Appl Math Optim 46:81–88 (2002)
2002 Springer-Verlag New York Inc.
Nonlinear Filtering with Fractional Brownian Motion
Department of Statistics, University of Michigan,
Ann Arbor, MI 48109-1092, USA
Abstract. Our objective is to study a nonlinear ﬁltering problem for the observa-
tion process perturbed by a Fractional Brownian Motion (FBM) with Hurst index
< H < 1. A reproducing kernel Hilbert space for the FBM is considered and a
“fractional” Zakai equation for the unnormalized optimal ﬁlter is derived.
Key Words. Nonlinear ﬁltering, Fractional Brownian motion, Reproducing kernel
Hilbert space, Stochastic differential equations.
AMS Classiﬁcation. 60H20, 60G15, 60G35.
The goal of nonlinear ﬁltering theory is to estimate a signal process (X
)(0 ≤ t ≤ T )
observed in the presence of an additive noise. Consider a complete probability space
(, F, P) and a family (F
of right-continuous increasing P-complete sub-σ -ﬁelds
of F. Let X = (X
, t ∈ [0, T ]) be a measurable, F
-adapted stochastic process with
values in a complete separable metric space S. The simplest model for the observation
) is given by
) ds + B
, 0 ≤ t ≤ T, (1)
) is a standard Brownian Motion (BM), and h ∈ C(S) satisﬁes
)(ω) ds < ∞ (P-a.s.). (2)
The classical model (1)–(2) can be written in the following form:
(ω) = F
(X (ω)) + B
(ω), t ∈ [0, T ], (3)