On partially observed jump diffusions. II: The filtering density (Q6606151)

The filtering theory is not only an attractive field, but also a very useful one in various other theoretical or practical problems in mathematics. This paper is the second part of a series of papers on filtering for partially observed jump diffusions \(Z=(X_t,Y_t)_{t\in[0,T]}\) given by a stochastic differential equation driven by Wiener processes and Poisson martingale measures when the coefficients of the equation satisfy appropriate Lipschitz and growth conditions. The aim in the present paper is to show, under fairly general conditions, that if the conditional distribution of \(X_0\) given \(Y_0\) has a density \(\pi_0\), such that it is almost surely in \(L_p\) for some \(p\ge 2\), then \(X_t\) for every \(t\) has a conditional density \(\pi_t\) given \((Y_s)_{s\in[0,t]}\), which belongs also to \(L_p\), almost surely for all \(t\). To do this, the paper is organized in seven sections. After an introduction into the topic in the first section, in the second one the main result (Theorem 2.1) is formulated and in the last section the proof is given. In the other sections, besides results obtained in the first part of this series [\textit{F. Germ} and \textit{I. Gyöngy}, ``On partially observed jump diffusions I. The filtering equations'', Preprint, \url{arXiv:2205.08286}], the necessary results for for the proof given in the last section of the paper are obtained.