Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.

http://ahvazdownload.persiangig.com/document/article/39.pdf

A wavelet tour of signal processing

Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition)

1. Introduction to wavelets 2. Review of Fourier theory and filters 3. Orthonormal transforms of time series 4. The discrete wavelet transform 5. The maximal overlap discrete wavelet transform 6. The discrete wavelet packet transform 7. Random variables and stochastic processes 8. The wavelet variance 9. Analysis and synthesis of long memory processes 10. Wavelet-based signal estimation 11. Wavelet analysis of finite energy signals Appendix. Answers to embedded exercises References Author index Subject index.

/pdf/wavelet-methods-for-time-series-analysis-1m2hg2nm53.pdf

Wavelet Methods for Time Series Analysis

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

The purpose of this chapter is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of McKean-Vlasov type. We tackle the characterization and construction of solutions of this special type of optimal control problem by means of forward-backward stochastic differential equations. Because of the presence of the distribution of the controlled state in the coefficients, the approach based on the Pontryagin stochastic maximum principle requires special attention. We provide a version of this maximum principle based on the differential calculus for functions of probability measures introduced and developed in Chapter 5 We test the results of the analysis on linear quadratic models and a few other models already considered in the framework of mean field games. Finally, we highlight the similarities and the differences between this problem and MFG problems with which it is often confused.

Optimal Control of SDEs of McKean-Vlasov Type

The goal of the present chapter is to present in a self-contained manner, elements of differential calculus and stochastic analysis over spaces of probability measures. Such a calculus will play a crucial role in the sequel when we discuss stochastic control of dynamics of the McKean-Vlasov type, and various forms of the master equation for mean field games. After reviewing the standard metric theory of spaces of probability measures, we introduce a notion of differentiability of functions of measures tailor-made to our needs. We provide a thorough analysis of its properties, and relate it to different notions of differentiability which have been used in the existing literature, in particular the geometric notion of Wasserstein gradient. Finally, we derive a first form of chain rule (Ito’s formula) for functions of flows of measures, and we illustrate its versatility on a couple of applications.

Spaces of Measures and Related Differential Calculus

The purpose of this note is to report on pedagogical experiments centered around the use of a dedicated program in the learning of mathematical statistics and the practice of data analysis. The authors developed a set of lectures notes with a strong emphasis on: Graphical analyses, random number generation, simulation techniques, resampling methods, dynamic illustration of regression diagnostics, robust methods ... . Most of these concepts can be presented to undergraduate students but no appropriate textbook existed. Also, such a pedagogical experiment could not be conceived without the use of a computer program for use as a companion to the lectures. No satisfactory solution could be found from the existing commercial or public softwares. We discuss in detail some of the most salient features of the experiment and we describe the tools which the authors developed in the process.

Learning Data Analysis and Mathematical Statistics with a Macintosh

The first part of the chapter gives a quick review of the classical parametric families of probability distributions and the statistical estimation of their parameters. We also review nonparametric density estimation, but our interest in financial data and heavy tail distributions prompts us to focus on quantile comparison. We introduce Q-Q plots as the main graphical tool to detect the presence of heavy tails. Because random simulation will be used throughout the book, the last part of the chapter presents the basics of Monte Carlo computations.

Univariate Data Distributions

We propose a new approach to mean field games with major and minor players. Our formulation involves a two player game where the optimization of the representative minor player is standard while the major player faces an optimization over conditional McKean-Vlasov stochastic differential equations. The definition of this limiting game is justified by proving that its solution provides approximate Nash equilibriums for large finite player games. This proof depends upon the generalization of standard results on the propagation of chaos to conditional dynamics. Because it is on independent interest, we prove this generalization in full detail. Using a conditional form of the Pontryagin stochastic maximum principle (proven in the appendix), we reduce the solution of the mean field game to a forward-backward system of stochastic differential equations of the conditional McKean-Vlasov type, which we solve in the Linear Quadratic setting. We use this class of models to show that Nash equilibriums in our formulation can be different from those of the formulations contemplated so far in the literature.

René Carmona

Papers

Optimal Control of SDEs of McKean-Vlasov Type

Spaces of Measures and Related Differential Calculus

Learning Data Analysis and Mathematical Statistics with a Macintosh

Univariate Data Distributions

A Probabilistic Approach to Mean Field Games with Major and Minor Players