Stochastic Analysis

Liber Amicorum for Moshe Zakai

Eddy Mayer-Wolf, Ely Merzbach, Adam Shwartz, Eddy Mayer-Wolf, Ely Merzbach, Adam Shwartz

• Publisher: Academic Press
• Year: 2014
• Print ISBN:
  9780124810051
• eBook ISBN:
  9781483218700
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Stochastic Analysis: Liber Amicorum for Moshe Zakai focuses on stochastic differential equations, nonlinear filtering, two-parameter martingales, Wiener space analysis, and related topics. The selection first ponders on conformally invariant and reflection positive random fields in two dimensions; real time architectures for the Zakai equation and applications; and quadratic approximation by linear systems controlled from partial observations. Discussions focus on predicted miss, review of basic sequential detection problems, multigrid algorithms for the Zakai equation, invariant test functions and regularity, and reflection positivity. The text then takes a look at a model of stochastic differential equation in Hubert spaces applicable to Navier Stokes equation in dimension 2; wavelets as attractors of random dynamical systems; and Markov properties for certain random fields. The publication examines the anatomy of a low-noise jump filter, nonlinear filtering with small observation noise, and closed form characteristic functions for certain random variables related to Brownian motion. Topics include derivation of characteristic functions for the examples, proof of the theorem, sequential quadratic variation test, asymptotic optimal filters, mean decision time, and asymptotic optimal filters. The selection is a valuable reference for researchers interested in stochastic analysis.

Table of contents

• Front Cover
• Stochastic Analysis: Liber Amicorum for Moshe Zakai
• Copyright Page
• Table of Contents
• Preface
• Foreword
• Publications by Moshe Zakai
• Chapter 1. Conformally Invariant and Reflection Positive Random Fields in Two Dimensions
• Chapter 2. Real time architectures for the Zakai equation and applications
• Chapter 3. Quadratic Approximation by Linear Systems Controlled From Partial Observations
• Chapter 4. A model of stochastic differential equation in Hilbert spaces applicable to Navier Stokes equation in dimension 2
• Chapter 5. Wavelets as Attractors of Random Dynamical Systems
• Chapter 6. Markov Properties for Certain Random Fields
• Chapter 7. The Anatomy of a Low-Noise Jump Filter: Part I
• Chapter 8. On the Value of Information in Controlled Diffusion Processes
• Chapter 9. Orthogonal Martingale Representation
• Chapter 10. Nonlinear Filtering with Small Observation Noise: Piecewise Monotone Observations
• Chapter 11. Closed Form Characteristic Functions for Certain Random Variables Related to Brownian Motion
• Chapter 12. Adaptedness and Existence of Occupation Densities for Stochastic Integral Processes in the Second Wiener Chaos
• Chapter 13. A Skeletal Theory of Filtering
• Chapter 14. EQUILIBRIUM IN A SIMPLIFIED DYNAMIC, STOCHASTIC ECONOMY WITH HETEROGENEOUS AGENTS
• Chapter 15. Feynman-Kac Formula for a Degenerate Planar Diffusion and an Application in Stochastic Control
• Chapter 16. On the Interior Smoothness of Harmonic Functions for Degenerate Diffusion Processes
• Chapter 17. The Stability and Approximation Problems in Nonlinear Filtering Theory
• Chapter 18. Wong-Zakai Corrections, Random Evolutions, and Simulation Schemes for SDE's
• Chapter 19. Nonlinear Filtering for Singularly Perturbed Systems
• Chapter 20. Smooth σ -Fields
• Chapter 21. Composition of Large Deviation Principles and Applications
• Chapter 22. Nonlinear Transformations of the Wiener Measure and Applications
• Chapter 23. Finite Dimensional Approximate Filters in the case of High Signal–to–Noise Ratio
• Chapter 24. A Simple Proof of Uniqueness for Kushner and Zakai Equations
• Chapter 25. Itô-Wiener expansions of holomorphic functions on the complex Wiener space
• Chapter 26. Limits of the Wong-Zakai Type with a Modified Drift Term
• Chapter 27. Donsker's δ-functions in the Malliavin calculus
• Chapter 28. Implementing Boltzmann Machines
• Chapter 29. Infinite Dimensionality Results for MAP Estimation