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 goal of this note is to illustrate the impact of a self-financing condition recently introduced by the authors. We present the analyses of two specific applications usually considered in more traditional models in financial mathematics. They include hedging European options with limit orders and the optimal behavior of market makers.

Applications of a New Self-Financing Equation

We consider a continuous model for transverse magnetization of spins diffusing in a homogeneous Gaussian random longitudinal field \(\{\lambda V(x);\, x \in {\Bbb R}^{d} \} \), where \(\lambda \) is the coupling constant giving the intensity of the random field. In this setting, the transverse magnetization is given by the formula \(M(t)={\Bbb E} \exp \{ -{\lambda}^{2} \int_{0}^{t} \int_{0}^{t} K(B_r-B_s) \; ds dr \} \), where \(\{B_t;\,t\ge 0\}\) is the standard process of Brownian motion and \(K(x)\) is the covariance function of the original random field \(V(x)\). We use large deviation techniques to show that the limit \(S(\lambda)=\lim_{t \rightarrow \infty} \frac{1}{t} \ln M(t)\) exists. We also determine the small \(\lambda\) behavior of the rate \(S(\lambda)\) and show that it is indeed decaying as conjectured in the physics literature.

Large deviations and exponential decay for the magnetization in a Gaussian random field

. For the purpose of Monte Carlo scenario generation, we propose a graphical model for the joint distribution of wind power and electricity demand in a given region. To conform with the practice in the electric power industry, we assume that point forecasts are provided exogenously, and concentrate on the modeling of the deviations from these forecasts instead of modeling the actual quantities of interest. We ﬁnd that the marginal distributions of these deviations can have heavy tails, feature which we need to handle before ﬁtting a graphical Gaussian model to the data. We estimate covariance and precision matrices using an extension of the graphical LASSO procedure which allows us to identify temporal and geographical (conditional) dependencies in the form of separate dependence graphs. We implement our algorithm on data made available by NREL, and we conﬁrm that the dependencies identiﬁed by the algorithm are consistent with the relative locations of the production assets and the geographical load zones over which the data were collected.

Joint Stochastic Model for Electric Load, Solar and Wind Power at Asset Level and Monte Carlo Scenario GenerationRen\'e Carmona \&Xinshuo Yang

This chapter is concerned with existence and uniqueness of classical solutions to the master equation. The importance of classical solutions will be demonstrated in the next chapter where they play a crucial role in proving the convergence of games with finitely many players to mean field games. We propose constructions based on the differentiability properties of the flow generated by the solutions of the forward-backward system of the McKean-Vlasov type representing the equilibrium of the mean field game on an L2-space. Existence of a classical solution is first established for small time. It is then extended to arbitrary finite time horizons under the additional Lasry-Lions monotonicity condition.

Classical Solutions to the Master Equation

The lion’s share of this chapter is devoted to the construction of equilibria for mean field games with a common noise. We develop a general two-step strategy for the search of weak solutions. The first step is to apply Schauder’s theorem in order to prove the existence of strong solutions to mean field games driven by a discretized version of the common noise. The second step is to make use of a general stability property of weak equilibria in order to pass to the limit along these discretized equilibria. We also present several criteria for strong uniqueness, in which cases weak equilibria are known to be strong.

René Carmona

Papers

Applications of a New Self-Financing Equation

Large deviations and exponential decay for the magnetization in a Gaussian random field

Joint Stochastic Model for Electric Load, Solar and Wind Power at Asset Level and Monte Carlo Scenario GenerationRen\'e Carmona \&Xinshuo Yang

Classical Solutions to the Master Equation

Solving MFGs with a Common Noise