Filtering in continuous time by least action

0. The reason I volunteered for the working group on continuous-time filtering problems was because it seemed to me that in the other areas probably the best you could hope to achieve was some improvement of existing ‘bread-and-butter’ SMC methods, whereas in continuous time there was much more scope for novelty. In particular, in continuous time you have all the tools of stochastic calculus, so it should be possible to use these before discretizing, as a way of getting further. These notes present one way of carrying out this approach; perhaps all which follows is well known, but from my brief study of the references we have been looking at, it looks like it could do with being better known if so! These notes are very preliminary, and will be expanded as time permits.