Proper dispersion state space models for stochastic volatility (Q2739864)

The author considers a state-space model \((X_n,Y_n)\), where the state \(X_n\) is unobservable, the state \(Y_n\) is observable and the conditional densities \(p(y_n |x_n)\) are defined by a proper dispersion model NEWLINE\[NEWLINEp(y,\mu,\lambda)=c(\lambda)|\bar V (y)|^{1/2} \exp\left(-2^{-1}\lambda d(y,\mu) \right)NEWLINE\]NEWLINE and NEWLINE\[NEWLINEp(x_{n+1} |y_n)\asymp (x_n^{(\lambda_n-2\alpha_n)/2\alpha_n}/c(x_n))\exp\left(2^{-1}(x\alpha_n^{-1}+x^{-1}\xi_n)\right).NEWLINE\]NEWLINE Some models of the parameters \(\mu\), \(\alpha_n\), \(\lambda_n\), \(\xi_n\) are discussed (regression-type, ARCH-type). Examples with inverse-Gaussian, simplex and hyperboloid densities \(p(y_n |x_n)\) are considered.