Stochastic Differential Equations and Processes : SAAP Tunisia October 7 9 2010 Preface 1.H. Schurz: Basic Concepts of Numerical Analysis of Stochastic Differential Equations Explained by Balanced Implicit Theta Methods 2.C.A. Tudor: Kernel Density Estimation Local Time and Chaos Expansion 3.W. Jedidi J. Almhana V. Choulakian R. McGorman: General Shot Noise Processes and Functional Convergence to Stable Processes 4.C. El Nouty: The Lower Classes of the Sub Fractional Brownian Motion 5.M. Erraoui and Y. Ouknine: On the Bounded Variation of the Flow of Stochastic Differential Equation 6.A. Ayache Q. Peng: Stochastic Volatility and Multifractional Brownian Motion 7.A. Gulisashvili J. Vives: Two sided Estimates for Distribution Densities in Models with Jumps 8.M. Lefebvre: Maximizing a Function of the Survival Time of a Wiener Process in an Interval. Selected papers submitted by participants of the international Conference “Stochastic Analysis and Applied Probability 2010” ( www.saap2010.org ) make up the basis of this volume. The SAAP 2010 was held in Tunisia from 7 9 October 2010 and was organized by the “Applied Mathematics & Mathematical Physics” research unit of the preparatory institute to the military academies of Sousse (Tunisia) chaired by Mounir Zili. The papers cover theoretical numerical and applied aspects of stochastic processes and stochastic differential equations. The study of such topic is motivated in part by the need to model understand forecast and control the behavior of many natural phenomena that evolve in time in a random way. Such phenomena appear in the fields of finance telecommunications economics biology geology demography physics chemistry signal processing and modern control theory to mention just a few. As this book emphasizes the importance of numerical and theoretical studies of the stochastic differential equations and stochastic processes it will be useful for a wide spectrum of researchers in applied probability stochastic numerical and theoretical analysis and statistics as well as for graduate students. To make it more complete and accessible for graduate students practitioners and researchers the editors Mounir Zili and Daria Filatova have included a survey dedicated to the basic concepts of numerical analysis of the stochastic differential equations written by Henri Schurz.