Mathematics of Information Technology and Complex Systems


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Research

Pod Filtering on a Manifold

This project involves the filtering of a pod sensor held within a bubble protruding from the regular exterior of an aircraft. While in flight, various forces are exerted on the pole such that the position of the pod is not stationary with respect to the aircraft. The pod's location is constrained to be on a manifold, based on the forces exerted on the pole. The goal of this project is to determine the conditional expectation of the pod's position given a series of observations regarding the pod's location in relation to the airplane.


Nonlinear filtering with fractional Brownian motion noise

We consider nonlinear filtering problem for the observation process corrupted by fractional Brownian motion noise. The signal process is assumed to be a Markov diffusion process. We obtain the Zakai equation and the Kushner-FKK equation in this setup. We also prove uniqueness of solution to these equations and robustness of the optimal filter.


Multi-target tracking problem

Consider three cases. Case I: the number of targets in a given region is known. Bayes formula and filtering equations are obtained, and the uniqueness of solution to these equations is discussed. We construct Markov chain approximations to implement the filter. Case II: the number of targets in given region is an unknown parameter. We try to apply statistical methods to estimate the number. We can use Bayesian model selection approach to choose the number. Case III: the number of targets in a given region is a counting process. We shall explore such complicated case.


Markov chain approximations to stochastic reaction diffusion equations

In the context of approximating stochastic reaction diffusion equations driven by Poisson noises, we extended the stochastic particle Markov chain approximation method developed by Thomas Kurtz, Ludwig Arnold, and Peter Kotelenez by using random time change arguments and by reducing state change rates. Our new algorithm is far more efficient. We established both the quenched law of large numbers for each fixed sample path of the Poisson source and the annealed law of large numbers while considering the Poisson source as a random medium for the Markov chains.

Markov chain approximations to filtering equations for reflecting diffusion processes

We have also found implementable approximate solutions to the Duncan-Mortensen-Zakai equations for reflecting diffusion processes by using the Markov chain approximations. Our Markov chains are constructed by employing a wide band observation noise approximation, dividing the signal state space into cells, and utilizing an emperical measure process estimation. This Markov chain method is demonstrated to outperform the branching particle filter and interacting particle filter methods on our simulated test problem, which is motivated by the fish farming industry.

Nonlinear filtering for diffusions in random environments

We conducted some research on nonlinear filtering for diffusions in random environments, which was motivated from tracking of a dinghy lost at sea. The motion of the dinghy will dramatically change due to random ocean surface wave propagation under bad weather conditions. The motion of the dinghy can be formally described by a stochastic differential equation with singular coefficients (the gradient of a multi-dimensional Levy's Brownian motion). By applying Dirichlet form theory, we have established some type of chaos expansion via multiple Wiener-Ito's integrals for the unnormalized pathspace measure-valued filtering processes. This opens the possibility of simulating filtering processes in random environments on a computer.

On generalizing the classical filtering equations to financial logstable models

It has been known for more than thirty years that there are solutions to the classical filtering equations under the mean-square finite energy condition. In a recent work, we gave a direct derivation of the Duncan-Mortensen-Zakai equation as well as the Kushner-Stratonovich equation under some conditions weaker than the usual finite energy condition by considering a martingale problem related to the unnormalized filter and using martingale representation theorem. For instance, our result allows use of filtering equations when the sensor function is linear and the signal (e.g., log-return of asset prices) has heavy tails like a Levy process.

Combined state and parameter estimation for partially observed nonlinear stochastic systems

Currently, we are doing some research on the combined state and parameter estimation for partially observed nonlinear stochastic systems. We plan to develop an implementable algorithm by combining parameter estimation and branching particle method, and prove convergence of the algorithm. Also, we are doing some research on multi-target tracking by using general filtering theory for measure-valued signal processes and stochastic particle Markov chain method.


Simulations:



  

  3 Object Tracking Simulation

  Description of 3 Object Tracking Simulation

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