We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a new projection method for the optimal filter infinite-dimensional stochastic partial differential equation (SPDE), based on the direct \(L^2\) metric and on a family of normal mixtures. This results in a new finite-dimensional approximate filter based on the differential geometric approach to statistics. We compare this new filter to earlier projection methods based on the Hellinger distance/Fisher metric and exponential families, and compare the \(L^2\) mixture projection filter with a particle method with the same number of parameters, using the Levy metric. We discuss differences between projecting the SPDE for the normalized density, known as Kushner–Stratonovich equation, and the SPDE for the unnormalized density known as Zakai equation. We prove that for a simple choice of the mixture manifold the \(L^2\) mixture projection filter coincides with a Galerkin method, whereas for more general mixture manifolds the equivalence does not hold and the \(L^2\) mixture filter is more general. We study particular systems that may illustrate the advantages of this new filter over other algorithms when comparing outputs with the optimal filter. We finally consider a specific software design that is suited for a numerically efficient implementation of this filter and provide numerical examples. We leverage an algebraic ring structure by proving that in presence of a given structure in the system coefficients the key integrations needed to implement the new filter equations can be executed offline.

We explain how Itô stochastic differential equations (SDEs) on manifolds may be defined using 2-jets of smooth functions. We show how this relationship can be interpreted in terms of a convergent numerical scheme. We also show how jets can be used to derive graphical representations of Itô SDEs, and we show how jets can be used to derive the differential operators associated with SDEs in a coordinate-free manner. We relate jets to vector flows, giving a geometric interpretation of the Itô–Stratonovich transformation. We show how percentiles can be used to give an alternative coordinate-free interpretation of the coefficients of one-dimensional SDEs. We relate this to the jet approach. This allows us to interpret the coefficients of SDEs in terms of ‘fan diagrams’. In particular, the median of an SDE solution is associated with the drift of the SDE in Stratonovich form for small times.