The general formulas for smoothing and prediction problems of mixed-type observations

The filtering, smoothing, and prediction problems for mixed-type states and observations (continuous plus discontinuous) are considered. Starting with the general models of stochastic systems (mixed-type of states and observations), the normalized equations of general stochastic equations using semimartingale decomposition techniques are derived. It is also shown that the corresponding unnormalized equations could be derived in two different ways (indirect and direct derivation). The same derivation methods result in similar structure in forms. Thus the optimal estimation problems are unified in these frameworks