Inside Volatility Filtering: Secrets of the Skew, 2nd Edition

Chapter 4The Quality Problem

Our interest in the WCE was motivated mainly by computational purposes.

—Lototsky, Mikuleviciusz, and Rozovskii

Introduction

A couple of new ideas are introduced in this chapter. The first one is presenting an “exact” solution in form of a series. Naturally the solution is not implementable in an exact manner given one would need an infinite number of terms. The second one advances the idea of using better quality observations. This idea responds to the statements made in previous chapters on the poor observability by simply using different kinds of observations.

An Exact Solution?

As we know, all filtering techniques so far present some kind of approximation. Indeed, the Kalman filter assumes Gaussian noise and linear transition and observation equations. The extended Kalman filter does a first-order Taylor approximation to deal with nonlinearity. The Unscented/Kushner filters use a polynomial approximation to represent the integration and as such assume that the nonlinear image of the Gaussian noise remains Gaussian. The particle filter resorts to a Monte-Carlo simulation as well as use of a finite number of particles to represent the state space.

Hence the question: Is there a known exact solution for the nonlinear filtering problem?

It seems the question has come up in mathematics several times and was addressed in the context of a spectral representation as mentioned in [190] for instance.

Nonlinear Filtering

With slightly different notations ...