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  Abstracts
Keynote Speakers

Nick Duffield
Tyrone Duncan
Gopi Kallianpur

Organizing Committee

Robert Elliott
Tom Kurtz

Contributing Speakers

Lahkdar Aggoun
Anna Amirdjanova
Doug Blount
Amarjit Budhiraja
Pavel Chigansky
Fred Daum
Pierre Del Moral
Nicole El Karoui
Vikram Krishnamurthy
Yoojung Lee
Hongwei Long
Ron Mahler
Laurent Miclo
Alexander Melnikov
Christina Popescu
Slyvain Rubenthaler
Joern Sass
Molly Scheffe
Barbara Torti
Allanus Tsoi
Frederi Viens
Jie Xiong
George Yin
Tim Zajic
Yong Zeng
Xun Yu Zhou


Keynote Speaker Abstracts

Nick Duffield

Title: Revealing the detail in network measurements

Abstract: 

Communications network providers collect diverse operational data from their networks for the purposes of planning, engineering, and control.  However, finding the right view of network behavior from this data is challenging: phenomena of interest are not always
directly observable; their presence may be revealed only through joining different data sets; and the enormous volumes of data present obstacles for management and the extraction of detail.

These talks will describe (i) tomographic inference of network usage and performance not directly observable; (ii) importance sampling to manage the analysis of large network data sets; and (iii) some emerging problem areas for network data analysis.

Tyrone Duncan

Title: Fractional Brownian Motion and Applications

Abstract: Fractional Brownian motion is a family of Gaussian processes indexed by the Hurst Parameter, H, in the interval (0,1).  For H=1/2 it is Brownian motion and for H1/2 the processes have a long range dependence.  A stochastic calculus is introduced for a fractional Brownian motion with H in (0,1) and values in a finite or an infinite dimensional Hilbert space. In the finite dimensional case, some stochastic differential equations are explicitly solved, and some parameter estimation, filtering and prediction problems are solved.  In the infinite dimensional case of cylindrical fractional Brownian motion, some linear stochastic partial differential equations are solved.

Gopi Kallianpur

Title: Lectures on Nonlinear Filtering Theory

Abstract: Lecture I. Stochastic integrals of Ito and Stratonovich. Multiple Wiener and Stratonovich Integrals (MWI and MSI).Definition of the Optimal
nonlinear Filter.  The Kallianpur-Striebel (KS) Bayes formula for conditional expectations.

LectureII. SDE's of the Optimal Filter: The Kushner-FKK Equation; the Zakai equation for the unnormalized conditional expectation.  Uniqueness of solution of the measure valued Zakai equation when the signal process X is Markov and the test functions belong to the domain of the restricted generator of X.(Solution of a martingale problem)
Robustness of the Optimal filter; Markov property and ergodicity of the nonlinear filter.
Hypoellipticity questions.

Lecture III.( III.1) Approximations of the solution of the Zakai equation using MWI and MSI expansions; (III.2) Applications to Stochastic Hydrodynamics.             
 

 

Organizing Committee Abstracts

Robert Elliott

Title: Financial Filtering

Abstract: Some applications of filtering to financial problems are described. These include volatility estimation, using hidden Markov filtering, volatility estimation using a non linear filter, price estimation to explain the 'smile' in option pricing, and calibration of linear commodity models using an extension of the Kalman filter.

Tom Kurtz

Title: Filtering models for network applications

Abstract: Monitoring of communications networks involves many situations in which inferences regarding the state of the network must be based on partial information. "Network tomography," for example, attempts to infer the behavior of internal portions of the network from measurements made at the edges, and intrusion detection systems attempt to identify the presence of illegitimate activity among large amounts of legitimate activity. This entirely speculative presentation will attempt to derive filtering models analogous to a variety of network related inference problems. In each setting one must model the "signal", the "observations", and the relationship between them and derive the corresponding filtering algorithm. Possible benefits of this approach include the recursive nature of filtering computations and the optimality of the conditional distribution computed. Difficulties include the possible high dimensionality of the state description.

 

Contributing Speaker Abstracts

Lahkdar Aggoun

Title: Discrete Time Model Tracking With Poisson Observations

Abstract: In this article M-ary detection filters for discrete time models with Poisson observations are derived. The models considered consist of a discrete-time Markov chain which determines the intensity of an observed Poisson random variable. For the proposed model we compute state estimation filters and M-ary detection filters.   

Anna Amirdjanova

Title: Nonlinear Filtering with Fractional Brownian Motion

Abstract: Our objective is to study a nonlinear filtering problem for the observation process perturbed by a Fractional Brownian Motion (FBM) with Hurst index 1/2<H<1. A reproducing kernel Hilbert space for the FBM is considered and a "fractional" Zakai equation for the unnormalized  optimal filter is derived.

Doug Blount

Title: Convergence to the measure-valued historical evolution of a Markov process

Abstract: In a 1993 paper Bhatt and Karandikar extended a martingale problem approach to weak convergence of processes by using a mapping technique to transfer the problem to a totally bounded(precompact) space. They assumed a separability condition on the domain of the limiting operator. otivated in part by their result we study convergence of a sequence of random historical measure-valued processes. The convergence results have applications to martingale problems,particle filters,Mckean-Vlasov models, and other applications. We generalize the results of Bhatt and Karandikar; and in addition we show that the mapping technique gives much simplified proofs under weaker assumptions for some basic results in the theory of weak convergence in metric spaces.

Amarjit Budhiraja

Title: Ergodic properties of the nonlinear filter

Abstract: In this work we study connections between various asymptotic properties of the nonlinear filter. It is assumed that the signal has a unique invariant probability measure. The key property of interest is expressed in terms of a relationship between the observation $\sigma$ field and the tail $\sigma$ field of the signal, in the stationary filtering problem. This property can be viewed as the permissibility of the interchange of the order of the operations of maximum and countable intersection for certain $\sigma$- fields. Under suitable conditions, it is shown that the above property is equivalent to various desirable properties of the filter such as

(a) uniqueness of invariant measure for the filter,
(b) uniqueness of invariant measure for the pair (signal, filter),
(c) a finite memory property of the filter ,
(d) a property of finite time dependence between the signal and observation $\sigma$ fields and
(e) asymptotic stability of the filter.Previous works on the asymptotic stability of the filter for a variety of filtering models then identify a rich class of filtering problems for which the above equivalent properties hold.

Pavel Chigansky

Title: Stability of the Wonham filter

Abstract: Stability of the nonlinear filter with respect to wrong initial condition is investigated, when the signal $X_t$ is a continuous time Markov chain valued in a finite alphabet and observation is $$ Y_t=\int_0^t h(X_s)ds+W_t $$ with Wiener process $W_t$ independent of $X_t$. We show that for egodic signal the filter is stabilized exponentially fast regardless of $h$, while for non ergodic signal stabilization requires certain observability of the system, expressed in terms of $h$ and transition intensity matrix of $X_t$. The ergodic case is of interest in connection to a gap, recently discovered in the proof of the classic result by H. Kunita, 1971.

Dan Crisan

Title: Branching Particle Systems and the  NonLinear Filtering Problem

Abstract: The talk will present a class of algorithms for solving the filtering problem in a continuous time set-up. The algorithms are based on the construction of certain systems of branching particles. This extends the classical Dawson-Watanabe construction of a super-process to a random environment. Their rates of convergence and optimality is discussed.
   
Fred Daum

Title: Curse of Dimensionality for Particle Filters

Abstract: The crucial issue with particle filters is the extremely high real time computational complexity, which can be 2 to 6 orders of magnitude more than the extended Kalman filter for the same accuracy for low dimensional examples. We derive a simple back-of-the-envelope formula that explains why a carefully designed PF should avoid the curse of dimensionality for certain classes of estimation problems (e.g., "vaguely Gaussian" problems).  However, the PF does not avoid the curse of dimensionality in general. We show experimental results with several PFs, including bells ∓ whistles that dramatically reduce computational complexity.  This new theory hinges on the fact that the volume of the d-dimensional unit sphere is an
amazingly small fraction of the volume of the unit d-dimensional cube, for large d.

Vikram Krishnamurthy

Title: Robust smoothers for Hidden Markov Models and Piecewise Linear Systems

Abstract: This talk presents continuous-time smoothers for functionals of the state of a Hidden Markov Model and piecewise linear systems. These smoothers are based on the robust formulation and do not require the use of two sided stochastic calculus. The smoothers can be used in the Expectation Maximization algorithm to compute maximum likelihood estimates of the parameters of the underlying system.

Pierre Del Moral

Title: On the genealogy and the increasing propagation of chaos for Feynman-Kac models

Abstract: A path-valued interacting particle systems model for the genealogical structure of genetic algorithms is presented. We connect the historical process and the distribution of the whole ancestral tree with a class of Feynman-Kac formulae on path space. We also present increasing and uniform versions of propagation of chaos for appropriate particle block size and time horizon. Applications to non linear filtering/smoothing and path estimation will also be presented   

Nicole El Karoui

Title: Pricing and hedging financial products with partial information

Financial markets have developed a lot of strategies to control risk induced by markets fluctuations. Traditional or sophisticated (exotics) financial products can be used by business companies or investors to transfer their financial risks to financial institutions. The history began with the paradigm of zero-risk introduced by Black, Scholes and Merton stating that any random amount to be paid in the future may be replicated by a dynamical portfolio. That is only if all agents in the market have a perfect knowledge on model parameters, in particular on the underlying volatility defined as the noise intensity of the underlying returns. The market is then said to be complete. When the volatility is stochastic, this pricing rule does not hold and a natural extension is the super-replication price defined as the price of the smallest replicating derivative dominating the given random amount in all scenarios. When super-replicating is too expensive, the seller has to introduce a risk measure of this exposure. The reservation price is the amount to be paid to make the transaction risk acceptable. When the option seller only observed dynamic of the underlying asset , the market is doubly incomplete due to the noise on the volatility and the partial observation. Using stochastic filtering theory, the problem may be transformed in a classical incomplete problem. Reservation price dynamics may be characterized through backward stochastic differential equations.

Nicole El Karoui

Title: Continuous trading with asymmetric information

We are dealing with the microstructure problem of price formation on a market with asymmetric information. In this Kyle, Back problem, the market is waiting for a public release of information on a speculative asset. An agent, the insider already knows the information. Other agents on the market are market makers setting prices competitively by trying to infer this information from orders and noise traders motivated only by rebalancing portfolio. The presence of noise traders makes impossible for the market makers to invert the price to know the informed trader's signal. Insider's orders are optimally chosen to maximize the expected utility of his final wealth. In the Gaussian case, Back finds an explicit expression for the pricing rule of market markers and for the insider strategy. In a two-point signal distribution, we characterize the non- linear equilibrium, and the property of the post announcement wealth.

Yoojung Lee

Title: Application of Filtering Methods to Intrusion Detection

Abstract: We suggest an alternative approach to modeling intrusion detection system. Our model formally describes the stochastic behavior of normal users and intruders in the network system. By applying the filtering method, we recursively compute a statistical distance which measures how far the current traffic is from the normal traffic. If the computed distance is above a threshold, which is chosen to control the false positive rate, the traffic will be considered anomalous and a flag will be raised. A detailed model for monitoring Unix system calls is introduced as an example. Our results are compared with those of Hofmeyr, Forrest, and Somayaji (Intrusion Detection Using Sequences of System Calls, 1998). They use the minimum Hamming distance as their primary statistic. From this example, we are able to demonstrate several advantages of our approach such as flexibility of modeling, intuitive interpretation of results, and computational efficiency.

Pierre Del Moral and Laurent Miclo

Title: On convergence of chains with occupational self-interactions

Abstract: We consider stochastic chains on abstract measurable spaces whose evolution at any given time depends on the present position and on the occupation measure created by the path up to this instant. This generalization of reinforced random walks enables us to impose conditions insuring or Lp, p³ 1, or a.s. convergence of the empirical measures toward some fixed point of a probability-valued dynamical system. We present two sets of hypotheses based on weak contraction properties, leading to two different proofs, but in both situations the rates of convergence are optimal in the examined level of generality.

Giovanna Nappo

Title: A singular filtering problem involving the Brownian motion local time

Abstract: We consider the filtering problem when the state process is a Brownian motion and the observation process is its local time at level 0. We use an approximation scheme, based on a suitable interpolation of the observed local time. The approximating filters convergence strongly  to the original filter. The explicit expression of the approximating filter can be computed in terms of the interpolated observation process evaluated at time t and the elapsed time from its last jump time. Then the original filter can be computed in terms of the local time evaluated at time t and the elapsed time from last visit to 0 of the reflected Brownian motion.
Finally, some connections with the Azéma martingale are discussed.

Hongwei Long

Title: Markov chain approximations to nonlinear filtering equations

Abstract: We consider direct Markov chain approximations to Duncan-Mortensen-Zakai
equations for continuous-space, continuous-time nonlinear filtering problems on regular, bounded domains. We prove that our approximations converge to the desired conditional distribution of the signal given the observation.   

Ron Mahler

Title: The "PHD" multi-target first-moment filter for bulk target tracking

Abstract: In certain applications it is sometimes not necessary to detect and track individual targets with high accuracy.  Ideally, one would use the theoretically optimal approach to precisely track all targets simultaneously, which is a suitable generalization of the recursive Bayes nonlinear filter.  Even in single-target problems, however, the optimal filter is computationally very challenging:  The computationally fastest approximate filter is the constant-gain Kalman filter.  This filter propagates a first-order statistical moment of the single-target system (the posterior expectation) in the place of the posterior distribution.  This
presentation describes an analogous strategy:  filtering equations that propagate a first-order statistical moment of the entire multitarget system. This moment, the probability hypothesis density (PHD), is the density function whose integral in any region of state space is the actual number of targets in the region.  The PHD filter is general enough to include target birth and death, target spawning, and Poisson false alarms.  In this paper we report on an implementation of a bulk tracking algorithm based on the PHD filter, using simulated data. 

Alexander Melnikov

Title: On filtering and finance

Abstract: It is supposed to discuss some leading ideas and facts which are important for both mathematical finance and filtering theory.

Christina Popescu

Title: Filtering and prediction of aeroelastic dynamics- a statistical approach

Abstract: In this study, the unscented filter is applied for predicting the response of a discrete time aeroelastic system with structural nonlinearities.
   
Sylvain Rubenthaler

Title: Stability and uniform particle approximation of nonlinear filters

Abstract: We propose a new approach to study the stability of the discrete time optimal
filter w.r.t its initial condition by introducing a "robust filter" which approximates the optimal filter uniformly in time (under the assumption, in certain cases, that the signal-to-noise ratio is big enough, no ergodicity assumption is needed). This approach allows us to prove
the uniform convergence of some particle filter to the optimal filter.

Joern Sass

Title: Portfolio Optimization with Partial Information: An HMM model

Abstract: We consider a financial market consisting of one deterministic bank account and one stock, driven by a geometric Brownian motion with a random drift process and constant volatility. Partial information means that the only available information are the prices. The investor's objective is to maximize the utility of the terminal wealth under the available information. In general, it is not possible to determine an optimal trading strategy. If the drift process is an Ornstein-Uhlenbeck process, the problem can be solved by Kalman filtering
(Lakner 1998).
 
We consider the case hat the drift process is a continuous time Markov chain. Because only the prices are observable, we have a hidden Markov model (HMM) for (the excess return of) the stock prices.

Using filtering results for Markov chains and applying Malliavin calculus  we can solve the optimization problem for general utility and present an explicit representation of the optimal trading strategy in terms of the unnormalized filter for the drift process. The strategy can be determined numerically.

Molly Scheffe

Title: Dynamic Magnetic Resonance Imaging: Challenges and Prospects

Abstract: This talk will survey some medical applications where it is desirable to estimate or predict parameters in dynamically changing MR images in real-time, such as image-guided
hyperthermia therapy for cancer.  Some recent filtering and signal-processing algorithms for dynamic MRI will be reviewed

Barbara Torti

Title: The filter of a continuous time random walk with respect to its local time

Abstract: We consider  the conditional law of a continuous time random walk Y(t), when the observation is its local time L(s) at level 0, up to time t. When Y(t) = V(Z(t)), where Z(t) is a renewal process and V(k) is a discrete time random walk, mutually independent, there exists a two parameter  family of  measures, characterizing the conditional law: namely the two parameters are on the value at time t of  the local time L and the time elapsed from last jump time of  L. When the renewal process is a Poisson process, then the process Q(t)=Y(t)+L(t) represents an M/M/1 queue, and we can compute the filter of  Q(t) w.r.t. the local time L. In the symmetric case, under a suitable rescaling, the previous characterization of the filter enables us to prove that the filters of  the rescaled queues converge to the filter of a reflected Brownian motion w.r.t. its local time. Finally we extend the last convergence result to the case when we can observe whether the queue is busy or idle.

Allanus Tsoi

Title: Linear Filtering under the Influence of Canonical Gaussian Noise

Abstract: We shall consider linear filtering when the noise term of the observation is a canonical Gaussian noise. The key tool is the semimartingale decomposition of the Gaussian process, which holds when the process is defined on the white noise space. We show that the mean square error is bounded by the solution of an ordinary differential equation.
   
Frederi Viens

Title: Branching and Interacting Particle Systems for Stochastic PDEs, Optimal Filtering of Stochastic Volatility, and Portfolio Optimization

Abstract: When stocks are assumed or observed to have stochastic volatility, this
quantity is notoriously difficult to estimate in practice; it is reasonable to work with the understanding that the volatility is partially observed via the discretely observed stock prices. In this incomplete information situation, we show that the problem of maximizing the expected
future wealth of a portfolio comprised of stock and a risk-free asset is closely related to a new type of optimal stochastic filtering for diffusion processes. A new particle method of del Moral, Jacod, and Protter for addressing this filtering issue is itself closely related to branching and interacting particle systems.

We will describe ongoing work which aims at using an explicit recursion relation for this new filter in order to derive some theoretical asymptotics of the maximum expected wealth, and of its corresponding optimal strategy, when the stochastic process driving the volatility is
fast-mean-reverting.

We will also describe a wide class of stochastic PDEs that can be approximated by a new type of branching and interacting particle systems, generalizing a known efficient particle method of Crisan, Gaines, and Lyons for classical non-linear stochastic filtering. We will discuss whether our particle system can be adapted to generalize the method of del
Moral, Jacod, and Protter, and whether it can be used for new stochastic volatility filtering problems going beyond the issue of discrete observations.

Hisao Watanabe

Title: Generalized solutions of linear parabolic partial differential equations

Abstract: Existence and uniquess theorems for parabolic stochsatic equations with space-time
white noise and related topics.

Jie Xiong

Title: Approximation to optimal filtering

Abstract: In this talk I will present two approaches in the approximation of nonlinear filtering.
One is to use the interacting particle system representation. Another is to deal with the SPDE
directly.   

George Yin

Title: Hybrid Filtering Problems: Discrete-time Model and Its Continuous-time Limit

Abstract: In this talk,  we study a class of hybrid discrete-time filtering problems that are modulated by Markov chains with nearly decomposable transition matrices. Aiming to reducing the complexity of the underlying problems, we show that a reduced system of filtering equations can be obtained by aggregating the states of each recurrent class into one state.  Weak convergence methods are used to obtain the desired limit system.

Tim Zajic

Title: Approximate propogation of the PHD by a particle system

Abstract: The probability hypothesis density (PHD) has been put forth as a means to perform tracking of multiple targets. The PHD is a first-order statistical moment of the multitarget system and, within a Bayesian framework, tracking is accomplished by computing appropriate prediction and observation update equations for the PHD. As in the single target case, the resulting equations are computationally intractable except in special cases. We report upon investigations into the implementation and performance of the approximate propagation of the PHD via a particle systems approach. In particular, a resampling step is performed and a novelty is the consideration of individual clusters of particles when performing this step.

Yong Zeng

Title: A micro-movement model of prices with Bayes estimation via filtering equation

Abstract: A general micro-movement model that describes the transactional price behavior is proposed. The model can be formulated as a filtering problem with counting process observations. The filtering equations are derived. A theorem on the convergence of
conditional expectation of the model is proven. A consistent recursive algorithm is constructed via the Markov chain approximation method to compute the approximate posterior and then the Bayes estimates. A simplified model and its recursive algorithm are constructed in detail. Simulations show that the computed Bayes estimates converge to their true values as expected. As an application, Bayes estimates for transaction prices of Microsoft are obtained and shown to outperform other estimates.

Xun Yu Zhou

Title: A Continuous-Time Markowitz World

Abstract: This talk reports some latest researches that attempt to faithfully extend Markowitz's Nobel-prize-winning mean--variance portfolio selection model to the continuous-time setting, using the approach of indefinite stochastic linear--quadratic control that has been developed very recently. Models with time-varying deterministic market parameters, random market parameters, and short-selling prohibition are respectively discussed. A related mean--variance hedging problem is also considered at last. In all the models explicit forms of efficient/optimal portfolios and efficient frontiers are presented. While many results in the continuous-time Markowitz world are analogous to their single-period counterparts, there are some results that are strikingly different.
   
   

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