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We use jets as a natural language to express geometric properties of SDEs.We show how jets can lead to intuitive representations of It\^o SDEs, including three different types of drawings.We explain that the mainstream choice of Fisk-Stratonovich-Mc Shane calculus for stochastic differential geometry is not necessary.We give a new geometric interpretation of the It\^o--Stratonovich transformation in terms of the 2-jets of curves induced by consecutive vector flows.You can contact me by following this link, and please be patient if I reply after a while or do not reply, the number of messages I receive is out of control, and goes on top of the huge amount of emails I receive in my professional accounts.Research papers in stochastic nonlinear filtering, probability, statistics, information geometry, and in several areas of quantitative finance.In the present paper, we construct new classes of scalar nonlinear filtering problems admitting finite--dimensional filters.

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We compare this method to earlier projection methods based on the Hellinger distance/Fisher metric and exponential families, and we compare the L2 mixture projection filter with a particle method with the same number of parameters.We discuss the forward Kolmogorov equation and the backward diffusion operator in geometric terms.In the one-dimensional case we consider percentiles of the solutions of the SDE and their properties.We construct a drift for the state equation such that the resulting nonlinear filtering problem admits a finite--dimensional filter evolving in the prescribed exponential family augmented by the observaton function and its square.We explain how Ito Stochastic Differential Equations on manifolds may be defined as 2-jets of curves and show how this relationship can be interpreted in terms of a convergent numerical scheme.We explain how this apparatus can be used for the nonlinear filtering problem, in relationship also to earlier projection methods based on the Fisher metric.