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International audience ; Motivated by the European sovereign debt crisis, we study the sovereign risk by analyzing the solvency and the sovereign bond yield and propose a hybrid model which takes into account the movement of the sovereign solvency and the impact of critical political events. This model combines the structural and the reduced-form approaches in the credit risk modelling and the sovereign default time can be decomposed into an accessible part with predictable components and a totally inaccessible part. As a consequence, the probability of default at a critical political event date is nonzero and the probability law admits atoms. We study this model in a generalized density framework to deduce the compensator process of default and show that the intensity process does not necessarily exist. We also apply the model to the valuation of sovereign bond and explain the significant jumps in the long-term government bond yield during the sovereign crisis.

International audience ; Motivated by the European sovereign debt crisis, we study the sovereign risk by analyzing the solvency and the sovereign bond yield and propose a hybrid model which takes into account the movement of the sovereign solvency and the impact of critical political events. This model combines the structural and the reduced-form approaches in the credit risk modelling and the sovereign default time can be decomposed into an accessible part with predictable components and a totally inaccessible part. As a consequence, the probability of default at a critical political event date is nonzero and the probability law admits atoms. We study this model in a generalized density framework to deduce the compensator process of default and show that the intensity process does not necessarily exist. We also apply the model to the valuation of sovereign bond and explain the significant jumps in the long-term government bond yield during the sovereign crisis.

International audience ; Motivated by the European sovereign debt crisis, we study the sovereign risk by analyzing the solvency and the sovereign bond yield and propose a hybrid model which takes into account the movement of the sovereign solvency and the impact of critical political events. This model combines the structural and the reduced-form approaches in the credit risk modelling and the sovereign default time can be decomposed into an accessible part with predictable components and a totally inaccessible part. As a consequence, the probability of default at a critical political event date is nonzero and the probability law admits atoms. We study this model in a generalized density framework to deduce the compensator process of default and show that the intensity process does not necessarily exist. We also apply the model to the valuation of sovereign bond and explain the significant jumps in the long-term government bond yield during the sovereign crisis.

This open access book covers the most cutting-edge and hot research topics and fields of post-quantum cryptography. The main purpose of this book is to focus on the computational complexity theory of lattice ciphers, especially the reduction principle of Ajtai, in order to fill the gap that post-quantum ciphers focus on the implementation of encryption and decryption algorithms, but the theoretical proof is insufficient. In Chapter 3, Chapter 4 and Chapter 6, author introduces the theory and technology of LWE distribution, LWE cipher and homomorphic encryption in detail. When using random analysis tools, there is a problem of "ambiguity" in both definition and algorithm. The greatest feature of this book is to use probability distribution to carry out rigorous mathematical definition and mathematical demonstration for various unclear or imprecise expressions, so as to make it a rigorous theoretical system for classroom teaching and dissemination. Chapters 5 and 7 further expand and improve the theory of cyclic lattice, ideal lattice and generalized NTRU cryptography. This book is used as a professional book for graduate students majoring in mathematics and cryptography, as well as a reference book for scientific and technological personnel engaged in cryptography research.

By drawing some parallels between the history of ideas and literary discourses of late modernity, this book traces the influence exerted by Immanuel Kant, either directly or through the mediation of Henri Bergson's intuitionism, Edmund Husserl's phenomenology, Max Dessoir's psychological aesthetics, Hans Vaihinger's als ob fictionalism, or Karl Popper's logical positivism. As the argument goes, background radiation of the Kantian revolution can be detected even in semiotic poetics, quantum probability, and complexity theory. An interdisciplinary approach seems appropriate when considering the works of a thinker who fused Newton's physico-mathematics with psychology and anthropology to form a new paradigm that opened vistas to integrated disciplinary fields, such as J.F. Herbart's psycho-physics, Wilhelm Wundt's physiological psychology, and William James's pragmatism. Enfolded selves, dissolution of selves into quanta of personality, and multiple hypothetical plots analogous to the Kantian thing in itself as an unfathomable matrix of possibilities are seen as latent effects of his deconstruction of rationalist metaphysics. Immanuel Kant of the Prolegomena( 59) emerges as the philosopher of the coastal mind, teased by the sense-world out of thought, yet not confined, but experiencing the thrill of being connected with things and possessed of the knowledge of the boundary

Introduction. The practical side of philosophy ; The history of epistemology ; Error avoidance and the foundations of knowledge ; A word about metaphysics -- Skepticism. The strange story of the idea of error avoidance ; Radical skepticism ; The case of mathematics and logic ; The skeptical arguments ; Skepticism in epistemology ; Practical implications -- Science. The critique of empiricism ; The probability theory of rational beliefs ; The minimum unexpectedness hypothesis on beliefs ; More on the paradoxes of empiricism ; Theories adjusted ad hoc ; Simplicity and Occam's razor ; Predictions and explanations ; Tradition ; Applications -- Ethics. Skepticism in ethics ; Toward a psychological theory of moral judgments ; Maximizing the welfare of humanity ; Liberalism as the default option ; Utilitarian versus Kantian ethics ; Sympathy ; Moral disputes ; More against reductionism in ethics -- Politics. The justification of sovereignty ; Toward a psychological theory of the morality of politics ; The fear of social disorders ; Political disputes ; The dissolution of the nation-state ; Democracy versus dictatorship -- Aesthetics. Skepticism in aesthetics ; Aesthetics and beauty ; Toward a psychological theory of beauty ; The evolution of art ; Non-beautiful art ; Beauty and pleasure ; Tragedy ; Repetitions (1) ; Repetitions (2) ; Pretty average faces ; Humor ; Non-reductionism and non-relativism ; Aesthetic disputes -- Conclusion. The next step ; Psychology ; The meaning of life.

Nowadays, representations are all the more often conceived as probability distri­butions. Although many authors continue to adhere to "conceptual" and "proposi­tional" idioms, meanwhile, there are cognitive theories that can explain the tran­scend of mental operations beyond the narrow framework of associations and reflexive learning to conceptual cognition. The article discusses two concepts based on the understanding of cognitions as probabilistic predictions: the theory of pre­dictive processing (PP) and the theory of functional systems (FS). The PP-theory relies heavily on the concepts of computation and representation. However, repre­sentations are considered here as sub-symbolic, and, accordingly, computations are understood as probabilistic (Bayesian) inference. The FS-theory believes that a cognitive system anticipates parameters of the expected result and strives to maintain a certain level of metabolism, as well as to minimize the degrees of freedom of the system. As it is shown, we don't have sufficient reasons to con­sider both theories equivalent. They are located at different levels of the Marr's scheme, use different mathematics, and, probably, are not equally falsifiable. Based on these considerations, it can be assumed that the explanations and pre­dictions of these two theories will differ.

Soumis à Philosophy Studies ; This paper discusses aspects of risk and uncertainty relevant in an interdisciplinary assessment of climate change policy. It opposes not only the objective approach versus the subjective approach, but also situations when precise probabilities are well founded versus situations of broader forms of error such as Knightian or deep uncertainty, incompleteness, vagueness. Additional human and social dimensions of ignorance: strategic uncertainties, surprises, values diversity, and taboos, are discussed. We argue that the broader forms of error affect all sciences, including those studying Nature. For these aspects the IPCC guidance notes provides an interdisciplinary unified approach on risk and uncertainty. This is a significant advance from a simple multidisciplinary justaposition of approaches. However, these guidance notes are not universal, they mostly omit the human and social dimensions of ignorance. ; Ce papier discute les divers aspects du risque et de l'incertitude pertinents dans le cadre de l'évaluation interdisciplinaire des politiques climatiques. Il marque non seulement l'opposition entre l'approche objective (qui voit les probabilités comme des degrés de vérité) et l'approche bayésienne (qui les voit comme des degrés de certitude), mais encore l'opposition entre les situations de risque (quand on dispose de probabilités précises et bien fondées) et les situations d'incertitude (des formes d'ignorance plus générale, comme l'incertitude au sens de Knight, l'incomplétude ou le vague). L'évolution des directives IPCC sur le risque et l'incertitude entre le troisième et le quatrième rapport peuvent se lire comme un mouvement s'écartant de la position objectiviste et probabiliste, pour inclure des aspects plus complexes de l'incertitude. Cependant, il reste encore des dimensions humaines comme l'ignorance stratégique, les surprises, les aspects métaphysiques, les taboos et l'incertitude épistémique qui manquent dans les directives IPCC.

Soumis à Philosophy Studies ; This paper discusses aspects of risk and uncertainty relevant in an interdisciplinary assessment of climate change policy. It opposes not only the objective approach versus the subjective approach, but also situations when precise probabilities are well founded versus situations of broader forms of error such as Knightian or deep uncertainty, incompleteness, vagueness. Additional human and social dimensions of ignorance: strategic uncertainties, surprises, values diversity, and taboos, are discussed. We argue that the broader forms of error affect all sciences, including those studying Nature. For these aspects the IPCC guidance notes provides an interdisciplinary unified approach on risk and uncertainty. This is a significant advance from a simple multidisciplinary justaposition of approaches. However, these guidance notes are not universal, they mostly omit the human and social dimensions of ignorance. ; Ce papier discute les divers aspects du risque et de l'incertitude pertinents dans le cadre de l'évaluation interdisciplinaire des politiques climatiques. Il marque non seulement l'opposition entre l'approche objective (qui voit les probabilités comme des degrés de vérité) et l'approche bayésienne (qui les voit comme des degrés de certitude), mais encore l'opposition entre les situations de risque (quand on dispose de probabilités précises et bien fondées) et les situations d'incertitude (des formes d'ignorance plus générale, comme l'incertitude au sens de Knight, l'incomplétude ou le vague). L'évolution des directives IPCC sur le risque et l'incertitude entre le troisième et le quatrième rapport peuvent se lire comme un mouvement s'écartant de la position objectiviste et probabiliste, pour inclure des aspects plus complexes de l'incertitude. Cependant, il reste encore des dimensions humaines comme l'ignorance stratégique, les surprises, les aspects métaphysiques, les taboos et l'incertitude épistémique qui manquent dans les directives IPCC.

Soumis à Philosophy Studies ; This paper discusses aspects of risk and uncertainty relevant in an interdisciplinary assessment of climate change policy. It opposes not only the objective approach versus the subjective approach, but also situations when precise probabilities are well founded versus situations of broader forms of error such as Knightian or deep uncertainty, incompleteness, vagueness. Additional human and social dimensions of ignorance: strategic uncertainties, surprises, values diversity, and taboos, are discussed. We argue that the broader forms of error affect all sciences, including those studying Nature. For these aspects the IPCC guidance notes provides an interdisciplinary unified approach on risk and uncertainty. This is a significant advance from a simple multidisciplinary justaposition of approaches. However, these guidance notes are not universal, they mostly omit the human and social dimensions of ignorance. ; Ce papier discute les divers aspects du risque et de l'incertitude pertinents dans le cadre de l'évaluation interdisciplinaire des politiques climatiques. Il marque non seulement l'opposition entre l'approche objective (qui voit les probabilités comme des degrés de vérité) et l'approche bayésienne (qui les voit comme des degrés de certitude), mais encore l'opposition entre les situations de risque (quand on dispose de probabilités précises et bien fondées) et les situations d'incertitude (des formes d'ignorance plus générale, comme l'incertitude au sens de Knight, l'incomplétude ou le vague). L'évolution des directives IPCC sur le risque et l'incertitude entre le troisième et le quatrième rapport peuvent se lire comme un mouvement s'écartant de la position objectiviste et probabiliste, pour inclure des aspects plus complexes de l'incertitude. Cependant, il reste encore des dimensions humaines comme l'ignorance stratégique, les surprises, les aspects métaphysiques, les taboos et l'incertitude épistémique qui manquent dans les directives IPCC.

Soumis à Philosophy Studies ; This paper discusses aspects of risk and uncertainty relevant in an interdisciplinary assessment of climate change policy. It opposes not only the objective approach versus the subjective approach, but also situations when precise probabilities are well founded versus situations of broader forms of error such as Knightian or deep uncertainty, incompleteness, vagueness. Additional human and social dimensions of ignorance: strategic uncertainties, surprises, values diversity, and taboos, are discussed. We argue that the broader forms of error affect all sciences, including those studying Nature. For these aspects the IPCC guidance notes provides an interdisciplinary unified approach on risk and uncertainty. This is a significant advance from a simple multidisciplinary justaposition of approaches. However, these guidance notes are not universal, they mostly omit the human and social dimensions of ignorance. ; Ce papier discute les divers aspects du risque et de l'incertitude pertinents dans le cadre de l'évaluation interdisciplinaire des politiques climatiques. Il marque non seulement l'opposition entre l'approche objective (qui voit les probabilités comme des degrés de vérité) et l'approche bayésienne (qui les voit comme des degrés de certitude), mais encore l'opposition entre les situations de risque (quand on dispose de probabilités précises et bien fondées) et les situations d'incertitude (des formes d'ignorance plus générale, comme l'incertitude au sens de Knight, l'incomplétude ou le vague). L'évolution des directives IPCC sur le risque et l'incertitude entre le troisième et le quatrième rapport peuvent se lire comme un mouvement s'écartant de la position objectiviste et probabiliste, pour inclure des aspects plus complexes de l'incertitude. Cependant, il reste encore des dimensions humaines comme l'ignorance stratégique, les surprises, les aspects métaphysiques, les taboos et l'incertitude épistémique qui manquent dans les directives IPCC.

[EN] We provide a full stochastic description, via the first probability density function, of the solution of linear-quadratic logistic-type differential equation whose parameters involve both continuous and discrete random variables with arbitrary distributions. For the sake of generality, the initial condition is assumed to be a random variable too. We use the Dirac delta function to unify the treatment of hybrid (discrete-continuous) uncertainty. Under general hypotheses, we also compute the density of time until a certain value (usually representing the population) of the linear-quadratic logistic model is reached. The theoretical results are illustrated by means of several examples, including an application to modelling the number of users of Spotify using real data. We apply the Principle Maximum Entropy to assign plausible distributions to model parameters ; This work has been supported by the Spanish Ministerio de Economa, Industria y Competitividad (MINECO) , the Agencia Estatal de Investigaci on (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM201789664P. Computations have been carried thanks to the collaboration of Raul San Julian Garces and Elena Lopez Navarro granted by European Union through the Operational Program of the European Regional Development Fund (ERDF) /European Social Fund (ESF) of the Valencian Community 2014-2020, grants GJIDI/2018/A/009 and GJIDI/2018/A/010, respectively ; Burgos-Simon, C.; Cortés, J.; López-Navarro, E.; Villanueva Micó, RJ. (2021). Probabilistic analysis of linear-quadratic logistic-type models with hybrid uncertainties via probability density functions. AIMS Mathematics. 6(5):4938-4957. https://doi.org/10.3934/math.2021290 ; S ; 4938 ; 4957 ; 6 ; 5 ; R. Goebel, R. G. Sanfelice, A. R. Teel, i>Hybrid Dynamical Systems: Modeling, Stability, and Robustness /i>, Princeton University Press, 2012. ; J. Lunze, F. Lamnabhi-Lagarrigue, i>Handbook of Hybrid Systems Control: Theory, Tools, Applications /i>, ser. Nonlinear Systems and Complexity, ...

[EN] A fractional forward Euler-like method is developed to solve initial value problems with uncertainties formulated via the Caputo fractional derivative. The analysis is conducted by using the so-called random mean square calculus. Under mild conditions on the data, the mean square convergence of the numerical method is proved. This type of stochastic convergence guarantees the approximations of the mean and the variance of the solution stochastic process, computed via the aforementioned numerical scheme, will converge to their corresponding exact values. Furthermore, from this probability information, we calculate reliable approximations to the first probability density function of the solution by taking advantage of the Maximum Entropy Principle. The theoretical analysis is illustrated by two examples. ; This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2017-89664-P and by the European Union through the Operational Program of the European Regional Development Fund (ERDF)/European Social Fund (ESF) of the Valencian Community 20142020, grants GJIDI/2018/A/009 and GJIDI/2018/A/010. ; Burgos, C.; Cortés, J.; Villafuerte, L.; Villanueva Micó, RJ. (2020). Mean square convergent numerical solutions of random fractional differential equations: Approximations of moments and density. Journal of Computational and Applied Mathematics. 378:1-14. https://doi.org/10.1016/j.cam.2020.112925 ; S ; 1 ; 14 ; 378 ; Slama, H., El-Bedwhey, N. A., El-Depsy, A., & Selim, M. M. (2017). Solution of the finite Milne problem in stochastic media with RVT Technique. The European Physical Journal Plus, 132(12). doi:10.1140/epjp/i2017-11763-6 ; Golmankhaneh, A. K., Arefi, R., & Baleanu, D. (2013). Synchronization in a nonidentical fractional order of a proposed modified system. Journal of Vibration and Control, 21(6), 1154-1161. doi:10.1177/1077546313494953 ; Lupulescu, V., O'Regan, D., & ur Rahman, G. (2014). Existence results for random fractional differential equations. Opuscula ...

Providing an introduction to mathematical analysis as it applies to economic theory and econometrics, this book bridges the gap that has separated the teaching of basic mathematics for economics and the increasingly advanced mathematics demanded in economics research today. Dean Corbae, Maxwell B. Stinchcombe, and Juraj Zeman equip students with the knowledge of real and functional analysis and measure theory they need to read and do research in economic and econometric theory. Unlike other mathematics textbooks for economics, An Introduction to Mathematical Analysis for Economic Theory and Econometrics takes a unified approach to understanding basic and advanced spaces through the application of the Metric Completion Theorem. This is the concept by which, for example, the real numbers complete the rational numbers and measure spaces complete fields of measurable sets. Another of the book's unique features is its concentration on the mathematical foundations of econometrics. To illustrate difficult concepts, the authors use simple examples drawn from economic theory and econometrics. Accessible and rigorous, the book is self-contained, providing proofs of theorems and assuming only an undergraduate background in calculus and linear algebra. Begins with mathematical analysis and economic examples accessible to advanced undergraduates in order to build intuition for more complex analysis used by graduate students and researchers Takes a unified approach to understanding basic and advanced spaces of numbers through application of the Metric Completion Theorem Focuses on examples from econometrics to explain topics in measure theory

"This book provides a comprehensive overview of calibration and validation techniques for traffic simulation models. It details the data required as an input for the calibration and validation processes and shows how to increase its applicability using data enhancement techniques. It presents an extensive overview of traffic estimation, calibration and validation techniques, taking into account the key factor of uncertainty. This book derives from the methods and tools for supporting the use, calibration and validation of traffic simulation models project of the European Union COST framework"--