The Follmer-Schweizer decomposition under incomplete information

n this paper we study the Föllmer–Schweizer decomposition of a square integrable random variable ξ with respect to a given semimartingale S under restricted information. Thanks to the relationship between this decomposition and that of the projection of ξ with respect to the given information flow, we characterize the integrand appearing in the Föllmer–Schweizer decomposition under partial information in the general case where ξ is not necessarily adapted to the available information level. For partially observable Markovian models where the dynamics of S depends on an unobservable stochastic factor X, we show how to compute the decomposition by means of filtering problems involving functions defined on an infinite-dimensional space. Moreover, in the case of a partially observed jump-diffusion model where X is described by a pure jump process taking values in a finite dimensional space, we compute explicitly the integrand in the Föllmer–Schweizer decomposition by working with finite dimensional filters. Finally, we use our achievements in a financial application where we compute the optimal hedging strategy under restricted information for a European put option and provide a comparison with that under complete information.