Filtering of a Markov jump process with counting observations
This paper deals with filtering of an \(R^d\)-valued Markov jump process \((X_s)\) given the observation process \(N_t\) that counts the jumps of \((X_s)\) up to time \(t\). The authors write the Kushner-Stratonovich (KS) equation for the normalized filter \(\pi_t\), and derive the Zakai equation for the unnormalized filter \(\sigma_t\), using the Girsanov theorem for point processes. They prove weak and strong uniqueness for the solution of (KS), under assumptions on the transition function \(\mu (x,dy)\) of \((X_t)\) and on the deterministic function \(\lambda\) that defines the intensity \(\lambda (X_t)\) of \((N_t)\). Strong uniqueness for the solution of the Zakai equation is shown to imply strong uniqueness for the solution of the Kushner-Stratonovich equation, under additional hypotheses on \(\mu (x,dy)\) and \(\lambda\). Finally, strong uniqueness for the solution of the Zakai equation is proved when \((X_t)\) takes values in a countable state space.
scientific article; zbMATH DE number 1485110