Random-weight particle filtering of continuous time processes

It is possible to implement importance sampling, and particle filter algorithms, where the importance sampling weight is random. Such random-weight algorithms have been shown to be efficient for inference for a class of diffusion models, as they enable inference without any (time discretization) approximation of the underlying diffusion model. One difficulty of implementing such random-weight algorithms is the requirement to have weights that are positive with probability 1. We show how Wald's identity for martingales can be used to ensure positive weights. We apply this idea to analysis of diffusion models from high frequency data. For a class of diffusion models we show how to implement a particle filter, which uses all the information in the data, but whose computational cost is independent of the frequency of the data. We use the Wald identity to implement a random-weight particle filter for these models which avoids time discretization error.