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Keynote Speaker Abstracts
Title: Revealing the detail in network measurements
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.
Title: Fractional Brownian Motion and Applications
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
Organizing Committee Abstracts
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.
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
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.
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.
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.
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.
Title: Stability of the Wonham filter
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.
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
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
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.
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.
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.
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.
Title: Markov chain approximations to nonlinear filtering equations
Abstract: We consider direct Markov chain approximations to
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
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.
Title: Filtering and prediction of aeroelastic dynamics- a statistical approach
Title: Stability and uniform particle approximation of nonlinear filters
Abstract: We propose a new approach to study the stability of
the discrete time optimal
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
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
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.
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
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
Title: Generalized solutions of linear parabolic partial differential equations
Abstract: Existence and uniquess theorems for parabolic
stochsatic equations with space-time
Title: Approximation to optimal filtering
Abstract: In this talk I will present two approaches in the
approximation of nonlinear filtering.
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.
Title: Approximate propogation of the PHD by a particle
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
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