State and Parameter Estimation from Boundary–Crossings

In this paper we consider the problem of estimating the state, and identifying parameters of a diffusion pro-cess, when the only available information is the cross-ing times of a boundary. By using a partial differ-ential equation approach related with the computa-tion of boundary–crossing probabilities, we derive fi-nite dimensional reconstructors (filters) for the state and Feynman Kac type functional of the state. These are then used to compute maximum likelihood param-eter estimates of the drift coefficient of the diffusion.