Controlled diffusion processes

This article gives an overview of the developments in controlled diffusion processes, emphasizing key results regarding existence of optimal controls and their characterization via dynamic programming for a variety of cost criteria and structural assumptions. Stochastic maximum principle and control under partial observations (equivalently, control of nonlinear filters) are also discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the segmentation of astronomical images via level-set methods

Astronomical images are of crucial importance for astronomers since they contain a lot of information about celestial bodies that can not be directly accessible. Most of the information available for the analysis of these objects starts with sky explorations via telescopes and satellites. Unfortunately, the quality of astronomical images is usually very low with respect to other real images and this is due to technical and physical features related to their acquisition process. This increases the percentage of noise and makes more difficult to use directly standard segmentation methods on the original image. In this work we will describe how to process astronomical images in two steps: in the first step we improve the image quality by a rescaling of light intensity whereas in the second step we apply level-set methods to identify the objects. Several experiments will show the effectiveness of this procedure and the results obtained via various discretization techniques for level-set equations.Comment: 24 pages, 59 figures, paper submitte

Sequential Monte Carlo pricing of American-style options under stochastic volatility models

We introduce a new method to price American-style options on underlying investments governed by stochastic volatility (SV) models. The method does not require the volatility process to be observed. Instead, it exploits the fact that the optimal decision functions in the corresponding dynamic programming problem can be expressed as functions of conditional distributions of volatility, given observed data. By constructing statistics summarizing information about these conditional distributions, one can obtain high quality approximate solutions. Although the required conditional distributions are in general intractable, they can be arbitrarily precisely approximated using sequential Monte Carlo schemes. The drawback, as with many Monte Carlo schemes, is potentially heavy computational demand. We present two variants of the algorithm, one closely related to the well-known least-squares Monte Carlo algorithm of Longstaff and Schwartz [The Review of Financial Studies 14 (2001) 113-147], and the other solving the same problem using a "brute force" gridding approach. We estimate an illustrative SV model using Markov chain Monte Carlo (MCMC) methods for three equities. We also demonstrate the use of our algorithm by estimating the posterior distribution of the market price of volatility risk for each of the three equities.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS286 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

The History of the Quantitative Methods in Finance Conference Series. 1992-2007

This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.

Optimal control of multiscale systems using reduced-order models

We study optimal control of diffusions with slow and fast variables and address a question raised by practitioners: is it possible to first eliminate the fast variables before solving the optimal control problem and then use the optimal control computed from the reduced-order model to control the original, high-dimensional system? The strategy "first reduce, then optimize"--rather than "first optimize, then reduce"--is motivated by the fact that solving optimal control problems for high-dimensional multiscale systems is numerically challenging and often computationally prohibitive. We state sufficient and necessary conditions, under which the "first reduce, then control" strategy can be employed and discuss when it should be avoided. We further give numerical examples that illustrate the "first reduce, then optmize" approach and discuss possible pitfalls