Deep Learning for the Benes Filter. (arXiv:2203.05561v1 [stat.ML])

The Benes filter is a well-known continuous-time stochastic filtering model
in one dimension that has the advantage of being explicitly solvable. From an
evolution equation point of view, the Benes filter is also the solution of the
filtering equations given a particular set of coefficient functions. In
general, the filtering stochastic partial differential equations (SPDE) arise
as the evolution equations for the conditional distribution of an underlying
signal given partial, and possibly noisy, observations. Their numerical
approximation presents a central issue for theoreticians and practitioners
alike, who are actively seeking accurate and fast methods, especially for such
high-dimensional settings as numerical weather prediction, for example. In this
paper we present a brief study of a new numerical method based on the mesh-free
neural network representation of the density of the solution of the Benes model
achieved by deep learning. Based on the classical SPDE splitting method, our
algorithm includes a recursive normalisation procedure to recover the
normalised conditional distribution of the signal process. Within the
analytically tractable setting of the Benes filter, we discuss the role of
nonlinearity in the filtering model equations for the choice of the domain of
the neural network. Further we present the first study of the neural network
method with an adaptive domain for the Benes model.



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