Mixture of conjugate prior distributions and large deviations for level crossing probabilities

In this paper we preset asymptotic estimates of level crossing probabilities from a Bayesian point of view, based on large deviation. For the Bayesian analysis we chose a finite mixture of conjugate prior distributions to model the uncertenty of the unknow parameters of two classes of stochastic process considered: the brownian motion and the compound Poisson process with upward jumps and negative drift. The estimate of level crossing probabilities are derived as a consequence of large deviation principles of posterior distributions.