This paper is concerned with using stochastic approximation and optimization methods for stock liquidation decision making and option pricing. For stock liquidation problem, we present a class of stochastic recursive algorithms, and make comparisons of performances using stochastic approximation methods and that of certain commonly used heuristic methods, such as moving averaging method and moving maximum method. Stocks listed in NASDAQ are used for making the comparisons. For option pricing, we design stochastic optimization algorithms and present numerical experiments using data derived from Berkeley Options Data Base. An important problem in these studies concerns the rate of convergence taking into consideration of bias and noise variance. In an effort to ascertain the convergence rates incorporating the computational efforts, we use a Liapunov function approach to obtain the desired convergence rates. Variants of the algorithms are also suggested.

We consider hidden Markov processes in discrete time with a finite state space $X$ and a general observation or read-out space $Y$, which is assumed to be a Polish space. It is well-known that in the statistical analysis of HMMs the so-called predictive filter plays a fundamental role. A useful result establishing the exponential stability of the predictive filter with respect to perturbations of its initial condition was given in "Exponential forgetting and geometric ergodicity in hidden Markov models" (F. LeGland, L. Mevel, Mathematics of control, signals and systems, 13(2000), pp. 63-93) in the case, when the assumed transition probability matrix was primitive. The main technical result of the present paper is the extension of the cited result by showing that the random constant and the deterministic positive exponent showing up in the inequality stating exponential stability can be chosen so that for any prescribed ${s\geq 0}$ the $s$-th exponential moment of the random constant is finite. An application of this result to the estimation of HMMs with primitive transition probabilities will be also briefly presented.

Factor models are used to condense high dimensional data consisting of many vari ables into a much smaller number of factors. Here we present an introductory survey to factor models for time series, where the factors represent the comovement between the single time series. Principal component analysis, linear dynamic factor models with idiosyncratic noise and generalized linear dynamic factor models are introduced and structural properties, such as identifiability, as well as estimation are discussed.

Estimation of the Hurst parameter provides information about the memory range orcorrelations (long vs. short) of processes (time-series). A new application for the Hurst parameter, real-time event detection, is identified. Hurst estimates using rescaled range, dispersional and bridgedetrended scaled windowed variance analyses of seizure time-series recorded from human subjects reliably detect their onset, termination and intensity. Detection sensitivity is unaltered by signal decimation and window size increases. The high sensitivity to brain state changes, ability to operate in real time and small computational requirements make Hurst parameter estimation using any of these three methods well suited for implementation into miniature implantable devices for contingent delivery of anti-seizure therapies.

Maintaining optical alignment between stations of a free-space optical link requires an active pointing mechanism to persistently aim an optical beam toward the receiving station with an acceptable accuracy. This mechanism ensures delivery of maximum optical power to the receiving station in spite of the relative motion of the stations. In the active pointing scheme proposed in the present paper, the receiving station estimates the center of the incident optical beam based on the output of a position-sensitive photodetector. The transmitting station receives this estimate via an independent communication link and uses it to accurately aim at the receiving station. The overall mechanism which implements this scheme can be described in terms of a diffusion process which modulates the rate of a doubly stochastic space-time Poisson process. At the receiving station, observation of the space-time process over a subset of $\mathbb{R}^2$ is provided in order to control the diffusion process. Our goal is to determine a control law, measurable with respect to the history of the space-time process, which minimizes a quadratic cost functional.