We study a mean-variance portfolio selection problem under a hidden Markov regime-switching Black-Scholes-Merton economy with parameter uncertainty. By exploiting the separation principle, we solve the mean-variance portfolio selection problem and the filtering/estimation problem separately. An explicit solution to the mean-variance problem is derived using the stochastic maximum principle. Robust filters of the chain and robust-based EM algorithm for unknown model parameters are developed.

In this paper, a new version of the quadrature Kalman filter (QKF) is developed theoretically and tested experimentally. We first derive the new QKF for nonlinear systems with additive Gaussian noise by linearizing the process and measurement functions using statistical linear regression (SLR) through a set of Gauss-Hermite quadrature points that parameterize the Gaussian density. Moreover, we discuss how the new QKF can be extended and modified to take into account specific details of a given application. We then go on to extend the use of the new QKF to discrete-time, nonlinear systems with additive, possibly non-Gaussian noise. A bank of parallel QKFs, called the Gaussian sum-quadrature Kalman filter (GS-QKF) approximates the predicted and posterior densities as a finite number of weighted sums of Gaussian densities. The weights are obtained from the residuals of the QKFs. Three different Gaussian mixture reduction techniques are presented to alleviate the growing number of the Gaussian sum terms inherent to the GS-QKFs. Simulation results exhibit a significant improvement of the GS-QKFs over other nonlinear filtering approaches, namely, the basic bootstrap (particle) filters and Gaussian-sum extended Kalman filters, to solve nonlinear non- Gaussian filtering problems.

We consider risk sensitive filtering and smoothing for a dynamical system whose output is a vector process in /spl Ropf//sup 2/. The components of the observation process are a Markov process observed through a Brownian motion and a Markov process observed through a Poisson process. Risk-sensitive filters for the robust estimation of an indirectly observed Markov state processes are given. These filters are stochastic partial differential equations for which robust discretizations are obtained. Computer simulations are given which demonstrate the benefits of risk sensitive filtering.

A discrete time Poisson process whose rate evolves as the square of the state of a linear Gaussian dynamical system is studied. An optimal filter is derived, yielding real-time estimates of the Poisson rate. Also a suboptimal filter based on an Edgeworth series expansion is derived.

A finite state Markov random field is observed in Gaussian noise. Changes of measures are defined under which all random variables of the signal are independent and uniformly distributed over the finite state space and all random variables of the observation are independent and N(0,1). The problem of estimating the most likely signal given the observations is treated in a related form by introducing probabilities over the possible signals.>

New finite-dimensional filters and smoothers that are related to the Wonham filter of a noisily observed Markov chain are obtained. In particular, finite-dimensional, recursive filters and smoothers are given for the number of jumps from state i to state j, for the occupation time of state i, and for a stochastic integral related to the drift in the observations. These filters allow easy application of the EM algorithm for the estimation of the parameters of the Markov chain and observation process.>

Semimartingale decompositions and prediction problems are discussed for a diffusion observed through a two-sided filtration. Enlarged filtrations as well as bilateral prediction are discussed in detail.>

Zakai forms for the prediction and smoothing equations are obtained which, like the Zakai filtering equation, are linear in the unnormalized conditional expectation.

The problem of representing a Markov process as a reverse-time Markov process is discussed. Diffusions and finite-state Markov processes are two important classes of Markov processes for which this problem has been considered. A gap in a paper of Castanon is pointed out which discusses diffusions; reverse-time finite and discrete-state Markov processes are then considered using martingale methods.