Deep Learning for the Benes Filter

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.