Optimal filtration in systems with noise modeled by a polynomial of fractional Brownian motion. (Ukrainian, English) Zbl 1119.60030

Mishura, Yu. S.; Posashkov, S. V.

Teor. Jmovirn. Mat. Stat. 73, 104-111 (2005); translation in Theory Probab. Math. Stat., Vol 73, 117-124 (2006).

The author deals with the signal process \(X_t\) and the noise process \(Y_t\) on \([0,T]\) described by the following stochastic equations:
\[ X_t=\eta+\int_0^ta(s,X_s)\,ds+\int_0^tp(W_s)\,dW_s, \]
\[ Y_t=\xi+\int_0^tA(s,X_s)\,ds+\int_0^tB(s)\,dV_s, \]
where \(V_s, W_s\) are fractional Brownian motions with the Hurst parameter \(H\in(0.5,1)\), \(A(s,x)\) and \(a(s,x)\) are functions continuous on \([0,T]\times\mathbb R\), \(p(x)= \sum_{i=1}^Na_ix^i\) is a polynomial, \(B(s)\) is a continuous function on \([0,T]\). The optimal filtering problem is considered. A system of equations for the optimal filtering is obtained based on the representation of fractional Brownian motion in terms of standard Brownian motion.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60G35	Signal detection and filtering (aspects of stochastic processes)
60G15	Gaussian processes
60H05	Stochastic integrals
93E11	Filtering in stochastic control theory

Keywords:

filtering; fractional Brownian motion; Hurst parameter