Linearized Filtering of Affine Processes Using Stochastic Riccati Equations

TL;DR

This paper introduces an efficient approximate filtering method for affine processes observed with noise, using stochastic Riccati equations to compute conditional characteristic functions, outperforming Gaussian approximations especially for complex processes.

Contribution

The paper develops a novel filtering approach based on solving Riccati differential equations, providing a tractable alternative to particle filtering for high-dimensional affine processes.

Findings

Method effectively computes conditional laws of affine processes.

Numerical experiments demonstrate superior performance over Gaussian approximations.

Applicable to complex models like Cox-Ingersoll-Ross and Wishart processes.

Abstract

We consider an affine process $X$ which is only observed up to an additive white noise, and we ask for its law, for some time $t > 0 $, conditional on all observations up to this time $ t $. This is a general, possibly high dimensional filtering problem which is not even locally approximately Gaussian, whence essentially only particle filtering methods remain as solution techniques. In this work we present an efficient numerical solution by introducing an approximate filter for which conditional characteristic functions can be calculated by solving a system of generalized Riccati differential equations depending on the observation and the process characteristics of the signal $X$. The quality of the approximation can be controlled by easily observable quantities in terms of a macro location of the signal in state space. Asymptotic techniques as well as maximization techniques can be directly applied to the solutions of the Riccati equations leading to novel very tractable filtering formulas. The efficiency of the method is illustrated with numerical experiments for Cox-Ingersoll-Ross and Wishart processes, for which Gaussian approximations usually fail.