. Additionally, we give some results for the scale function and the optimal dividends problem. In the final chapter, we propose an expectation maximization (EM) algorithm similar to that in Lee and Lin (2009) for fitting the univariate distribution to data. This algorithm is implemented and numerical results on the goodness of fit to sample data and on the optimal dividends problem are presented.

In this study, general geometric Levy market models are considered. Since these models are, in general, incomplete, that is, all contingent claims cannot be replicated by a self-financing portfolio consisting of investments in a risk-free bond and in the stock, it is suggested that the market should be enlarged by artificial assets based on the power-jump processes of the underlying Levy process. Then it is shown that the enlarged market is complete and the explicit hedging portfolios for claims whose payoff function depends on the prices of the stock and the artificial assets at maturity are derived. Furthermore, the portfolio optimization problem is considered in the enlarged market. The problem consists of choosing an optimal portfolio in such a way that the largest expected utility of the terminal wealth is obtained. It is shown that for particular choices of the equivalent martingale measure in the market, the optimal portfolio only consists of bonds and stocks. This corresponds to completing the market with additional assets in such a way that they are superfluous in the sense that the terminal expected utility is not improved by including these assets in the portfolio.