"Nonlinear Stochastic Stability Study of Radar Range Tracker Behavior"
Eyad H. Abed
University of Maryland
abed@isr.umd.edu

The end-game of a radar range tracker encounter with a target and decoy is studied within the framework of nonlinear stochastic stability. Analysis and simulation are used to obtain insights into the relative influence of various system parameters on the nature and timing of the radar's commitment to a specific target.

 

"Singularly Perturbed Markov Chain and Applications to Control Problems"
Grazyna Badowski
University of Maryland
grazyna@math.wayne.edu

This work is concerned with asymptotic properties of singularly perturbed Markov chains in discrete time with finite-state space. We study asymptotic expansion of the probability distribution vector and derive a mean square estimate on a sequence of occupation measures. The state space of the underlying Markov chain can be decomposed into several groups of recurrent states and a group of transient states. By treating the states within each recurrent class as a single state, an aggregated process is defined. It is shown that its continuous-time interpolation converges to a continuous-time Markov chain. In addition, it is shown that a sequence of suitably rescaled occupation measures converges to a switching diffusion process weakly. Moreover, the convergence of probability vectors under the weak topology of L^2 is obtained. Applications to optimal controls will also be mentioned.

 

"Information Based Control Theory"
John Baillieul
Boston University
johnb@bu.edu

This talk will describe recent results in controlling distributed arrays of physical devices through communications networks. There is an emerging theory which prescribes the way in which control designs must be tailored to account for the channel data capacity of feedback loops.

 

"Stochastic Dynamic Models for TCP, Extensions and Marking: "A Control Perspective""
John Baras
ISR, University of Maryland
baras@isr.umd.edu

We investigate statistical properties of real TCP traces in order to develop mathematical/analytical stochastic models for performance evaluation, network dimensioning and queuing control. Using control theoretic ideas we develop stochastic differential equation models for the distribution of windows in TCP. We use these model to investigate the behavior of RED in bottleneck links with many flows. We analyze both RED and marking with these methods and we derive optimal properties of the marking function from a control perspective. Finally, we develop analytical models that can account for fractal and multi-fractal behavior.

 

"A New Martingale Approach for Stochastic Regression Models"
Bernard Bercu
Universite Paris-Sud
Bernard.Bercu@math.u-psud.fr

We establish a new strong law of large numbers for powers of martingale transforms. It enables us to deduce the convergence of moments in the almost sure central limit theorem for martingales. Some statistical applications are also provided. The first one deals with the estimation of moments for linear regression models without control. The second one is devoted to the adaptive control of parametric nonlinear autoregressive models.

 

"Analysis of Dependent Defaults via Intensity Based Approach, with Applications to Valuation of Basket Credit Derivatives"
Tomasz Bielecki
Northeastern Illinois University
t-bielecki@neiu.edu

In this talk we shall discuss the issue of evaluating certain functionals of several random times using intensity based approach. In particular, we shall provide calculations and formulas for the functionals corresponding to the minimum, and to the maximum of two random times. A martingale technique is typically used for such calculations. Respective formulas may be applied to valuation of the first-to-default and to valuation of the last-to-default basket credit derivatives, or, in more generality, to valuation of the ith-to-default basket credit derivatives. It will be shown that the assumption of conditional independence of random times leads to considerable simplification in calculations (whenever such an assumption may be justified).

 

"Learning Wardrop Equilibria in Stochastic Networks"
(Joint work with P.R. Kumar)
Vivek Borkar
Northeastern Illinois University
borkar@tifr.res.in

This talk will present the analysis of a learning algorithm for flow control in a stochastic network based on delay information. It is an event-driven, distributed, asynchronous recursion with multiple timescales. The main result is that it learns a 'Wardrop' equilibrium in a special case and a 'Cesaro-Wardrop' equilibrium in general.

 

"Stochastic Lagrangian Adaptive LQG Control"
Peter E. Caines and David Levanony
McGill University
Ben Gurion University
peterc@cim.mcgill.edu
levanony@ee.bgu.ac.il

Click Here for Abstract (PDF file)

 

"QoS Measures in Wireless Networks via Stochastic Optimal Control"
Charalambos D. Charalambous
University of Ottawa
chadcha@site.uottawa.ca

This talk will introduce a stochastic optimal control framework which can be used to address the QoS in wireless networks, which are subject to multipath fading and long distance transmission. The first part of the talk will introduce certain stochastic differential equations, which emerged from modeling large scale shadowing due to long distance transmission of signals, and scattering of signals due to multipath. The second part of the talk will address the power control problem using a stochastic optimal control framework, subject to various constraints. Finally, the connection between the power control problem, which is a physical layer issue, and admission control, which is a network layer issue will be presented.

 

"Boundaries of Open Orbits"
David Elliott
ISR, University of Maryland
delliott@isr.umd.edu

For real-analytic vector fields f and g on Euclidean n-space, the diffusion described by the (Stratonovich) stochastic differential equation dx = f(x)dt + g(x)dw lives on the orbits of the corresponding control system dx/dt= f(x) + u(t) g(x). These orbits, or accessible sets, have open components if and only if the Lie rank of the system is full a.e.. A neglected problem is to characterize the boundaries of the open components, where the Lie rank drops below n. We discuss the bilinear case, where I conjecture that these boundaries are given by hyperplanes and quadric cones, easily computed.

 

"Regime Switching and American Options"
Robert Elliott and John Buffington
University of Calgary, Canada
University of Adelaide, Australia
R.Elliott@ualberta.ca

It is well known that in the Black Scholes framework the solution of the pricing problem for an American option reduces to solving a free boundary problem. We consider a more general model in which the dynamics of the price processes depend on the state of the economy and can switch between different models following a hidden Markov chain. A new method is introduced to price European options in this framework and it is shown that the analog of the Black-Scholes equation is a coupled system of parabolic equations. The approximate solution of Barone-Adesi and Whaley is extended to this situation.

 

"Max-plus Stochastic Processes and Control"
Wendell H. Fleming
Brown University
whf@cfm.brown.edu

This lecture concerns processes which are max-plus counterparts of Markov diffusion processes, and with control of max-plus stochastic processes. The max-plus counterparts of backward and forward PDEs for Markov diffusions are first order PDEs of Hamilton-Jacobi-Bellman type. For max-plus stochastic control problems, the minimum max-plus expected cost is equal to the upper Elliott-Kalton value of an associated differential game.

 

"Adaptation of Real-Time Seizure Detection Algorithm"
Mark G. Frei (1), Shane M. Haas (1,2), Ivan Osorio (1,3)
(1) Flint Hills Scientific, L.L.C.
(2) Massachusetts Institute of Technology
(3) Department of Neurology, Kansas University Medical Center
frei@sound.net

The time-varying dynamics and non-stationarity of epileptic seizures makes their detection difficult. Osorio et. al. in ([1]) proposed an adaptable algorithm, however, that has had great success in detecting seizures. In this work, we present a methodology to adapt this algorithm, further improving its performance.

We begin with an overview of the original detection algorithm's architecture, motivate the problem in light the task at hand, and describe the degrees of freedom in algorithm flexibiltiy that we will focus on in the adaptation procedure.

The adaptation consists of generating multiple candidate digital filters using various techniques from signal processing, defining a practical optimization criteria, and using this criteria to select the best filter candidate. Coupled within the procedure is the selection of a corresponding optimal percentile value for use in the nonlinear (order statistic) filtering step that follows in the algorithm.

Finally, we provide examples illustrating the advantages of the adapted method over the generic detection algorithm.

Reference: [1] I. Osorio and M.G. Frei and S.B. Wilkinson, Real-Time Automated Detection and Quantitative Analysis of Seizures and Short-Term Prediction of Clinical Onset, Epilepsia 39(6) 615-627, 1998.

 

"Randomization Methods in Optimization and Stochastic Adaptive Control"
Laszlo Gerencser (coauthored by Zs. Vago and H. Hjalmarsson)
MTA SZTAKI
gerencser@szatki.hu

We consider discrete-time fixed gain stochastic approximation processes that are defined in terms of a random field that is identically zero at the same point theta. Under appropriate technical conditions the estimator sequence is shown to converge to theta with geometric rate almost surely. This result is in striking contrast to classical stochastic approximation theory where the typical convergence rate is n^{-1/2}. Experimental results on the behavior of the top Lyapunov-exponent will be presented. The talk is motivated by the study of simultaneous perturbation stochastic approximation methods applied to noise-free problems and to multivariable direct adaptive control.

"Capacity of the Multiple-Input, Multiple-Output Poisson Channel"
Shane Haas
shaas@MIT.EDU

This paper examines the Shannon capacity of the single-user, multiple-input, multiple-output (MIMO) Poisson channel with peak and average transmit power constraints. The MIMO Poisson channel is a good model for the physical layer of a multi-aperture optical communication system. We first derive the exact capacity of the multiple-input, single-output (MISO) Poisson channel as a corollary to the single-input, single-output (SISO) Poisson channel capacity derived in [1], [2], and [3]. We then derive simple upper and lower bounds on the MIMO capacity that coincide in a number of special cases. The MIMO capacity is bounded below by that of the MISO channel created by adding the MIMO channel outputs, and is bounded above by that of parallel, independent MISO channels. Next, we derive the exact MIMO Poisson channel capacity. This capacity has a closed form solution only in certain special cases. Numerical evaluation of the MIMO capacity, however, shows that it is very close to the parallel channel upper bound. We conclude by calculating the average capacity of the multi-aperture wireless optical channel with ergodic log-normal fading.

[1] Y.M. Kabanov, "The capacity of a channel of the Poisson type", Theory Probab. Appl., 1978, vol. 23, pp. 143-147

[2] M.A. Davis, "Capacity and cutoff rate for Poisson-type channels", IEEE Trans. Inform. Theory, 1980, IT-26, pp.710-715

[3] A.D. Wyner, "Capacity and error exponent for the direct detection photon channel--part 1", IEEE Trans. Inform. Theory, 1988, vol. 34(6), pp.1449-1461, November

 

"Portfolio Optimization with Jump-Diffusions and Quasi-Deterministic Processes"
Floyd B. Hanson
UI Chicago
hanson@uic.edu

Modeling, analysis and computations are discussed for a portfolio optimization application in a stochastic environment with diffusions, Poisson jumps and quasi-deterministic jumps. While diffusions are appropriate for moderate background fluctuations, jump processes are more appropriate for the crashes or important large fluctuations, whether the random rare event modeled by space-time Poisson processes or the scheduled event with random response modeled by quasi-deterministic processes. However, the analytic and computational complexity of a problem with jumps is much greater than those with just diffusions, since the jumps can lead to non-local and non-autonomous behavior in the PDE for stochastic dynamic programming. This talk is based on papers of Rishel (1999) and of Hanson and Westman (2001).

 

"A Numerical Method for Optimal Stopping Using Linear and Non-Linear Programming"
Kurt Helmes
Humboldt University of Berlin
helmes@wiwi.hu-berlin.de

We present a numerical method for solving stopping time problems of diffusion and jump-diffusion processes. The method is based on a linear programming (LP) formulation of such problems. The resulting infinite dimensional programs are approximated by appropriately chosen finite dimensional linear and non-linear optimization problems. We illustrate the method by analyzing several examples, e.g. (1) the sequential testing of two simple hypotheses on the mean of Brownian motion, (2) a dection problem involving a Wiener process, and (3) the pricing of a perpetual "Russian option."

 

"Numerical Approximation for an Investment Problem"
Daniel Hernandez-Hernandez
Centro de Investigacion en Matematica
dher@cimat.mx

In this talk we examine numerical techniques for an optimal investment model in which the main objective is to maximize the long term growth rate of expected utility of wealth. The model considers the case when the mean returns of the assets are affected by economic factors.

 

"Chaos expansion of local time of fractional Brownian motions"
Yaozhong Hu
University of Kansas
hu@math.ku.edu

In this talk a chaos expansion of local time of fractional Brownian motions and geometric fractional Brownian motions will be obtained. We will also establish generalized Ito formula for fractional and geometric fractional Brownian motions.

 

"The ODE Method & Spectral Theory of Markov Operators"
Jianyi Huang
University of Illinois
jhuang@control.csl.uiuc.edu

In this talk we develop extensions of the ODE method for channel estimation. An extension of the V-uniform ergodic theorem yields a general verification theorem, and a non-local generalization of the classical ODE approach.

 

"Kalman-type filter for nonparametrical on-line estimation"
Rafail Z. Khasminskii
Wayne State University
rafail@math.wayne.edu

Usually people use kernal or projection type estimators for nonparametrical estimation problems. We propose to use Kalman's ideas for its. It is proven that this approach gives the optimal rate convergence of risks among all possible estimators and has useful on-line and recursive properties, which are not achievable by known before estimators.

 

"Guaranteed Optimization of Dynamical Systems"
Faina M. Kirillova
National Academy of Science of Belarus
kirill@nsys.minsk.by

An approach to the solution of the classical synthesis problem for optimal control systems is under consideration. Methods of constructing bounded stabilizing feedbacks which provide given degrees of stability or oscillation are justified and tasted by computer. The results are used to solve problems of regulation such as the transfer an object from one operating conditions to a new regime and following stabilization and the problem of realizing given motions. Examples from mechanics are given.

 

"Particle Filters and Navigation"
P. S. Krishnaprasad
University of Maryland
krishna@isr.umd.edu

In this talk, based on joint work with Babak Azimi-Sadjadi, I discuss recent progress in nonlinear filtering via the method of particle evolution. Modern numerical integration methods for stochastic differential equations and projection of empirical distributions in suitable families yield filters that are effective approximations to the ideal filters. Applications to global positioning (GPS) and inertial navigation (INS) draw attention to important subclasses of problems that merit further study.

 

³Scaling Laws for Wireless Networks: How Much Traffic Can They Carry?²
P. R. Kumar
University of Illinois
prkumar@uiuc.edu

Wireless networks have to operate over a shared communication medium.  Transmissions can therefore interfere with other transmissions.  What then is the traffic carrying capacity of wireless networks when they are operated optimally, i.e., when nodes can choose the timing and range/power level of their transmissions, and the multi-hop routes to their destinations?

Under some models of current technology it is shown that as the number n of nodes in the network increases, the traffic that can be carried by the entire network increases only as n power 1/2.

 

"Estimation and Filtering in Hidden Markov Models, with Applications to Adaptive Control"
Tze Leung Lai
Stanford University
lait@stat.stanford.edu

Hidden Markov models were introduced in the 1960's and have become an important class of models in engineering, economics and bioinformatics. In this talk, we review some of these applications and describe some recent work concerning efficient estimation, filtering and smoothing problems in these models. Implementation that strikes a good balance between statistical efficiency and computational complexity will also be discussed. Applications to adaptive control of stochastic systems with time-varying parameters will also be considered.

 

"Parametric selection to minimize effect of worst case bounded model uncertainty or disturbance noise"
E. Bruce Lee
University of Minnesota
eblee@ece.umn.edu

We will give an explicit quantification of uncertainty for second order linear control systems and detailed design charts for third order linear control systems. This enables direct parameter selection to handle the worst case situation ­ the L₁­norm or L₂-gain of the system provide the quantification basis.

 

³Small Noise Asymptotics in Partially Observed Nonlinear Systems, and Applications to Statistis²
(joint work with Bo Wang)
Francois LeGland
Francois.Le_Gland@irisa.fr

Large deviations techniques have been used to prove convergence of the log-likelihood function in a partially observed nonlinear system depending on an unknown parameter, and consistency of the maximum likelihood estimate, in the small noise asymptotics.  In this talk, we will prove convergence in distribution of the score function (i.e. the derivative of the log-likelihood function w.r.t. the unknown parameter) to a Gaussian random vector, and asymptotic normality of the maximum likelihood estimate.  We will show also how a simple chi-square test can be used to detect a small change.

 

"LQ Control of Backward Stochastic Differential Equations"
Andrew Lim
Columbia University
lim@ieor.columbia.edu

A backward stochastic differential equation (BSDE) is an Ito type SDE for which a random terminal condition has been specified. These equations have been studied intensely in recent years, motivated in large by applications from mathematical finance and mathematical economics; for example, large investor problems, stochastic differential utility, contingent claim replication. On the other hand, research on BSDEs has focused, almost exclusively, on uncontrolled BSDEs and the problem of optimal control of BSDEs has been relatively unexplored. Apart from theoretical interest, these problems offer much by way of applications. For example, an investor who has the option of adding funds (obtained, for example, as income from another source) into a replicating portfolio could be interested in solving such a control problem in order to determine the optimal way that these funds should be added. In this talk, I will be presenting a complete solution to the problem of minimizing a quadratic cost when the dynamics are given by a linear BSDE.

 

³Operator Self-Similar Processes²
Mihaela T. Matache
University of Nebraska
vmatache@unomail.unomaha.edu

Operator Self-Similar (OSS) Processes are a topic at the intersection of the theory of probabilities and functional analysis.  The scalar-valued version of this notion is called self-similar processes.  Fractional Brownian Motions are probably the most popular class of examples.  For vector-valued processes the phenomenon of ³self-similarity² is induced by scaling families of linear operators, for which reason the term Operator Self-Similar is used.  This class of processes has been studied so far only in real, finite-dimensional, Euclidean spaces.  The talk will briefly report on the status of the theory in that framework, and present original results obtained by the speaker who initiated the study of OSS Processes valued in arbitrary Banach spaces.  Besides techniques belonging to the theory of stochastic processes and classical functional analysis, her results are based mainly on the theory of one-parameter semigroups of linear operators.

 

"Error Analysis of a Max-Plus Algorithm for a First-Order HJB Equation"
William McEneaney
University of California at San Diego
wmcenean@math.ucsd.edu

We consider max-plus based algorithms for the solution of nonlinear H(sub)infinity problems. This class of algorithms has been described for several problem types such as nonlinear H•(sub)infinity filtering, nonlinear H(sub)infinity• control and nonlinear H(sub)infinity_• control under partial information.  Previous treatments have been oriented towards the general introduction of the algorithms. The corresponding error analysis is just beginning.  In this paper, we both approach the error analysis for such an algorithm, and demonstrate convegence.  The errors are due to both the truncation of the basis expansion and computation of the matrix whose eigenvector one computes.

 

"The Variational Approach to Nonlinear Filtering and Stochastic Hamiltonian Systems"
Sanjoy Mitter
M.I.T.
Mitter@lids.mit.edu

In this talk I present a variational approach to nonlinear filtering using ideas from statistical mechanics. I discuss the relationship of this work to earlier work of Bismut, Davis, and my own unpublished work.

 

"Ergodic Stochastic Control Related to Portfolio Optimization"
Hideo Nagai
Osaka University
nagai@sigmath.es.osaka-u.ac.jp

We shall talk about ergodic stochastic control problems relating to risk-sensitive portfolio optimization. By taking up a factor model we consider the problems maximizing risk-sensitized long-run expected growth rate of the capital an investor possesses. We formulate them as the ergodic stochastic control problems. We discuss with construction of the optimal strategies by using the solutions of Bellman equations of ergodic control.

 

"Degenerate Variance Ergodic Control"
Daniel Ocone
Rutgers University
ocone@math.rutgers.edu

We discuss some ergodic (stationary) stochastic control problems using variance control which is allowed to degenerate.

 

"A Risk-sensitive Generalization of Maximum A Posterior Probability (MAP) Estimation for Hidden Markov Models"
Vahid Ramezani and S. I. Marcus
University of Maryland
rvahid@isr.umd.edu

A risk-sensitive generalization of Maximum A Posterior Probability (MAP) estimation for partially observed Markov chains is presented. Using a change of measure technique, a cascade filtering scheme for the risk-sensitive state estimation is introduced. Structural results, the influence of the availability of information, mixing and non-mixing dynamics, and the connection with other risk-sensitive estimation methods are considered. A qualitative analysis of the sample paths clarifies the underlying mechanism.

 

"Cost Cumulant Control for Protection of Civil Structures"
Michael K. Sain
University of Notre Dame
sain.1@nd.edu

Cost cumulant control has an interpretation in terms of managing the value of a combination of the cumulants of a traditional control performance index. The advantage of this interpretation is that it offers genuine and useful new opportunities in applications, such as protecting civil structures from earthquakes, winds, and seas which are well modeled in the stochastic sense. The control application considered here is the first of the American Society of Civil Engineer's benchmark problems for protection of civil structures from earthquakes (http://www.nd.edu/~quake), recently studied by numerous authors using a variety of different control algorithms. It is found that the performance results of cost cumulant control design are able to exceed the best of the performances associated with the published prior work, available on the website, and to do so with only a minimum of control design effort over and above the baseline design included in the Benchmark description. The paper will discuss methods for addressing k cumulants of the cost, and the presentation will show practical calculations involving controllers using up to four cumulants.

 

"Bayesian adaptive control of partially observed Markov processes"
Lukasz W. Stettner
Institute of Mathematics Polish Academy of Sciences
stettner@impan.gov.pl

Assume that a controlled Markov process has a transition operator depending on a parameter, which we consider as a random variable with a known law. The state process is partially observed. We are looking for an adaptive procedure which is nearly selfoptimizing for the average cost per unit time functional. There are various problems related to our adaptive problem. First of all, the ergodic behaviour of controlled partially observed Markov processes is known only for particular models. Adaptive control technics developed by V. Borkar require uniform (with respect to all set of parameters) law of large numbers for a certain martingale, which is very hard to verify in the case with partial observation. In this paper an approach for completely observed Markov processes studied in the papers of G. Di Masi and L. Stettner will be adapted to a partially observed case. In the partially observed case an identifiability of the unknown parameter (true transition operator) is a major problem. Studying the limits of conditional laws of the unknown parameter we can obtain coincidence of the values of the unnormalized filters over the whole state space. Since we shall need a recurrence of the controlled filtering process, we mainly restrict ourselves to study the following models: finite state Markov process with a noisy observation of each state, a model with complete observation when the process enters a recurrent set, a model with regeneration in which the moment of regeneration is observable.

 

"Portfolio Optimization in Markets having Stochastic Rates"
Richard H. Stockbridge
Univerisity of Wisconsin-Milwaukee
University of Kentucky
stockbri@uwm.edu

The Merton problem of optimizing the expected utility of consumption for a portfolio consisting of a bond and N stocks is considered when changes in the bond's interest rate, in the mean return rate and in the volatility for the stock price processes, are modeled by a finite-state Markov chain. This paper establishes an equivalent linear programming formulation of the problem. Two cases are considered. The first model assumes that these coefficients are known to the investor whereas the second model investigates a partially observed model in which the mean return rates and volatilities for the stocks are not directly observable.

 

"Approximate solutions of perturbed SDEs"
Jordan Stoyanov
University of Newcastle, UK
jordan.stoyanov@ncl.ac.uk

We deal with classes of SDEs perturbed by small or large parameters and without or with internal random noises. The solution processes are quite complicated and our goal is to find simpler processes which approximate well the original ones. Models with explicit limiting processes will be described. Also moment problems, uniqueness and non-uniqueness, related to the distributions of the solution processes will be discussed.

 

"The L-Transform in Poisson Noise Calculus"
Allanus Tsoi
University of Missouri
tsoi@math.missouri.edu

Duncan, Hu, and Pasik-Duncan have recently developed a stochastic calculus for the FBM. Hu and Oksendal have constructed a fractional white noise calculus based on the work of Duncan, Hu, and Pasik-Duncan, and applied their theory to finance. A Breton published a paper on filtering and parameter estimation in a linear system driven by a FBM. The role of Gaussian white noise calculus looks promising in many applied areas. In this talk we shall look on Poisson white noise and its L-Transform, which has a similar role to Hida's S-Transform in the Gaussian case.

 

"Probabilistic Compartment Model for Cancer Subject to Chemotherapy"
John Westman
UCLA
jwestman@math.ucla.edu

A compartmental model is presented for the evolution of cancer based on the characteristics of the cells present. The model is expanded to account for intrinsic or natural occurring and acquired drug resistance. This model can be explored to see the evolution of drug resistance starting from a single cell. Numerical studies are presented illustrating the palliative nature of chemotherapeutic treatments. Numerical results for traditional treatment schedules are presented. An alternate managed care schedule for treatments is presented increasing the life expectancy and quality of life for the patient. A framework for the managed care is presented that uses stochastic exit and stopping time optimal control problems subject to small Gaussian disturbances and large discrete fluctuations in conjunction with a receding time horizon. A key feature of the managed schedule is that information for a particular patient can be used resulting in a personalized schedule of treatments.

 

"Classification of Maximal Rank of Finite Dimensional Estimation Algebras in Nonlinear Filtering"
Stephen S.-T. Yau
University of Illinois at Chicago
yau@uic.edu

The Kalman-Bucy filter is widely used in modern industry. Despite of its usefulness, the Kalman-Bucy filter is not perfect.  One of the weakness is that it needs a Gaussian assumption for the initial data. The other weakness is that it requires the drift term of a linear function. Brockett and Mitter proposed independently using Lie algebraic method to solve DMZ equation for nonlinear filtering. This method allows the initial condition be modeled by an arbitrary distribution. It applies well also to nonlinear dynamic. However, in the Lie algebraic method, one has to know explicitly the structure of the estimation algebra. In 1983, Brockett proposed to classify all finite dimensional estimation algebras. In this talk, we shall report our recent results on classfication of finite dimensional estimation algebras of maximal rank with arbitrary state space dimension.

 

"Some Recent Results on Adaptive Filtering and Applications"
George Yin
Wayne State University
gyin@math.wayne.edu

Due to their applications to wireless communication and blind multiuser detection in DS/CDMA systems, adaptive filtering algorithms have received much attention lately. In this talk, we discuss some of the recent results on the asymptotic properties of sign-error and sign-regressor algorithms. The results to be reported include some of our joint work with H-F. Chen, V. Krishnamurthy, and C. Ion.

 

"Nonlinear Filtering:  A Hybrid Approximation Scheme
Qing Zhang
University of Georgia
qingz@math.uga.edu

This paper is concerned with nonlinear filtering schemes for systems which allow non-Gaussian noise. Using the most probable trajectory (MPT) approach, a finite-dimensional recursive hybrid filtering scheme is derived. By appropriately selecting a switching process, a linear hybrid system can be obtained that approximates the original nonlinear system. Then the MPT approach is used to obtain the hybrid filtering schemes for the nonlinear systems. Numerical experiments are carried out and reported.

 

"Indefinite Stochastic LQ Control and Applications"
Xun Yu Zhou
The Chinese University of Hong Kong
xyzhou@se.cuhk.edu.hk

In this talk we present the recent development on indefinite stochastic LQ control theory and its applications in finance.

 

"Numerical Approximation of a Coupled Stochastic Partial Differential Equation Model for Simulation of Neurons"
Peter J. Zimmer
West Chester University
pzimmer@wcupa.edu

A coupled stochastic partial differential equation model for the CA3 region of the hippocampus is developed. This model consists of a lattice of nodes, each describing a subnetwork consisting of a group of prototypical excitatory pyramidal cells and a group of prototypical inhibitory interneurons connected via on/off excitatory and inhibitory synapses. The nodes communicate using simple rules to simulate the diffusion of extracellular potassium. The stochastic term is an additive time-space white noise modeling the stimulation of the nodes from terms not included in the model. A numerical approximation is developed and simulations are analyzed.