A NOTE ON STATE ESTIMATION FROM DOUBLY STOCHASTIC POINT….pdfVIP

STUDIA UNIV. “BABES¸–BOLYAI”, MATHEMATICA, Volume XLVI, Number 1, March 2001 A NOTE ON STATE ESTIMATION FROM DOUBLY STOCHASTIC POINT PROCESS OBSERVATION DANG PHUOC HUY AND TRAN JUNG THAO 0. Introduction In this note we study a state estimation of a Markovian semimartingale from a doubly stochastic point process observation. All stochastic processes below are supposed to be defined on a filtered prob- ability space (Ω, F , (F ) , P ) where (F ) is a filtration satisfying usual conditions. t t≥0 t Consider a state estimation problem where the signal process is a real-valued continuous semimartingale X that is also a Markov process given by Xt = X0 + t H ds + B , t ∈ R+ , (0.1) s t 0 where Ht is a continuous process and Bt is a standard Brownian motion, and the observation is a doubly stochastic point process N driven by X : N is a point t t t process of intensity λ = λ(X ) where λ is a nonnegative boolean function. t t Denote by Zu the process exp(iuX ). We want to investigate the best state t t estimation π (Zu ) = E [Zu |FN ] (0.2) t t t t where F N is the natural filtration of the process N i.e. F N = σ (N , s ≤ t). In the t