A partially observed jump diffusion \(Z=(X_t,Y_t)_{t\in [0,T]}\) given by a stochastic differential equation driven by Wiener processes and Poisson martingale measures is considered when the coefficients of the equation satisfy appropriate Lipschitz and growth conditions. Under general conditions it is shown that the conditional density of the unobserved component \(X_t\) given the observations \((Y_s)_{s\in [0,t]}\) exists and belongs to \(L_p\) if the conditional density of \(X_0\) given \(Y_0\) exists and belongs to \(L_p\).

中文翻译：

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关于部分观察到的跳跃扩散 II：过滤密度

当方程的系数满足时，考虑由维纳过程和泊松鞅测度驱动的随机微分方程给出的部分观察到的跳跃扩散\ (Z=(X_t,Y_t)_{t\in [0,T]}\)适当的 Lipschitz 和生长条件。在一般条件下，表明给定观测值\((Y_s)_{s\in [0,t]}\) 的未观测分量 \ (X_t\)的条件密度存在且属于\(L_p\)，如果给定\ (Y_0\) 的 \ (X_0\)的条件密度存在并且属于\(L_p\)。