Dissertations / Theses on the topic 'Discrete time'

In this thesis, nonlinearly perturbed stochastic models in discrete time are considered. We give algorithms for construction of asymptotic expansions with respect to the perturbation parameter for various quantities of interest. In particular, asymptotic expansions are given for solutions of renewal equations, quasi-stationary distributions for semi-Markov processes, and ruin probabilities for risk processes.

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Manuscript. Paper 5: Manuscript. Paper 6: Manuscript.

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
PROGRAMA DE SUPORTE À PÓS-GRADUAÇÃO DE INSTS. DE ENSINO
O trabalho tem como objetivo ser uma introdução ao estudo de mercados financeiros tratados em tempo discreto com horizonte finito, bem como a dinâmica dos ativos financeiros principais. Descrevemos os tipos de ativos negociados em nosso mercado, dando enfoque aos contratos. Elaboraremos a hipótese central do modelo, a ausência de arbitragem e assim mostraremos como poderemos encontrar um preço correto ou então apresentaremos um intervalo de preços para os contratos. Posteriormente, mostraremos resultados gerais relativos à correta precificação de contratos, usando para isso os instrumentos de processos estocásticos e martingais. Apresentaremos alguns exemplos a título de ilustração.
The dissertation aims to be an introduction to the study of financial markets in discrete time with finite horizon, as well as the dynamics of the main financial assets. We describe the types of assets traded in the market, focusing on contracts. We will elaborate the central hypothesis of the model, the absence of arbitrage and thus show how we can find a correct price or, at least, a range of prices of the contracts. Subsequently, we will show general results regarding how to find correct prices for contracts, using the machinery of stochastic processes and martingales.As an illustration, we present some examples.

There are many situations in which one can preview future reference signals, or future disturbances. Optimal Preview Control is concerned with designing controllers which use this preview to improve closed-loop performance. In this thesis a general preview control problem is presented which includes previewable disturbances, dynamic weighting functions, output feedback and nonpreviewable disturbances. It is then shown how a variety of problems may be cast as special cases of this general problem; of particular interest is the robust preview tracking problem and the problem of disturbance rejection with uncertainty in the previewed signal. The general preview problem is solved in both the Fh and Beo settings. The H2 solution is a relatively straightforward extension ofpreviously known results, however, our contribution is to provide a single framework that may be used as a reference work when tackling a variety of preview problems. We also provide some new analysis concerning the maximum possible reduction in closed-loop H2 norm which accrues from the addition of preview action. The solution to the Hoo problem involves a completely new approach to Hoo preview control, in which the structure of the associated Riccati equation is exploited in order to find an efficient algorithm for computing the optimal controller. The problem tackled here is also more generic than those previously appearing in the literature. The above theory finds obvious applications in the design of controllers for autonomous vehicles, however, a particular class of nonlinearities found in typical vehicle models presents additional problems. The final chapters are concerned with a generic framework for implementing vehicle preview controllers, and also a'case study on preview control of a bicycle.

Thesis (M.S. in Operations Research) Naval Postgraduate School, December 1999.
Thesis advisors, Donald P. Gaver, Patricia A. Jacobs. Includes bibliographical references (p. 91). Also available online.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1999.
Includes bibliographical references (leaves 117-118).
Worldwide, the continuously growing air traffic induces a need for new ATM concepts to be defined. One possibility is using a more decentralized system predicated mainly around free routings (Free Flight), for a more flexible management of airspace. The present study first highlights the discrepancies and inefficiencies of the current best flightplan optimizing software that use the Cost Index concept before departure. It then investigates techniques to perform enhanced flight-plan optimizations en-route, with algorithms that are less complex than using the Cost Index. The long-haul flight leg that is considered through the simulations is London (UK) - Boston (MA, USA), flown on a constant flight level. This study shows that running another optimization at the Top of Climb point reduces the average delay at destination from 6.9 minutes to 5.0 minutes. Then, the more futuristic method of considering discrete flight-plan optimizations, while en-route using updated weather forecasts, provides results that are more interesting. If the weather forecasts and the optimizations are done simultaneously every 3-hour or 1.5-hour, the average delay respectively becomes 2.6 minutes or 2.0 minutes. The second part of this work investigates ways of performing a Linear Program to fly a route close to a 4D-trajectory. This study provides ways of determining the exact weight values for the different state variables used in the cost function to minimize.
by Francois Le Sellier.
S.M.

As Cyber Physical Systems (CPS) are getting more complex and safety critical, Model Based Design (MBD), which consists of building formal models of a system in order to be used in verification and correct-by-construction code generation, is becoming a promising methodology for the development of the embedded software of such systems. This design paradigm significantly reduces the development cost and time while guaranteeing better robustness, capability and correctness with respect to the original specifications, when compared with the traditional ad-hoc design methods. SIMULINK has been the most popular tool for embedded control design in research as well as in industry, for the last decades. As SIMULINK does not have formal semantics, the application of the model based design methodology and tools to its models is very limited. In this thesis, we present a semantic translator that transform discrete time SIMULINK models into SIGNAL programs. The choice of SIGNAL is motivated by its polychronous formalism that enhances synchronous programming with asynchronous concurrency, as well as, by the ability of its compiler of generating deterministic multi thread code. Our translation involves three major steps: clock inference, type inference and hierarchical top-down translation. We validate the semantic preservation of our prototype tool by testing it on different SIMULINK models.
Master of Science

This thesis is a collection of three essays on financial econometrics with a common background in ultra-high frequency modeling of market activity. In the first essay, we propose an accurate and fast-to-estimate forecasting model for discrete valued time series with long memory and seasonality.1 The modelling is achieved with an autoregressive conditional Poisson process that features seasonality and heterogeneous autoregressive components (whence the acronym SHARP: Seasonal Heterogeneous AutoRegressive Poisson). Motivated by the prominent role of the bid-ask spread as a transaction cost for trading, we apply the SHARP model to forecast the bid-ask spreads of a large sample of NYSE equity stocks. [...]

In this study, we present the implementation and experimental analysis of a modeling approach for analyzing tactical level land combat to generate information for weapon and ammunition planning. The discrete-time stochastic model (DSM), which can handle small and moderately large force levels, is based on single shot kill probabilities. Forces are assumed to be heterogeneous on both sides, and both directed and area fire types are modeled by means of combinatorial analysis. DSM considers overkills and can handle noncombat loss and engagement processes, discrete reinforcements, force combinations and divisions. In addition to experimenting with DSM, we estimate attrition rate coefficients used in Lanchester combat models, such that the two models will yield similar figures for force levels throughout the combat.

Un ordinateur quantique (OQ), i.e. utilisant les ressources de la physique Q, superposition et intrication, pourrait fournir un gain exponentiel de temps de calcul. Une simulation utilisant ces ressources est appelée simulation Q (SQ). L’avantage des SQs sur les simulations classiques est bien établi au niveau théorique, i.e. software. Leur avantage pratique requiert un hardware Q. L’OQ, sous-entendu universel (cf. plus bas), n’a pas encore vu le jour, mais les efforts en ce sens sont croissants et variés. Aussi la SQ a-t-elle déjà été illustrée par de nombreuses expériences de principe, grâce à des calculateurs ou simulateurs Qs de taille réduite. Les marches Qs (MQs) sont des schémas de SQ particulièrement étudiés, étant des briques élémentaires pour concevoir n’importe quel algorithme Q, i.e. pour le calcul Q universel. La présente thèse est un pas de plus vers une simulation des théories Qs des champs basée sur les MQs à temps discret (MQTD). En effet, il est montré, dans certains cas, comment les MQTD peuvent simuler, au continu, l'action d'un champ de jauge Yang-Mills sur de la matière fermionique, et la rétroaction de cette-dernière sur la dynamique du champ de jauge. Les schémas proposés préservent l’invariance de jauge au niveau de la grille d’espace-temps, i.e. pas seulement au continu. Il est proposé (i) des équations de Maxwell sur grille, compatibles avec la conservation du courant sur la grille, et (ii) une courbure non-abélienne définie sur la grille. De plus, il est montré comment cette matière fermionique à base de MQTD peut être couplée à des champs gravitationnels relativistes du continu, i.e. des espaces-temps courbes, en dimension 1+2
A quantum (Q) computer (QC), i.e. utilizing the resources of Q physics, superposition of states and entanglement, could fournish an exponential gain in computing time. A simulation using such resources is called a Q simulation (QS). The advantage of QSs over classical ones is well established at the theoretical, i.e. software level. Their practical benefit requires their implementation on a Q hardware. The QC, i.e. the universal one (see below), has not seen the light of day yet, but the efforts in this direction are both growing and diverse. Also, QS has already been illustrated by numerous experimental proofs of principle, thanks too small-size and specific-task Q computers or simulators. Q walks (QWs) are particularly-studied QS schemes, being elementary bricks to conceive any Q algorithm, i.e. to achieve so-called universal Q computation. The present thesis is a step more towards a simulation of Q field theories based on discrete-time QWs (DTQWs). Indeed, it is shown, in certain cases, how DTQWs can simulate, in the continuum, the action of Yang-Mills gauge fields on fermionic matter, and the retroaction of the latter on the gauge-field dynamics. The suggested schemes preserve gauge invariance on the spacetime lattice, i.e. not only in the continuum. In the (1+2)D Abelian case, consistent lattice equivalents to both Maxwell’s equations and the current conservation are suggested. In the (1+1)D non-Abelian case, a lattice version of the non-Abelian field strength is suggested. Moreover, it is shown how this fermionic matter based on DTQWs can be coupled to relativistic gravitational fields of the continuum, i.e. to curved spacetimes, in several spatial dimensions

Lattice Boltzmann methods are a fully discrete model and numerical method for simulating fluid dynamics, historically they have been developed as a continuation of lattice gas systems. Another route to a lattice Boltzmann system is a discrete approximation to the Boltzmann equation. An analysis of lattice Boltzmann systems is usually performed from one of these directions. In this thesis the lattice Boltzmann method is presented ab initio as a fully discrete system in its own right. Using the Invariant Manifold hypothesis the microscopic and macroscopic fluid dynamics arising from such a model are found. In particular this analysis represents a validation for lattice Boltzmann methods far from equilibrium. Far from equilibrium, at high Reynolds or Mach numbers, lattice Boltzmann methods can exhibit stability problems. In this work a conditional stability theorem for lattice Boltzmann methods is established. Furthermore several practical numerical techniques for stabilizing lattice Boltzmann schemes are tested.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2004.
Includes bibliographical references (p. 65-66).
In this thesis, we derive an exact deterministic nonlinear observer to compute the continuous-time states of inertial navigation system based on partial discrete measurements, the so-called strapdown problem. Nonlinear contraction theory is used as the main analysis tool. The hierarchical structure of the system physics is sytematically exploited and the use of nonlinear measurements, such as distances to time-varying reference points, is discussed. Effects of bounded errors on model and measurements are quantified, and can be used for active measurement selection. Work on vehicle state computation is carried out by using a similar observer design method. Finally, the approach is used to compute the head orientation of a simulated planar hopping robot, where the information provided by the observer is used for head stabilization and obstacle jump.
by Yong Zhao.
S.M.

Wigner distribution is considered to be one of the most powerful tools for time-frequency analysis of rumvstationary signals. Wigner distribution is a bilinear signal transformation which provides two dimensional time-frequency characterization of one dimensional signals. Although much work has been done recently in signal analysis and applications using Wigner distribution, not many synthesis methods for Wigner distribution have been reported in the literature.

This thesis is concerned with signal synthesis from discrete-time Wigner distribution and from discrete-time pseudo-Wigner distribution and their applications in noise filtering and signal separation. Various algorithms are developed to reconstruct signals from the modified or specified Wigner distribution and pseudo-Wigner distribution which generally do not have a valid Wigner distributions or valid pseudo-Wigner distribution structures. These algorithms are successfully applied to the noise filtering and signal separation problems.

Master of Science

Feedback is used to control systems whose open-loop behaviour is uncertain. Over the last twenty years a mature theory of robust control has been developed for linear multivariable systems in continuous time. But most practical control systems have constraints such as saturation limits on the actuators, which render the closed-loop nonlinear. Most of the modern controllers are also implemented digitally using computers.The study of this research is divided in two directions: the stability analysis of discrete-time Lur’e systems and the synthesis of static discrete-time anti-windup schemes. With respect to stability analysis, the main contributions of this thesis are the derivations of new LMI-based stability criteria for the discrete-time Lur’e systems with monotonic, slope-restricted nonlinearities via the Lyapunov method. The criteria provide convex stability conditions via LMIs, which can be efficiently computed via convex optimization methods. They are also extended to the general case that includes the non-diagonal MIMO nonlinearities. The importance of extending them to the general case is that it can eventually be applied to the stability analysis of several optimization-based controllers such as an input-constrainedmodel predictive control (MPC), which is inherently discrete. With respect to synthesis, the contribution is the convex formulation of a static discrete-time anti-windup scheme via one of the Jury-Lee criteria (a discrete-time counterpart of Popov criterion), which was previously conjectured to be unachievable. The result is also in the form of LMI, and is extended to several existing static anti-windup schemes with open-loop stable plants.

>Magister Scientiae - MSc
This dissertation studies the computation methods of pricing of Asian options. Asian options are options in which the underlying variable is the average price over a period of time. Because of this, Asian options have a lower volatility and this render them cheaper relative to their European counterparts. Asian options belong to the so-called path-dependent derivatives; they are among the most difficult to price and hedge both analytically and numerically. In practice, it is only discrete Asian options that are traded, however continuous Asian options are used for studying purposes. Several approaches have been proposed in the literature, including Monte Carlo simulations, tree-based methods, Taylor’s expansion, partial differential equations, and analytical ap- proximations among others. When using partial differential equations for pricing of continuous time Asian options, the high dimensionality is problematic. In this dissertation we focus on the discrete time methods. We start off by explaining the binomial tree method, and our last chapter presents the very exciting and relatively simple method of Tsao and Huang, using Taylor approximations. The main papers that are used in this dissertation are articles by Jan Vecer (2001); LCG Rogers (1995); Eric Benhamou (2001); Gianluca Fusai (2007); Kamizono, Kariya and Nakatsuma (2006) and Tsao and Huang (2007). The author has provided computations, including graphs and tables dispersed over the different chapters, to demonstrate the utility of the methods. We observe various parameters of influence such as correlation, volatility, strike, etc. A further contribution by the author of this dissertation is, in particular, in Chapter 5, in the presentation of the work of Tsao et al. Here we have provided slightly more detailed explanations and again some further computational tables.

In this thesis we propose four novel continuous-time return smoothing models. Borrowing from the private commercial real estate literature, we start with a benchmark model for return smoothing in discrete time. The benchmark model is translated into continuous time, taking us from an autoregressive moving average (ARMA) specification for the smoothed holding period return to an Ornstein-Uhlenbeck (OU) specification for the instantaneous continuously compounding rate of return on the smoothed asset price. The model is then extended to allow for the possibility of predictability in the underlying “true” return. In a second line of investigation, we propose an alternative continuous-time return smoothing model in which we keep the OU smoothing mechanism, but replace the instantaneous smoothed return with the smoothed detrended log price. This model leads to unrealistic autocorrelations in the smoothed return, and we address this with an extension that introduces a higher-order smoothing equation described by a continuous-time autoregressive moving average (CARMA) process—the continuous-time analogue of the ARMA process. We show that each of our four models belongs to a general framework for linear return smoothing in continuous time in which a CARMA process governing the underlying “true” asset price is overlaid with a CARMA-type smoothing equation that summarises the market mechanism whereby the “true” price is transformed into a market-observed, smoothed price. In each model, as with the general framework, the noise in the “true” price is represented by a Lévy process, allowing for non-normality and sample path discontinuities. To quantify the effect the smoothing models have on holding period returns, we develop a common set of smoothing metrics. These metrics are then computed for each of the models (including the discrete-time benchmark model), and form a basis on which the models can be compared. We also rely on the autocorrelation function in appraising the impact of return smoothing. We comment on the ability of the continuous-time smoothing models to reproduce stylised statistical properties commonly associated with smoothed returns, such as a reduction in return variance and an increase in return autocorrelation. Additionally, we develop the theory needed to operationalise the smoothing metrics for the smoothing framework in general.

Statistical inference for discrete-valued time series has not been developed as systematically as traditional methods for time series generated by continuous random variables. This Ph.D. dissertation deals with time series models for discrete-valued processes. In particular, Chapter 2 is devoted to a comprehensive overview of the literature about observation-driven models for discrete-valued time series. Derivation of stochastic properties for these models is presented. For the inference, general properties of the quasi maximum likelihood estimator (QMLE) are discussed, followed by an illustrative application. In Chapter 3, a general class of observation-driven time series models for discrete-valued processes is introduced. Stationarity and ergodicity are derived under easy-to-check conditions, which can be directly applied to all the models encompassed in the framework. Consistency and asymptotic normality of the QMLE are established, with the focus on the exponential family. Finite sample properties of the estimators are investigated through a Monte Carlo study and illustrative examples are provided. The framework introduced in the paper provides a self-contained background that relates different models developed in the literature as well as novel specifications and makes them fully applicable in practice. Discrete responses are commonly encountered in real applications and are strongly connected to network data. The specification of suitable network autoregressive models for count time series is an important aspect which is not covered by the existing literature. In Chapter 4, we consider network autoregressive models for count data with a known neighborhood structure. The main methodological contribution is the development of conditions that guarantee stability and valid statistical inference. We consider both cases of fixed and increasing network dimension and we show that quasi-likelihood inference provides consistent and asymptotically normally distributed estimators. The work is complemented by simulation results and a data example.

L'obbiettivo della presente Tesi é di considerare la specificazione di mod­elli di pricing in tempo discreto (in generale, incompleti) con variabili latenti, al fine di sfruttare i vantaggi derivanti da tale contesto a tempo discreto e al fine di fornire una descrizione completa degli aspetti storici e neutrali al rischio dei prezzi dei titoli. Negli ultimi anni osserviamo un importante sviluppo di modelli di pric­ing in tempo discreto, dove la modellizzazione secondo il principio dello Stochastic Discount Factor (SDF) e la caratterizzazione della distribuzione condizionale delle variabili di stato tramite la trasformata di Laplace sem­brano fornire risultati promettenti. Più precisamente, la caratterizzazione generale di modelli di pricing in tempo discreto, usando questo tipo di approccio, e dove é assunta una speci­ficazione Compound Autoregressive (CAR ovvero affine) per le variabili di stato [vedi Darolles, Gourieroux, Jasiak (2002)], é stata proposta da Gourier­oux e Monfort (2003) e Gourieroux, Monfort e Polimenis (2002, 2003); in questi articoli viene presentata la metodologia generale di pricing e vengono specificati modelli per la Struttura a Termine e per il Rischio di Credito. Il tempo discreto é un contesto naturale per sviluppare modelli di valoriz­zazione volti a future implementazioni econometriche; infatti, i dati storici sono campionati con frequenza discreta, le transazioni finanziarie sono tipi­camente registrate a intervalli temporali discreti, la stima di parametri e i test statistici implicano dati a tempo discreto e le previsione sono fatte a orizzonti discreti. Un secondo e importante vantaggio che si ha nel lavorare in tempo dis­creto emerge quando consideriamo la classe di processi affini per applicazioni finanziarie. La classe di processi affini in tempo discreto (processi CAR) [pro­posti, come indicato sopra, da Darolles, Gourieroux, Jasiak (2002)] é molto più ampia della classe equivalente in tempo continuo proposta da Duffie, Fil­ipovic and Schachermayer (2003) : tutti i processi affini in tempo continuo campionati a istanti temporali discreti sono CAR, mentre esiste un ampio numero di processi CAR senza un processo equivalente in tempo continuo. Questa é una conseguenza del problema di embedding che caratterizza la classe affine in tempo continuo : tali processi devono essere infinitamente decomponibili, mentre tale condizione non é necessaria in tempo discreto [vedi Darolles, Gourieroux and Jasiak (2002) and Gourieroux, Monfort and Polimenis (2002)]. Nella Tesi sfrutteremo il contesto a tempo discreto anche per introdurre processi Non-Gaussiani e Non-Markoviani come le Misture di Processi Con­dizionatamente Gaussiani. Per quanto riguarda l'utilizzo della trasformata di Laplace condizionale per descrivere la distribuzione storica e neutrale al rischio delle variabili di stato, si osservi come in molte applicazione economico-finanziarie siamo por­tati in modo naturale a dover calcolare la trasformata di Laplace di tali variabili di stato. Alcuni esempi possibili sono i seguenti : (a) optimal port­folio problems (CARA utility functions, Markowitz), (b) asset pricing by thè certainty equivalence principle (CARA utility functions), (c) discrete time derivative pricing and term strueture models with exponential-affine SDFs, (d) panel duration models, (e) extreme risk [see Darolles, Gourieroux and Jasiak (2002) for details]. Vedremo, inoltre, che la trasformata di Laplace é uno strumento molto utile per caratterizzare anche la distribuzione storica e neutrale al rischio di Misture di Processi Condizionatamente Gaussiani. Per finire, la necessità di prendere in considerazione le fonti di rischio ril­evanti nell'influenzare il titolo da valorizzare, porta a considerare lo Stochas­tic Discount Factor (SDF) come strumento per caratterizzare la procedura di pricing : lo SDF é una variabile casuale (chiamata anche Pricing Kernel o State Price Deflator) che sintetizza sia l'attualizzazione temporale che la correzione per il rischio, e che porta a specificare, conseguentemente, una procedura di valorizzazione che fornisce una modellizzazione completa degli aspetti storici e neutrali al rischio. Il tempo discreto implica in generale un contesto a mercato incompleto e una molteplicità di formule di pricing; il problema della molteplicità viene ridotto imponendo una forma particolare allo SDF; il Pricing Kernel viene specificato, infatti, secondo una forma esponenziale-affine che si é dimostrata utile in molte circostanze e che troviamo sovente in letteratura [vedi Lu­cas (1978), Gerber e Shiu (1994), Stutzer (1995, 1996), Buchen e Kelly (1996), Buhlmann et al. (1997, 1998), Polimenis (2001), Gourieroux e Mon­fort (2002)]. Inoltre, uno SDF con una forma esponenziale-affine presenta proprietà tecniche interessanti : tale approccio infatti, che é basato sulla trasformata di Esscher in un contesto dinamico a tempo discreto, perme­tte di selezionare una misura martingale (di pricing) equivalente che riflette, nella formula di pricing, le diverse fonti di rischio da valorizzare. Ora, il contesto a tempo discreto, assieme ai principi di modellizzazione dello SDF esponenziale-affine e della transformata di Laplace, costituiscono gli strumenti usati nei tre capitoli fondamentali della Tesi. La Tesi analizza il ruolo che l'introduzione di variabili latenti può avere, in questa classe di modelli di pricing a tempo discreto, nello specificare metodologie di valorizzazione complete e coerenti rispetto alle indicazioni empiriche. Nei Capitoli 2 e 3 l'obbiettivo, infatti, é quello di specificare metodologie per la valorizzazione di prodotti derivati in grado di prendere in considerazione i tipici fenomeni di skewness e excess kurtosis che osserviamo nella distribuzione dei rendimenti di titoli azionari, e di riuscire quindi a repli­care le volatilità implicite di Black e Scholes (BS) e superfici di volatilità im­plicita con forme di smile a volatility skew coerenti con l'evidenza empirica1. Qui, le variabili latenti introducono cambiamenti di regime nella dinamica del titolo sottostante (cambiamenti, per esempio, tra un mercato a regime di alta e bassa volatilità) ovvero, introducono nella distribuzione storica del rendimento rischioso fenomeni come medie e varianze stocastiche. Nel Capi­tolo 4, vengono proposti modelli affini bifattoriali per la struttura a termine dei tassi di interesse (in tempo discreto) con variabili latenti; l'obbiettivo é quello di ottenere famiglie di possibili strutture a termine con forme più prossime (rispetto ai modelli unifattoriali in tempo discreto e continuo) a quelle osservate. In questo caso, le variabili latenti introducono parametri stocastici (continui e discreti) nella dinamica del fattore (tasso di interesse a breve scadenza) responsabile della forma della struttura a termine nei modelli unifattoriale. In altre parole, si vogliono definire procedure di pricing capaci di pren­dere in considerazione, in modo coerente e utile, le fonti di rischio descritte dai cambiamenti di regime e dai parametri stocastici; vogliamo specificare metodologie di valorizzazione non basate su ipotesi arbitrarie (spesso us­ate in letterature) come, per esempio, la neutralità al rischio degli investi­tori o la natura idiosincratica del rischio, e vogliamo derivare formule di pricing che hanno una forma analitica o che sono facili da implementare. L'organizzazione della Tesi é indicata nel prossimo paragrafo. 1 Questi due capitoli corrispondono a due articoli scritti con Henri Bertholon e Alain Monfort. Sintesi dei capitoli Nel CAPITOLO 1 viene inizialmente presentato il principio di modelliz­zazione dello SDF, e come é legato alla Law of One Price e al principio di Absence of Arbitrage Opportunity; successivamente, vengono descritti gli strumenti base che caratterizzano i modelli di pricing a tempo discreto sviluppati nella tesi : lo SDF esponenziale-affine, e la rappresentazione della distribuzione condizionale delle variabili di stato tramite la trasformata di Laplace considerando come esempio i processi CAR. Nel CAPITOLO 2 proponiamo una nuova procedura di valorizzazione di opzioni Europee che porta ad una generalizzazione della formula di Black e Scholes [utile, quindi, dal punto di vista delle istituzioni finanziarie]; in particolare, ci focalizziamo sulle due fonti fondamentali di cattiva specifi­cazione dell'approccio BS, ovvero l'assenza di Gaussianità e la dinamica. Gli strumenti utilizzati sono le misture in tempo discreto di processi condizion­atamente gaussiani, cioè processi {yt} tali che yt+1 é gaussiano condizion­atamente ai propri valori passati e al valore presente zt+ì di un white noise non osservabile a valori discreti. Forniamo (in un semplice caso statico) le simulazioni di volatilità implicita di BS e di superfici di volatilità implicita, e osserviamo l'abilità delle procedure di pricing che proponiamo nel repli­care smiles e volatility skews coerenti con l'evidenza empirica. Per quanto riguarda le superfici di volatilità implicita, il modello statico mostra qualche limite che é superato, con una dinamica di tipo Regime-Switching attribuita a zt+i, nel Capitolo 3. Il CAPITOLO 3 presenta una naturale evoluzione del precedente capitolo; infatti, prende in considerazione il caso in cui la variabile latente zt+\ non sia più un white noise ma, tipicamente, una Catena di Markov. Più precisa­mente, presentiamo il modello General Switching Regime per il pricing di derivati, applicato ai casi di opzioni Europee e path dependent. Studiamo inoltre le condizioni sotto le quali c'è una trasmissione di causalità (assenza di causalità istantanea, assenza di causalità, indipendenza), esistente tra la madia e la varianza stocastica, dal mondo storico al mondo neutrale al ris­chio. A questo scopo separiamo la dinamica della media e della varianza (in un caso di Hidden Markov Chain) usando due distinte variabili latenti (zit+i , Z2t+i), dove sia zu+i che Zit+\ possono prendere J possibili valori, e dove la prima variabile latente descrive la dinamica della media mentre la seconda quella della varianza. Lo scopo del CAPITOLO 4 é di introdurre parametri stocastici e cambi­amenti di regime nei modelli affini unifattoriali per la struttura a termine presentati da Gourieroux, Monfort and Polimenis (2002) [GMP (2002)], al fine di estendere la dinamica del tasso a breve termine e di ampliare, con­seguentemente, la ricchezza di curve della struttura a termine che tali modelli sono in grado di riprodurre. Vengono studiati diversi modelli alternativi e vengono presentate le simulazioni sulle possibili struttura a termine che essi sono in grado di replicare; in particolare, le strutture a termine ottenute mostrano forme con gobbe verso l'alto e verso il basso, forme con diversi gradi curvatura e con due mode. Per finire, presentiamo in problema dell' in­dividuazione di mimicking factors [un vettore Rt = (rf;i+2,..., rt,t+n) di tassi di interesse con differenti maturity] per i parametri stocastici e i cambiamenti di regime : questo é un problema interessante, dal punto di vista statistico, data l'osservabilità dei tassi di interesse. The aim of the thesis is to consider, as a new research direction, the specification of discrete time pricing models (in general incomplete) with latent variables, in order to exploit the advantages coming from the discrete time framework and in order to give a complete description of historical and risk-neutral aspects of asset prices. In the last years, we observe an important development of asset pric­ing models in discrete time, where the use of the Stochastic Discount Factor (SDF) modeling principle and the characterization of the state variables con­ditional distributions by means of the Laplace transform seem promising. More precisely, the general discrete time characterization of asset pricing models, using this kind of approach, and where a compound autoregressive (affine or CAR) specification for the state variables is assumed [see Darolles, Gourieroux, Jasiak (2002)], has been proposed by Gourieroux and Monfort (2003) and Gourieroux, Monfort and Polimenis (2002, 2003); in these pa­pers the general pricing methodology and the specifications of Affine Term Structure models, along with the Credit Risk Analysis, are presented. The discrete time is a natural framework to develop pricing models for fu­ture econometric implementations, given that all historical data are sampled discretely, financial transactions are typically recorded at discrete intervals, parameter estimation and hypothesis testing involve discrete data records, and forecasts are produced at discrete horizons. A second important advantage to work in discrete time emerges when we consider the class of affine processes for financial applications. The class of discrete time affine (CAR) processes [proposed, as indicated above, by Darolles, Gourieroux and Jasiak (2002)] is much larger than the equivalent continuous time class proposed by Duffie, Filipovic and Schachermayer (2003) : all continuous time affine processes sampled at discrete points are CAR, while there exists a large number of CAR processes without a continuous time counterpart. This is a consequence of the embedding problem that characterizes the continuous time class : these processes have to be infinitely decomposable, and this decomposition condition is not necessary in discrete time [see Darolles, Gourieroux and Jasiak (2002) and Gourieroux, Monfort and Polimenis (2002) for details]. In this Thesis, we will also exploit the discrete time framework in order to introduce non-Gaussian and non-Markovian processes like, for instance, the Mixtures of Conditionally Normal Processes. With regard to the use of the conditional Laplace transform to describe the historical and risk-neutral (pricing) distribution of the state variables, we observe that in many financial and economic applications we are naturally lead to determine the Laplace transform of the processes of interest; possible examples are the followings : (a) optimal portfolio problems (CARA utility functions, Markowitz), (b) asset pricing by the certainty equivalence princi­ple (CARA utility functions), (c) discrete time derivative pricing and term structure models with exponential-affine SDFs, (d) panel duration models, (e) extreme risk [see Darolles, Gourieroux and Jasiak (2002) for details]. In this Thesis we will see that the Laplace transform is also very convenient for the class of Mixtures of Conditionally Normal Processes. Finally, the need to take into account the relevant sources of risk that influence the asset one wants to price, lead to consider a Stochastic Discount Factor (SDF) approach to characterize the pricing procedure : the SDF is a random variable (called also Pricing Kernel or State Price Deflator) which summarizes both the time discounting and the risk correction, and which specifies, consequently, a pricing procedure that gives a complete modelisa­tion of the historical and risk-neutral (pricing) aspects. Given that discrete time implies in general an incomplete market frame­work and a multiplicity of asset pricing formulas, the multiplicity problem is reduced by imposing a special structure on the SDF; in particular, it is pos­sible to consider for the pricing kernel an exponential-affine function of the state variables which has proved useful in many circumstances and that we find frequently in the literature [see Lucas (1978), Gerber and Shiu (1994), Stutzer (1995, 1996), Buchen and Kelly (1996), Buhlmann et al. (1997, 1998), Polimenis (2001), Gourieroux and Monfort (2002)]. Moreover, a SDF with an exponential-affine form presents interesting technical properties : it is the Esscher transform approach, in a dynamic discrete time framework, which gives the possibility to select an equivalent martingale (pricing) mea­sure that reflects, in the pricing formula, the different sources of risks to be priced. Now, the discrete time framework, along with the exponential-affine SDF modeling principle and the Laplace transform approach, constitute the in­struments used in the three core chapters of the Thesis. The Thesis analyzes the role that the introduction of latent variables could play, in this class of discrete time pricing models, for the specification of com­plete and coherent, with respect to the empirical evidence, pricing methodolo­gies. In Chapters 2 and 3 the purpose is, indeed, to specify derivative pricing methodologies able to take into account time-varying stock returns skewness and excess kurtosis, that is, pricing procedure able to replicate phenomena like implied Black and Scholes volatilities and implied volatility surfaces with smile and volatility skew shapes coherent with empirical studies2. Here, the latent variables are regimes of the underlying risky asset (switches, for in­stance, between a low volatility and a high volatility regime of the market), that is, they introduce phenomena like stochastic means and variances in the historical dynamics of the stock return underlying the derivative product. In Chapter 4, the Thesis proposes discrete time two-factor affine term structure models with latent variables, able to obtain families of possible term struc­tures with shapes closer (with respect to one-factor continuous and discrete time models) to the observed ones. In this case the latent variables intro­duce discrete and continuous stochastic parameters in the dynamics of the factor (the short term interest rate) that explains the term structure of the univariate models. In other words, we want to define pricing procedure able to take into account in a coherent and useful way the sources of risk described by the switching of regimes and by the stochastic parameters; we want to spec­ify pricing procedure not characterized by arbitrary assumptions, frequently used in the literature, like, for instance, the risk-neutrality of the investors or the idiosyncratic nature of the risk. In addition, we want to provide pricing formulas which have an analytical form or which are simple to implement. The organization of the Thesis is detailed below. 2They correspond to two papers written with Henri Bertholon and Alain Monfort. Outline of the chapters In CHAPTER 1 first we present the SDF modeling principle, and its rela­tions with the Law of One Price and the Absence of Arbitrage Opportunity principle, then we consider the basic tools characterizing the discrete time pricing models developed in this Thesis : the exponential-affine SDF, the conditional Laplace transform description of the future uncertainty and the CAR processes. In CHAPTER 2 we propose a new European option pricing procedure which lead to a generalization of the Black and Scholes pricing formula [and, therefore, useful for financial institutions]. We focus on two impor­tant sources of misspecification for the Black-Scholes approach, namely the lack of normality and the dynamics. The basic tools are the mixtures of discrete time conditionally normal processes, that is to say processes {yt} such that yt+i is gaussian conditionally to its past values and the present value zt+i of a discrete value unobservable white noise process. We provide (in a static framework) simulations of implied Black-Scholes volatilities and implied volatilities surfaces, and we observe the ability of the proposed as­set pricing methodology to replicate smiles and volatility skews coherent with empirical results. With regard to implied volatility surfaces, the static model shows some limit which is overcome, with a Regime-Switching dynamics for Zt+i, in Chapter 3. CHAPTER 3 presents a natural evolution of the previous chapter, that is, it considers the case where the latent variable zt+1 is no more a white noise but, typically a Markov chain. More precisely, we present the derivative pricing General Switching Regime model applied to the cases of European and path dependent options. We also study the conditions under which there is a transmission of causality relations (absence of instantaneous casuality, absence of causality, independence), existing between the stochastic mean and variance, from the historical to the risk-neutral world. For this purpose we separate the dynamics of these two moments (in the case of a Hidden Markov Chain) using two distinct latent variables (zit+i , 22t+i)> where both Z\t+\ and Z2t+\ can take J values and where the first latent variable describe the dynamics of the mean while the second one describe the dynamics of the variance. The aim of CHAPTER 4 is to introduce stochastic parameters and switch­ing regimes in the one-factor Affine Term Structure Models proposed by Gourieroux, Monfort and Polimenis (2002) [GMP (2002) hereafter], in order to extend the dynamics of the short rate and to improve, consequently, the richness of shapes of the term structure they are able to replicate. Different models are studied and simulations of the possible term structures we are able to replicate are presented; in particular, the provided term structures show shapes with bumps both upwards and downwards, shapes with different degrees of curvature and with two modes. Finally, we present the problem to find mimicking factors [a vector Rt (rt)t+2,..., ru+n) of interest rates at different maturities] for stochastic parameters and switching regimes : this is an interesting problem, from a statistical point of view, because of the observability of the interest rates.

Some problems of' statistical inference for discrete-valued time series are investigated in this study. New statistical theories and methods are developed which may aid us in gaining more insight into the understanding of discrete-valued time series data. The first part is concerned with the measurement of the serial dependence of binary time series. In early studies the classical autocorrelation function was used, which, however, may not be an effective and informative means of revealing the dependence feature of a binary time series. Recently, the autopersistence function has been proposed as an alternative to the autocorrelation function for binary time series. The theoretical autopersistence functions and their sample analogues, the autopersistence graphs, are studied within a binary autoregressive model. Some properties of the autopcrsistencc functions and the asymptotic properties of the autopersistence graphs are discussed, justifying that the antopersistence graphs can be used to assess the dependence feature. Besides binary time series, intcger-vall1ed time series arc perhaps the most commonly seen discrete-valued time series. A generalization of the Poisson autoregression model for non-negative integer-valued time series is proposed by imposing an additional threshold structure on the latent mean process of the Poisson autoregression. The geometric ergodicity of the threshold Poisson autoregression with perburbations in the latent mean process and the stochastic stability of the threshold Poisson autoregression are obtained. The maximum likelihood estimator for the parameters is discussed and the conditions for its consistency and asymptotic normally are given as well. Furthermore, there is an increasing need for models of integer-valued time series which can accommodate series with negative observations and dependence structure more complicated than that of an autoregression or a moving average. In this regard, an integer-valued autoregressive moving average process induced by the so-called signed thinning operator is proposed. The first-order model is studied in detail. The conditions for the existence of stationary solution and the existence of finite moments are discussed under general assumptions. Under some further assumptions about the signed thinning operators and the distribution of the innovation, a moment-based estimator for the parameters is proposed, whose consistency and asymptotic normality are also proved. The problem of conducting one-step-ahead forecast is also considered based on hidden Markov chain theory. Simulation studies arc conducted to demonstrate the validity of the theories and methods established above. Real data analysis such as the annual counts of major earthquakes data are also presented to show their potential usefulness in applications.
published_or_final_version
Statistics and Actuarial Science
Doctoral
Doctor of Philosophy

This thesis investigates the finite wordlength effects in one-dimensional discrete-time wavelet transform (DTWT). A MATLAB DTWT model is written based on an efficient algorithm suggested by the Multiresolution Analysis (MRA). This model allows users to investigate the overall non-ideal effects of a MRA-based DTWT system by specifying round-off method for computation, the number of quantization bits employed in representing filter coefficient and internal computational results. Further, the possibility of discarding the high frequency portion of a signal in DTWT signal reconstruction phase (Selective Subband Reconstruction) is shown practical with the presence of finite wordlength effects. The feasibility of using either direct structure filter or lattice structure filter to realize a MRA-based DTWT system is also considered in this thesis. Simulation results show that the lattice structure filters have superior magnitude and phase responses in most scenarios under investigation to its direct structure-based counterparts. Moreover, lattice structure filter demands relatively less computation than direct structure filter in the same filter length and DTWT configuration. In conclusion, the lattice structure filter is a better alternative than the direct structure filter for MRA-based DTWT system implementation.

In this thesis we study the asymptotic behaviour of the solution of a discrete time renewal equation depending on a small perturbation parameter. In particular, we construct asymptotic expansions for the solution of the renewal equation and related quantities. The results are applied to studies of quasi-stationary phenomena for regenerative processes and asymptotics of ruin probabilities for a discrete time analogue of the Cramér-Lundberg risk model.

Regarding the relationships among different insurance claims, especially in non-life insurance, the dependence behaviour in various models has been studied extensively. In this thesis, some discrete-time risk models with dependence structures would be investigated. One traditional discrete-time risk model is the time series risk model, in which the dependence would be on two aspects: time correlated claims and dependent business classes. A general vector (multivariate) autoregressive moving average (VARMA) model would be adopted to analyze the ruin probability of a surplus process. An upper bound for the ruin probability is derived for the general order of multivariate time series models in claims. Simulation studies are carried out for model comparison for finite time ruin probabilities. Another class of risk model is the compound binomial risk model, where the dependence structure would be based on the existence of a so-called by-claim in the claim process. The by-claim could be incurred in the same period as the main insurance claim, or it would be incurred in the next period, depending on a certain probability. A randomized dividend payment scheme with some fixed threshold value in surplus level would also be considered in this thesis. A methodology is discovered to obtain the Gerber-Shiu expected penalty function for the extended model. The final model investigated in this thesis is the periodic time series risk model. The periodic structure of the model gives a practical interpretation of the business cycle, in which there are high season and low season for the business. Some lower order periodic time series models are considered for the claim structures.
published_or_final_version
Statistics and Actuarial Science
Doctoral
Doctor of Philosophy

This paper presents a characterization of complete controllability for the class of discrete-time recurrent neural networks. We prove that complete controllability holds if and only if the rank of the control matrix equals the state space dimension. (author's abstract)
Series: Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
Os métodos tradicionais de avaliação de projetos vem sendo questionados por não considerarem possíveis incertezas associadas ao investimento. Neste contexto, a Teoria das Opções Reais busca aplicar o conceito de opções a ativos reais, com a finalidade de agregar o valor da flexibilidade gerencial aos métodos tradicionais de avaliação de investimentos. A avaliação por Opções Reais é considerada complexa devido à difícil modelagem de incertezas e das flexibilidades, além da necessidade de se ter mercados completos. Este estudo busca incorporar a flexibilidade gerencial à avaliação de projetos através do uso de Árvores Binomiais de Decisão, com probabilidades neutras ao risco, para a avaliação por Opções Reais em Tempo Discreto. Utilizamos programação dinâmica para a aplicação desta metodologia, a qual é computacionalmente intensa, porém de solução simples e intuitiva. A aplicação prática foi realizada através da valoração da opção de expandir e da opção de abandonar enfrentada por uma empresa de Tecnologia.
The traditional methods of Valuation are being questioned as they do not consider possible uncertainties related to investment decisions. In this scenario, Real Options Theory applies option`s concept to real assets, aiming to add the value of managerial flexibility to traditional Valuation techniques. The evaluation for Real Options is considered complex due to the difficulty of modeling uncertainties and flexibilities, beyond the need to have complete markets. This work aims to add the managerial flexibility to Valuation by binomial lattice and decision tree techniques, with risk neutral probabilities, in a discrete time approach to evaluation for Real Options. Using dynamic programming to apply this method, which is computationally intense, but simple and intuitive. The practical application consists in valuing an option to expand and to abandon faced by an IT company.

This thesis explores the stochastic features of models of ecological systems in discrete and in continuous time. Our interest lies in models formulated at the microscale, from which a mesoscopic description can be derived. The stochasticity present in the models, constructed in this way, is intrinsic to the systems under consideration and stems from their finite size. We start by exploring a susceptible-infectious-recovered model for epidemic spread on a network. We are interested in the case where the connectivity, or degree, of the individuals is characterised by a very broad, or heterogeneous, distribution, and in the effects of stochasticity on the dynamics, which may depart wildly from that of a homogeneous population. The model at the mesoscale corresponds to a system of stochastic differential equations with a very large number of degrees of freedom which can be reduced to a two-dimensional model in its deterministic limit. We show how this reduction can be carried over to the stochastic case by exploiting a time-scale separation in the deterministic system and carrying out a fast-variable elimination. We use simulations to show that the temporal behaviour of the epidemic obtained from the reduced stochastic model yields reasonably good agreement with the microscopic model under the condition that the maximum allowed degree that individuals can have is not too close to the population size. This is illustrated using time series, phase diagrams and the distribution of epidemic sizes. The general mesoscopic theory used in continuous-time models has only very recently been developed for discrete-time systems in one variable. Here, we explore this one-dimensional theory and find that, in contrast to the continuous-time case, large jumps can occur between successive iterates of the process, and this translates at the mesoscale into the need for specifying `boundary' conditions everywhere outside of the system. We discuss these and how to implement them in the stochastic difference equation in order to obtain results which are consistent with the microscopic model. We then extend the theoretical framework to make it applicable to systems containing an arbitrary number of degrees of freedom. In addition, we extend a number of analytical results from the one-dimensional stochastic difference equation to arbitrary dimension, for the distribution of fluctuations around fixed points, cycles and quasi-periodic attractors of the corresponding deterministic map. We also derive new expressions, describing the autocorrelation functions of the fluctuations, as well as their power spectrum. From the latter, we characterise the appearance of noise-induced oscillations in systems of dimension greater than one, which have been previously observed in continuous-time systems and are known as quasi-cycles. Finally, we explore the ability of intrinsic noise to induce chaotic behaviour in the system for parameter values for which the deterministic map presents a non-chaotic attractor; we find that this is possible for periodic, but not for quasi-periodic, states.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 325-333).
The cost of implementation of discrete-time filters is often strongly dependent on the number of non-zero filter coefficients or the precision with which the coefficients are represented. This thesis addresses the design of sparse and bit-efficient filters under different constraints on filter performance in the context of frequency response approximation, signal estimation, and signal detection. The results have applications in several areas, including the equalization of communication channels, frequency-selective and frequency-shaping filtering, and minimum-variance distortionless-response beamforming. The design problems considered admit efficient and exact solutions in special cases. For the more difficult general case, two approaches are pursued. The first develops low-complexity algorithms that are shown to yield optimal or near-optimal designs in many instances, but without guarantees. The second focuses on optimal algorithms based on the branch-and-bound procedure. The complexity of branch-and-bound is reduced through the use of bounds that are good approximations to the true optimal cost. Several bounding methods are developed, many involving relaxations of the original problem. The approximation quality of the bounds is characterized and efficient computational methods are discussed. Numerical experiments show that the bounds can result in substantial reductions in computational complexity.
by Dennis Wei.
Ph.D.

本文データは平成22年度国立国会図書館の学位論文(博士)のデジタル化実施により作成された画像ファイルを基にpdf変換したものである
Kyoto University (京都大学)
0048
新制・論文博士
博士(工学)
乙第9308号
論工博第3138号
新制||工||1047(附属図書館)
UT51-96-T528
(主査)教授 長谷川 利治, 教授 茨木 俊秀, 教授 金澤 正憲
学位規則第4条第2項該当

Thesis (M.A.)--University of North Carolina at Chapel Hill, 2008.
Title from electronic title page (viewed Dec. 11, 2008). "... in partial fulfillment of the requirements for the degree of Master of Political Science in the Department of Political Science." Discipline: Political Science; Department/School: Political Science.