Conditional Gauss–Hermite Filtering With Application to Volatility Estimation

The conditional Gauss-Hermite filter (CGHF) utilizes a decomposition of the filter density by conditioning on an appropriate part of the state vector. In contrast to the usual Gauss-Hermite filter (GHF) it is only assumed that the terms in the decomposition can be approximated by Gaussians. Due to the nonlinear dependence on the condition, quite complicated densities can be modeled, but the advantages of the normal distribution are preserved. For example, in models with multiplicative noise occuring in Bayesian estimation, the joint density of state and variance parameter strongly deviates from a bivariate Gaussian, whereas the conditional density can be well approximated by a normal distribution. As in the GHF, integrals in the time and measurement updates are computed by Gauss-Hermite quadrature. Alternatively, the unscented transform can be used, leading to a conditional unscented Kalman filter (CUKF).