Parameter identification for partially observed diffusions

TL;DR: In this article, the identification problem of drift and dispersion parameters for a class of partially observed systems governed by Ito equations is formulated as a deterministic control problem in which the unnormalized conditional density (solution of the Zakai equation) is treated as the state, the unknown parameters as controls, and the likelihood ratio as the objective functional.

Abstract: In this paper, we consider the identification problem of drift and dispersion parameters for a class of partially observed systems governed by Ito equations. Using the pathwise description of the Zakai equation, we formulate the original identification problem as a deterministic control problem in which the unnormalized conditional density (solution of the Zakai equation) is treated as the state, the unknown parameters as controls, and the likelihood ratio as the objective functional. The question of existence of elements in the parameter set that maximize the likelihood ratio is discussed. Further, using variational arguments and the Gateaux differentiability of the unnormalized density on the parameter set, we obtain the necessary conditions for optimal identification.