Nonlinear Systems

Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, as highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods. The aim of this Special Issue is to highlight papers that show the dynamics, control, optimization and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. Potential topics include, but are not limited to: Stability analysis of discrete and continuous dynamical systems; Nonlinear dynamics in biological complex systems; Stability and stabilization of stochastic systems; Mathematical models in statistics and probability; Synchronization of oscillators and chaotic systems; Optimization methods of complex systems; Reliability modeling and system optimization; Computation and control over networked systems

Generalized Volterra-Wiener and surrogate data methods for complex time series analysis

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (leaves 133-150).This thesis describes the current state-of-the-art in nonlinear time series analysis, bringing together approaches from a broad range of disciplines including the non-linear dynamical systems, nonlinear modeling theory, time-series hypothesis testing, information theory, and self-similarity. We stress mathematical and qualitative relationships between key algorithms in the respective disciplines in addition to describing new robust approaches to solving classically intractable problems. Part I presents a comprehensive review of various classical approaches to time series analysis from both deterministic and stochastic points of view. We focus on using these classical methods for quantification of complexity in addition to proposing a unified approach to complexity quantification encapsulating several previous approaches. Part II presents robust modern tools for time series analysis including surrogate data and Volterra-Wiener modeling. We describe new algorithms converging the two approaches that provide both a sensitive test for nonlinear dynamics and a noise-robust metric for chaos intensity.by Akhil Shashidhar.M.Eng

Essays on Numerical Evaluation of Derivatives

In general, this thesis contemplates the potential of series expansion methods in evaluating financial derivatives. Recently, instead of relying on closed-form solutions, the usage of numerical methods became increasingly popular. The contribution of the thesis at hand is three-folded: First, we present and analyze a new method to price European options that are written on a single underlying asset by introducing Gabor series methods into option pricing. The resulting procedure shows to be a very robust pricing tool with a special strength in calculating short-term contracts. Second, we dedicate our attention to multi-asset derivatives. Multi-asset contracts are notoriously hard to deal with in case the user demands both, advanced stochastic processes and fast evaluation. Compared to single underlying derivatives, these multi-asset exotic options are studied to a much lower degree. We focus on European multi-asset options as well as on discrete barrier multi-asset options. To the best of our knowledge, the valuation of multi-asset barrier options in terms of multi-dimensional Fourier series methods has not been addressed in literature before. Third, as recent events in the market for credit risk have shown, there is a need for further research in methods to evaluate credit derivatives. Therefore, we focused on extending the standard Gaussian factor model to price synthetic collateralized debt obligations. The new models are able to cope with a wide range of market conditions. However, given a crisis as severe as the financial crisis of 2008, questions, such as market liquidity, that are outside the scope of pure modeling overlay the approximation quality. Nevertheless, the models presented are flexible instruments to price synthetic collateralized debt obligations while still staying in the intuitive framework of a factor model

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

Numerical Computation of First-Passage Times of Increasing Levy Processes

Let $\{D(s), s \geq 0\}$ be a non-decreasing L\'evy process. The first-hitting time process $\{E(t) t \geq 0\}$ (which is sometimes referred to as an inverse subordinator) defined by $E(t) = \inf \{s: D(s) > t \}$ is a process which has arisen in many applications. Of particular interest is the mean first-hitting time $U(t)=\mathbb{E}E(t)$. This function characterizes all finite-dimensional distributions of the process $E$. The function $U$ can be calculated by inverting the Laplace transform of the function $\widetilde{U}(\lambda) = (\lambda \phi(\lambda))^{-1}$, where $\phi$ is the L\'evy exponent of the subordinator $D$. In this paper, we give two methods for computing numerically the inverse of this Laplace transform. The first is based on the Bromwich integral and the second is based on the Post-Widder inversion formula. The software written to support this work is available from the authors and we illustrate its use at the end of the paper.Comment: 31 Pages, 7 sections, 11 figures, 2 table

Book of Abstracts

USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud São Carlos, Brasil. 3-7 february 2014

Option pricing and risk management: analytic approaches with GARCH-Lévy dynamics

This Ph.D. thesis considers making some contributions to the asset pricing and financial risk management literature. First of all it offers some dynamics in the area of asset pricing which are practically implement able for pricing European style options. More precisely it considers blending GARCH type non-Markovian dynamics with Levy type Markovian innovations to offer analytic valuation of European style derivatives (at this initial stage). Revealing the mathematical underpinnings- required to replace conditional Gaussian innovations in G ARCH option pricing models by innovations coming from some Levy processes( with one sided and both sided jumps)-is the main focus. The necessity for this arises from the fact that the non-normal (Levy) innovations are crucial as heteroskedasticity alone doesn't suffice to capture the option smirk and the analytic valuation is highly expected because it makes the model practically implementable. Thus besides incorporating non-normality particular attention is paid to analytic valuation as well; though the Monte Carlo techniques can be readily applied for the proposed dynamics. However an approximation is required to uphold the analytic pricing, especially for innovations coming from Levy processes which are not Subordinator. These dynamics are capable of overcoming many deficiencies of benchmark Black-Scholes model and can be used to price other derivatives such as Credit, Interest rate, Commodity, Weather etc. The approach is built on a discrete time continuous state space and upholds the no-arbitrage principle of derivative pricing through the use of conditional Esscher transform to configure Equivalent :tviartingale Measure(EMl'vI). Similar to the existing literature, established for GARCH with normal innovations, existence of EMM provides de-facto evidence in support of no-arbitrage argument. Besides the main focus this research has made some complementary contributions to the option pricing literature. Since J.P.Morgan introduced RiskMetrics in 1994, the normal quantile based VaR has been considered as industry standard for risk management. However VaR itself has inherent inconsistencies which are exacerbated under the assumption of normality. The second part of this thesis considers two frequently referred approaches to non-normality in risk management : extreme value(EV) approach and Levy approach. The idea is to reveal the relative performance of various risk measures under full density based Levy approach and solely tail observation based EV approach. We provide empirical evidence which confirms that though purely tail based risk measures value-at-risk (VaR) and its coherent version expected shortfall (ES) are well comparable under both approaches, entire spectrum based spectral risk measure (SRM) is misleading for EV approach. Backtesting risk measure VaR is considered under both approaches. We plan to improve the computational efficiency of estimation of Levy coherent risk measures through application of characteristic function based FRFT. Our ultimate goal is to see whether the conditional moment generating functions -developed for GARCH-Levy models in the first part of this thesis- can be adapted to the characteristic function based FRFT technique in order to estimate the risk measures in analytic fashion

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science

Applied Mathematics and Fractional Calculus

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia

A Bayesian approach to robust identification: application to fault detection

In the Control Engineering field, the so-called Robust Identification techniques deal with the problem of obtaining not only a nominal model of the plant, but also an estimate of the uncertainty associated to the nominal model. Such model of uncertainty is typically characterized as a region in the parameter space or as an uncertainty band around the frequency response of the nominal model. Uncertainty models have been widely used in the design of robust controllers and, recently, their use in model-based fault detection procedures is increasing. In this later case, consistency between new measurements and the uncertainty region is checked. When an inconsistency is found, the existence of a fault is decided. There exist two main approaches to the modeling of model uncertainty: the deterministic/worst case methods and the stochastic/probabilistic methods. At present, there are a number of different methods, e.g., model error modeling, set-membership identification and non-stationary stochastic embedding. In this dissertation we summarize the main procedures and illustrate their results by means of several examples of the literature. As contribution we propose a Bayesian methodology to solve the robust identification problem. The approach is highly unifying since many robust identification techniques can be interpreted as particular cases of the Bayesian framework. Also, the methodology can deal with non-linear structures such as the ones derived from the use of observers. The obtained Bayesian uncertainty models are used to detect faults in a quadruple-tank process and in a three-bladed wind turbine