Filtering and control with information increasing (Q1594515)

A filtering problem is considered where the observation filtration is enlarged with a future information. \textit{I. Pikovsky} and \textit{I. Karatzas} [Adv. Appl. Probab. 28, No. 4, 1095-1122 (1996; Zbl 0867.90013)] studied this problem under complete observation. In the present paper, the observation is partial, i.e., a diffusion process \((x_t)\) is studied, but what is observed is a randomized function \((y_t)\). In the linear case, the filter equation is obtained using the technique of enlargement of filtration developed in ``Grossissements de filtrations: exemples et applications'' (ed. by Th. Jeulin and M. Yor, 1985). The enlarged filtering problem remains linear and is dealt with Kalman-Bucy filtering. Next, the associated linear regulator problem with increased information is considered and solved. Such a problem can arise in the financial market where an investor has received an information about the future, through the knowledge of a random variable adapted to the \(\sigma\)-field \({\mathcal F}_1\), if \(1\) is the time when he wants to optimize his profit.