Abstract
We consider the problem of filtering an unseen Markov chain from noisy observations, in the presence of uncertainty regarding the parameters of the processes involved. Using the theory of nonlinear expectations, we describe the uncertainty in terms of a penalty function, which can be propagated forward in time in the place of the filter. We also investigate a simple control problem in this context.
Highlights
Filtering is a common problem in many applications
We consider the problem of filtering an unseen Markov chain from noisy observations, in the presence of uncertainty regarding the parameters of the processes involved
We consider a simple setting in discrete time, where the underlying process is a finite-state Markov chain
Summary
Filtering is a common problem in many applications. The essential concept is that there is an unseen Markov process, which influences the state of some observed process, and our task is to approximate the state of the unseen process using a form of Bayes’ theorem. Under the assumption that the underlying process is a finite-state Markov chain, a general formula to calculate the filter can be obtained (the Wonham filter Wonham (1965)) These results are well known, in both discrete and continuous time (see Bain and Crisan (2009) or Cohen and Elliott (2015) Chapter 21 for further general discussion). We are interested in allowing the level of uncertainty in the filtered state to be endogenous to the filtering problem, arising from the uncertainty in parameter estimates and process dynamics We model this uncertainty in a general manner, using the theory of nonlinear expectations, and concern ourselves with a description of uncertainty for which explicit calculations can be carried out, and which can be motivated by considering statistical estimation of parameters. We apply this to building a dynamically consistent expectation for random variables based on future states, and to a general control problem, with learning, under uncertainty
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