A Particle Implementation of the Recursive Mle for Partially Observed Diffusions

In this paper, the problem of identifying a hidden Markov model (HMM) with general state space, e.g. a partially observed diffusion process, is considered. A particle implementation of the recursive maximum likelihood estimator for a parameter in the transition kernel of the Markov chain is presented. The key assumption is that the derivative of the transition kernel w.r.t. the parameter has a probabilistic interpretation, suitable for Monte Carlo simulation. Examples are given to show that this assumption is satisfied in quite general situations. As a result, the linear tangent filter, i.e. the derivative of the filter w.r.t. the parameter, is absolutely continuous w.r.t. the filter and the idea is to jointly approximate the (prediction) filter and its derivative with the empirical probability distribution and with a weighted empirical distribution associated with the same and unique particle system. Application to the identification of a stochastic volatility model is presented.