Adaptive particle filters with applications

In this thesis, we study the problem of estimating a Markov chain X (signal) from its noisy partial information Y, when the transition probability kernel depends on some unknown parameters. Our goal is to compute the conditional distribution process $P${lcub}X_n|Y_n,…, Y₁{rcub}, referred to hereafter as the optimal filter . We rewrite the system, so that the kernel is now known but the uncertainty is transferred to the initial conditions. We show that, under certain conditions, the optimal filter will forget any erroneous initialization. So, starting with a ‘good’ prior distribution on the parameters, the filter will ultimately choose the correct value. This can also be seen as an asymptotic stability result, for systems that allow for non-ergodic signals. As a consequence, we can use Monte-Carlo particle filters adaptively, i.e. so that they simultaneously estimate the parameters and the signal. Finally, we give some examples of applications in finance (stochastic volatility estimation) and in blind source separation.