Optimal Filtering for Time-Varying Stochastic System With Delay and Multiplicative Noise

This paper pays attention to the problem of optimal filtering for linear continuous time-varying It ô stochastic system with multiple delayed measurements and multiplicative noise. The stochastic analysis and calculus of stochastic variables are employed to analyze and design the optimal filtering. For the It ô stochastic continuous-time system with multiple delayed measurements and multiplicative noise, a delay is first transferred from the state to integral term with the Brownian motion in measurements by solving the stochastic equation of multiplicative noise. Then, based on the delay-free state in measurements and the independent increment characteristics of the Brownian motion the optimal filter is derived through the calculation of the conditional expectation. It should be stressed that the optimal filter follows directly from the manipulation of the performance. Finally, a price-volatility feedback rate model in mathematical finance is chosen to demonstrate the design of the optimal filter via the proposed approach in this paper.