Optimal Filtering for ${\rm It}{math\hat{o}}$ -Stochastic Continuous-Time Systems With Multiple Delayed Measurements

This technical note focuses on optimal filtering for Itô stochastic continuous-time systems with multiple delayed measurements. Stochastic analysis and calculus of stochastic variables are the main tools employed for the analysis and design. For an Itô-stochastic system, its stochastic differential and integral have a significant place and are different from that for other stochastic systems owing to the Wiener or the Brownian process. In this technical note, an Itô stochastic continuous-time system with multiple delayed measurements is first reduced to an equivalent system with delay free measurements by solving stochastic equation via the non-singularity of the transition matrix instead of reorganizing the innovation. Then, based on the delay-free measurements the optimal filter is derived through calculation of the conditional expectation. It is should be stressed that the optimal filter follows directly from the manipulation of the performance. Finally, a short interest rate model in mathematical finance is chosen to demonstrate the design of the optimal filter via the approach proposed in the technical note.