Disertacijoje ištirti trupmeniškai integruotų tiesinių laiko eilučių modelių su nestacionaria ilgąja atmintimi dalinių sumų ribiniai skirstiniai ir tam tikros statistikos, susijusios su dalinių sumų procesais. Philippe, Surgailis, Viano 2006 ir 2008 m. darbuose apibrėžė kintančius laike trupmeniškai integruotus filtrus su baigtine dispersija ir nagrinėjo jų dalinių sumų ribinius skirstinius. Disertacijoje ištirti tokių procesų dalinių sumų ribiniai skirstiniai, kai dispersija begalinė, laikant, kad inovacijos priklauso α–stabilaus dėsnio traukos sričiai (čia 1<α<2). Įrodyta, kad dalinių sumų procesas konverguoja į tam tikrą α–stabilų savastingąjį procesą su nestacionariais pokyčiais. Surgailis, Teyssière, Vaičiulis 2008 m. darbe įvedė pokyčių santykių arba IR (= Increment Ratio) statistiką ir parodė, kad IR statistika gali būti naudojama tikrinti neparametrinėms hipotezėms apie stacionariosios laiko eilutės ilgąją atmintį bei ilgosios atminties parametrą d. Disertacijoje apibendrinti šių autorių gauti rezultatai, t. y. įrodyta IR statistikos centrinė ribinė teorema ir gauti poslinkio įverčiai, kai stebiniai aprašomi tiesiniu laiko eilutės modeliu su trendu. Praplėsta laiko eilučių klasė, kuriai IR statistika yra pagrįsta, t. y. konverguoja į vidurkį. / In the thesis is studied the limit distribution of partial sums of certain linear time series models with nonstationary long memory and certain statistics which involve partial sums processes. Philippe, Surgailis, Viano (2006, 2008) introduced time-varying fractionally integrated filters and studied the limit distribution of partial sums processes of these filters under finite variance set-up. In the thesis is studied the limit distribution of partial sums processes of infinite variance time-varying fractionally integrated filters. We assume that the innovations belong to the domain of attraction of an α-stable law (1<α<2) and show that the partial sums process converges to some α-stable self-similar process. In the thesis is studied the limit of the Increment Ratio (IR) statistic for Gaussian observations superimposed on a slowly varying deterministic trend. The IR statistic was introduced in Surgailis, Teyssière, Vaičiulis (2008) and its limit distribution was studied under the assumption of stationarity of observations. The IR statistic can be used for testing nonparametric hypotheses about d-integrated (-1/2 < d <3/2) behavior of the time series, which can be confused with deterministic trends and change-points. This statistic is written in terms of partial sums process and its limit is closely related to the limit of partial sums. In particularly, the consistency of the IR statistic uses asymptotic independence of distant partial sums, the fact is established in the... [to full text]

In the thesis is studied the limit distribution of partial sums of certain linear time series models with nonstationary long memory and certain statistics which involve partial sums processes. Philippe, Surgailis, Viano (2006, 2008) introduced time-varying fractionally integrated filters and studied the limit distribution of partial sums processes of these filters under finite variance set-up. In the thesis is studied the limit distribution of partial sums processes of infinite variance time-varying fractionally integrated filters. We assume that the innovations belong to the domain of attraction of an α-stable law (1<α<2) and show that the partial sums process converges to some α-stable self-similar process. In the thesis is studied the limit of the Increment Ratio (IR) statistic for Gaussian observations superimposed on a slowly varying deterministic trend. The IR statistic was introduced in Surgailis, Teyssière, Vaičiulis (2008) and its limit distribution was studied under the assumption of stationarity of observations. The IR statistic can be used for testing nonparametric hypotheses about d-integrated (-1/2 < d <3/2) behavior of the time series, which can be confused with deterministic trends and change-points. This statistic is written in terms of partial sums process and its limit is closely related to the limit of partial sums. In particularly, the consistency of the IR statistic uses asymptotic independence of distant partial sums, the fact is established in the... [to full text] / Disertacijoje ištirti trupmeniškai integruotų tiesinių laiko eilučių modelių su nestacionaria ilgąja atmintimi dalinių sumų ribiniai skirstiniai ir tam tikros statistikos, susijusios su dalinių sumų procesais. Philippe, Surgailis, Viano 2006 ir 2008 m. darbuose apibrėžė kintančius laike trupmeniškai integruotus filtrus su baigtine dispersija ir nagrinėjo jų dalinių sumų ribinius skirstinius. Disertacijoje ištirti tokių procesų dalinių sumų ribiniai skirstiniai, kai dispersija begalinė, laikant, kad inovacijos priklauso α–stabilaus dėsnio traukos sričiai (čia 1<α<2). Įrodyta, kad dalinių sumų procesas konverguoja į tam tikrą α–stabilų savastingąjį procesą su nestacionariais pokyčiais. Surgailis, Teyssière, Vaičiulis 2008 m. darbe įvedė pokyčių santykių arba IR (= Increment Ratio) statistiką ir parodė, kad IR statistika gali būti naudojama tikrinti neparametrinėms hipotezėms apie stacionariosios laiko eilutės ilgąją atmintį bei ilgosios atminties parametrą d. Disertacijoje apibendrinti šių autorių gauti rezultatai, t. y. įrodyta IR statistikos centrinė ribinė teorema ir gauti poslinkio įverčiai, kai stebiniai aprašomi tiesiniu laiko eilutės modeliu su trendu. Praplėsta laiko eilučių klasė, kuriai IR statistika yra pagrįsta, t. y. konverguoja į vidurkį.

This PhD thesis aims to study financial processes which have semi-heavy-tailed marginal distributions and may exhibit memory. The traditional Black-Scholes model is expanded to incorporate memory via an integral operator, resulting in a class of market models which still preserve the completeness and arbitragefree conditions needed for replication of contingent claims. This approach is used to estimate the implied volatility of the resulting model. The first part of the thesis investigates the semi-heavy-tailed behaviour of financial processes. We treat these processes as continuous-time random walks characterised by a transition probability density governed by a fractional Riesz- Bessel equation. This equation extends the Feller fractional heat equation which generates a-stable processes. These latter processes have heavy tails, while those processes generated by the fractional Riesz-Bessel equation have semi-heavy tails, which are more suitable to model financial data. We propose a quasi-likelihood method to estimate the parameters of the fractional Riesz- Bessel equation based on the empirical characteristic function. The second part considers a dynamic model of complete financial markets in which the prices of European calls and puts are given by the Black-Scholes formula. The model has memory and can distinguish between historical volatility and implied volatility. A new method is then provided to estimate the implied volatility from the model. The third part of the thesis considers the problem of classification of financial markets using high-frequency data. The classification is based on the measure representation of high-frequency data, which is then modelled as a recurrent iterated function system. The new methodology developed is applied to some stock prices, stock indices, foreign exchange rates and other financial time series of some major markets. In particular, the models and techniques are used to analyse the SET index, the SET50 index and the MAI index of the Stock Exchange of Thailand.

This thesis concerns measuring, understanding and modelling Internet traffic. We first study the origins of the statistical properties of Internet traffic, in particular its scaling behaviour, and propose a constructive model of packet traffic with physically motivated parameters. We base our analysis on a large amount of empirical data measured on different networks, and use a so called semi-experimental approach to isolate certain features of traffic we seek to model. These results lead to the choice of a particular Poisson cluster process, known as Bartlett-Lewis point process, for a new packet traffic model. This model has a small number of parameters with simple networking meaning, and is mathematically tractable. It allows us to gain valuable insight on the underlying mechanisms creating the observed statistics. / In practice, Internet traffic measurements are limited by the very large amount of data generated by high bandwidth links. This leads us to also investigate traffic sampling strategies and their respective inversion methods. We argue that the packet sampling mechanism currently implemented in Internet routers is not practical when one wants to infer the statistics of the full traffic from partial measurements. We advocate the use of flow sampling for many purposes. We show that such sampling strategy is much easier to invert and can give reasonable estimates of higher order traffic statistics such as distribution of number of packets per flow and spectral density of the packet arrival process. This inversion technique can also be used to fit the Bartlett-Lewis point process model from sampled traffic. / We complete our understanding of Internet traffic by focusing on the small scale behaviour of packet traffic. To do so, we use data from a fully instrumented Tier-1 router and measure the delays experienced by all the packets crossing it. We present a simple router model capable of simply reproducing the measured packet delays, and propose a scheme to export router performance information based on busy periods statistics. We conclude this thesis by showing how the Bartlett-Lewis point process can model the splitting and merging of packet streams in a router.

Financial processes may possess long memory and their probability densities may display heavy tails. Many models have been developed to deal with this tail behaviour, which reflects the jumps in the sample paths. On the other hand, the presence of long memory, which contradicts the efficient market hypothesis, is still an issue for further debates. These difficulties present challenges with the problems of memory detection and modelling the co-presence of long memory and heavy tails. This PhD project aims to respond to these challenges. The first part aims to detect memory in a large number of financial time series on stock prices and exchange rates using their scaling properties. Since financial time series often exhibit stochastic trends, a common form of nonstationarity, strong trends in the data can lead to false detection of memory. We will take advantage of a technique known as multifractal detrended fluctuation analysis (MF-DFA) that can systematically eliminate trends of different orders. This method is based on the identification of scaling of the q-th-order moments and is a generalisation of the standard detrended fluctuation analysis (DFA) which uses only the second moment; that is, q = 2. We also consider the rescaled range R/S analysis and the periodogram method to detect memory in financial time series and compare their results with the MF-DFA. An interesting finding is that short memory is detected for stock prices of the American Stock Exchange (AMEX) and long memory is found present in the time series of two exchange rates, namely the French franc and the Deutsche mark. Electricity price series of the five states of Australia are also found to possess long memory. For these electricity price series, heavy tails are also pronounced in their probability densities. The second part of the thesis develops models to represent short-memory and longmemory financial processes as detected in Part I. These models take the form of continuous-time AR(∞) -type equations whose kernel is the Laplace transform of a finite Borel measure. By imposing appropriate conditions on this measure, short memory or long memory in the dynamics of the solution will result. A specific form of the models, which has a good MA(∞) -type representation, is presented for the short memory case. Parameter estimation of this type of models is performed via least squares, and the models are applied to the stock prices in the AMEX, which have been established in Part I to possess short memory. By selecting the kernel in the continuous-time AR(∞) -type equations to have the form of Riemann-Liouville fractional derivative, we obtain a fractional stochastic differential equation driven by Brownian motion. This type of equations is used to represent financial processes with long memory, whose dynamics is described by the fractional derivative in the equation. These models are estimated via quasi-likelihood, namely via a continuoustime version of the Gauss-Whittle method. The models are applied to the exchange rates and the electricity prices of Part I with the aim of confirming their possible long-range dependence established by MF-DFA. The third part of the thesis provides an application of the results established in Parts I and II to characterise and classify financial markets. We will pay attention to the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX), the NASDAQ Stock Exchange (NASDAQ) and the Toronto Stock Exchange (TSX). The parameters from MF-DFA and those of the short-memory AR(∞) -type models will be employed in this classification. We propose the Fisher discriminant algorithm to find a classifier in the two and three-dimensional spaces of data sets and then provide cross-validation to verify discriminant accuracies. This classification is useful for understanding and predicting the behaviour of different processes within the same market. The fourth part of the thesis investigates the heavy-tailed behaviour of financial processes which may also possess long memory. We consider fractional stochastic differential equations driven by stable noise to model financial processes such as electricity prices. The long memory of electricity prices is represented by a fractional derivative, while the stable noise input models their non-Gaussianity via the tails of their probability density. A method using the empirical densities and MF-DFA will be provided to estimate all the parameters of the model and simulate sample paths of the equation. The method is then applied to analyse daily spot prices for five states of Australia. Comparison with the results obtained from the R/S analysis, periodogram method and MF-DFA are provided. The results from fractional SDEs agree with those from MF-DFA, which are based on multifractal scaling, while those from the periodograms, which are based on the second order, seem to underestimate the long memory dynamics of the process. This highlights the need and usefulness of fractal methods in modelling non-Gaussian financial processes with long memory.