Filtering Using Nonlinear Expectations
Technical Report,05 Jun 2015,04 Mar 2016
The University of Adelaide Adelaide Australia
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Filtering is the recursive estimation of signals observed in noise, a topic of importance in signal processing and other fields which involve the extraction of information from noisy data. The proposal was to investigate filtering using nonlinear, in particular sublinear, expectations. When there is uncertainty about the correct probability describing the noise, a supremum over a class of possible probabilities can be considered, giving a sublinear expectation. In continuous time Peng introduced a G-expectation which is related to a modified Brownian motion given by the solution of a nonlinear heat equation. Whilst being an interesting concept this definition involves difficult technicalities. An alternative definition of G-Brownian motion uses ideas from stochastic control and considers a supremum over a set of diffusion coefficients. The first paper completed gives a solution to estimating a Markov chain observed in Gaussian noise when the variance of the noise is unkown. This paper is accepted for the IEEE Transactions on Automatic Control, an A journal. The second paper considers the related problem in continuous time. The methods used include stochastic control when the control parameter influences the diffusion coefficients and Nash equilibria from game theory, because the different components of the diffusion can be considered as being controlled by different players. This paper is under a second review for the SIAM Journal on Control and Optimization, an A journal. A short third paper discusses how to estimate a change in the transition dynamics of a noisily observed Markov chain. The change point time is hidden in a hidden Markov chain, so a second level of discovery is involved. This paper is accepted for Communications in Stochastic Analysis.
- Statistics and Probability
- Operations Research