Signal sampling is the study of approximating continuous signals with discrete ones and vice versa. What if the signal you are trying to recover is random, but you have a model for that randomness, and can thus assign likelihoods (posterior probabilities even) to some sample paths.? Now you are sampling a stochastic process.
This is a particular take on a classic inverse problem that arises in many areas, framed how electrical engineers frame it. In the presence of observation noise they might also frame it in terms of state filtering/smoothing. There is a brief summary of that framing in Draščić (2016).
I am especially interested in this in the context of non-Gaussian-process models, because everything more or less works already for stationary gaussian processes. If you consider Lévy noise driving a linear SDE there is some work done, under the heading of sparse stochastic processes.
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