Estimation and filtering of hidden semi Markov models, event based filters and stochastic control.

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The central objective of the work will be the derivation of new algorithms for signal processing, in particular the estimation of information hidden in noisy signals. This will build on previous contributions by the applicant. Three directions will be pursued. The first will develop new estimation algorithms for hidden semi Markov models, extending our previous work on hidden Markov models. The second will consider event based filtering problems when the observed process is received only when the hidden signal changes by more than a specified amount or when the signal process crosses certain levels. The third will discuss the optimal control of a noisily observed Markov chain using backward stochastic differential equations.***In addition to signal processing, speech processing and other areas, an important application of hidden Markov models has been to genome and protein sequencing. However, when modelling discrete sequences, for example in discrete time or in biological applications, if the hidden process is a Markov chain its (random) occupation time in any state is a geometrically distributed random variable. In many areas of applications, including queuing theory, reliability and maintenance, survival analysis, performance evaluation, biology, DNA analysis, and genome sequencing, it seems more general occupation times should be considered. This leads us to consider semi-Markov models. We shall consider 'hidden' semi-Markov models, that is situations where the semi-Markov chain is not observed directly but modulates a second, observed, process.  We shall develop for these models the results found in our book and previous publications.  ***In the digital world, continuous-time signals must be sampled. Traditionally, they are sampled uniformly in time. The term 'event-based sampling' can refer to the traditional uniform time step sampling, (sometimes called Riemann sampling), but it usually means that samples are taken in response to a priori defined events, such as when the signal changes by more than a specified amount, (send-on-delta), or when it crosses specified levels, (Lebesgue sampling). The objectives of the work will be to obtain new implementable filters. The theory of event-based sampling is more involved but there are many practical benefits including cheaper sensors, reduced communication costs and less data to process.  ***The third line of research will use our recent results on backward stochastic differential equations to investigate partially observed stochastic control problems. The adjoint process is described by a backward stochastic differential equation; for partially observed problems this is a backward stochastic partial differential equation. Initially we consider this problem for the control of a partially observed Markov chain where a system of backward stochastic ordinary differential equations will give criteria which determine an optimal control.