State and parameter estimation from boundary-crossings

We consider the problem of estimating the state, and identifying parameters of a diffusion process, when the only available information is the crossing times of a boundary. By using a partial differential equation approach related with the computation of boundary-crossing probabilities, we derive finite dimensional reconstructors (filters) for the state and Feynman Kac type functionals of the state. These are then used to compute maximum likelihood parameter estimates of the drift coefficient of the diffusion.