# Stochastic Analysis

## Description

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

Invited Speakers to the Conference in Honor of Moshe Zakai are Denoted by a star*
Preface
Foreword
Publications by Moshe Zakai
Conformally Invariant and Reflection Positive Random Fields in Two Dimensions
Real Time Architectures for the Zakai Equation and Applications
Quadratic Approximation by Linear Systems Controlled from Partial Observations
A Model of Stochastic Differential Equation in Hilbert Spaces Applicable to Navier Stokes Equation in Dimension 2
Wavelets as Attractors of Random Dynamical Systems
Markov Properties for Certain Random Fields
The Anatomy of a Low-Noise Jump Filter: Part I
On the Value of Information in Controlled Diffusion Processes
Orthogonal Martingale Representation
Nonlinear Filtering with Small Observation Noise: Piecewise Monotone Observations
Closed Form Characteristic Functions for Certain Random Variables Related to Brownian Motion
Adaptedness and Existence of Occupation Densities for Stochastic Integral Processes in the Second Wiener Chaos
A Skeletal Theory of Filtering
Equilibrium in a Simplified Dynamic, Stochastic Economy with Heterogeneous Agents
Feynman-Kac Formula for a Degenerate Planar Diffusion and an Application in Stochastic Control
On the Interior Smoothness of Harmonic Functions for Degenerate Diffusion Processes
The Stability and Approximation Problems in Nonlinear Filtering Theory
Wong-Zakai Corrections, Random Evolutions, and Simulation Schemes for SDE’s
Nonlinear Filtering for Singularly Perturbed Systems
Smooth σ-Fields
Composition of Large Deviation Principles and Applications
Nonlinear Transformations of the Wiener Measure and Applications
Finite Dimensional Approximate Filters in the Case of High Signal-To-Noise Ratio
A Simple Proof of Uniqueness for Kushner and Zakai Equations
Itô-Wiener Expansions of Holomorphic Functions on the Complex Wiener Space
Limits of the Wong-Zakai Type with a Modified Drift Term
Donsker’s δ-Functions in the Malliavin Calculus
Implementing Boltzmann Machines
Infinite Dimensionality Results for MAP Estimation