Optimal Filtration for Systems with Fractional Brownian Noises Udc 519.21

Teor Imovr.taMatem.Statist. Theor. Probability and Math. Statist. Vip. 72, 2005 No. 72, 2006, Pages 135–144 S 0094-9000(06)00671-5 Article electronically published on September 5, 2006 OPTIMAL FILTRATION FOR SYSTEMS WITH FRACTIONAL BROWNIAN NOISES UDC 519.21 S. V. POSASHKOV Abstract. We consider the problem of the optimal filtration for systems with the noise being a multivariate fractional Brownian motion. We partially solve the problem of the optimal filtration for nonlinear systems. The system of equations for the optimal filtration is obtained in the case of linear systems. 1. Introduction Systems governed by the fractional Brownian motion form a generalization of systems governed by the standard Brownian motion. We consider a real-valued process Xt treated as an input signal and a process Yt treated as an observation (the signal distorted by the noise). The processes Xt and Yt are defined by the following system of equations: X = η + t a(s, X ) ds + N t b (s) dV i,t∈ [0,T], (1.1) t 0 s i=1 0 i s t t ∈ Yt = ξ + 0 A(s, Xs) ds + 0 B(s) dVs,t[0,T], ∈ i i ∈ where V =(Vt,t [0,T]) and V =(Vt ,t [0,T]), i =1,...,N, are fractional ∈ 1 ∈ 1 Brownian motions with Hurst parameters H ( 2 , 1) and H1,...,HN [ 2 , 1), re- spectively. The coefficients A and a are continuous functions on [0,T] × R, while the coefficients B and bi,i =1,...,N, are continuous functions on [0,T] such that B van- ishes nowhere. The random initial data (η, ξ) do not depend on the fractional Brownian 1 N motions (V ,...,V ,V), and the pair (X, ξ) has a given distribution µ(X,ξ). Assume that Y is observed and one wants to estimate the process X. So we deal with the classical problem of filtration of a signal X at a moment t from the values of the process Y observed until the moment t. It is well known that the conditional expectation πt(X) with respect to the σ-algebra Yt = σ({Ys,s ∈ [0,t]}) is the solution of the problem of filtration (πt(X) is called the optimal filter for this problem). The solution of this problem is presented in books [1] and [2] for the case of noise being the standard Brownian motion. The classical theory of filtration is extended in [3] to the case of noise being a fractional Brownian motion. The filtration for linear systems with one-dimensional fractal Brownian noises is studied in [4]. This paper is devoted to the problem of the optimal filtration for systems with a multivariate fractional Brownian noise. The solution of this problem is based on the representation of the process X in terms of the corresponding family of semimartingales. 2000 Mathematics Subject Classification. Primary 60G35; Secondary 60G15, 60H05, 60J65. Key words and phrases. Problem of filtration, fractional Brownian motion. c 2006 American Mathematical Society 135 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 136 S. V. POSASHKOV The main properties of the fractional Brownian motion are briefly discussed in Sec- tion 1. These properties allow one to use classical methods of filtration theory. The problem of the optimal filtration is partially solved in Section 2 for system (1.1). The system of equations for the optimal filter is obtained in Section 3 for linear systems. 2. Main properties 2.1. Fractional Brownian motion. In what follows we assume that all random variables and processes are defined on a common stochastic basis (Ω, F, Ft, P) satisfying standard assumptions. The P-completion of the flow of σ-algebras generated by a stochastic process is taken as the natural flow of σ-algebras for this process. Let T>0 be fixed. A stochastic process V =(Vt,t ∈ [0,T]) is called normalized fractional Brownian motion with Hurst parameter H ∈ (1/2, 1) if (1) V is a Gaussian process with continuous trajectories and stationary increments; 2 2H ∈ (2) V0 =0,E Vt =0,andE Vt = t for all t [0,T]. The standard Brownian motion corresponds to the case of H =1/2. Any fractional Brownian motion is not a semimartingale; thus one cannot apply the classical theory of the stochastic integral for the fractional Brownian motion. Nevertheless the integration with respect to the fractional Brownian motion can be defined for some deterministic functions. Let the functions f and g be defined on [0,T]. Put T T 2H−2 (2.1) f,gH = H(2H − 1) f(s)g(t)|s − t| ds dt. 0 0 H Then the space L2 of classes of equivalent measurable functions f on [0,T] such that f,fH < ∞ is a Hilbert space equipped with the scalar product ·, ·H . The corre- −→ H spondence 1I[0,T ] VT can be extended to the isometry between L2 and the Gaussian ∈ T ∈ 2 space generated by random variables Vt, t [0,T]. The integral 0 f(s) dVs for f LH ∈ 2 is defined as the image of f for this isometry. For all f,g LH ,wehave T T (2.2) E f(s) dVs g(s) dVs = f,gH , 0 0 where f,g H is defined by (2.1). t ∈ 2 The process 0 f(s) dVs is defined for f LH as follows: t T f(s) dVs = 1[0,t)(s)f(s) dVs,t∈ [0,T]. 0 0 The process V is not a semimartingale. Nevertheless, an integral transform of V , t ∗ ∗ (2.3) Vt = kt (s) dVs, 0 ∗ is a martingale, where kt is a nonrandom kernel such that ∗ −1 1/2−H − 1/2−H − (2.4) kt (s)=kH s (t s) , 0 <s<t,kH =2HΓ(3/2 H)Γ(H +1/2). ThecovariancematrixofV ∗ is given by Γ(3/2 − H) (2.5) ψ (t)=V ∗ = t2−2H . H t 2HΓ(3 − 2H)Γ(H +1/2) Moreover the flow of σ-algebras generated by the process V ∗ coincides with the flow of σ-algebras generated by the process V up to null sets. The process V ∗ is called the fundamental martingale of a fractional Brownian motion V (see [5]). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OPTIMAL FILTRATION 137 More details concerning the fractional Brownian motion can be found in [5]. 2.2. The optimal filtration for nonlinear systems with fractional Brownian motion. The problem of filtration is considered in [3] for the case of a fractional Brow- nian noise. It is shown in [3] that equations for the optimal filter πt(X)=E[Xt|Yt], Yt = σ{Ys,s ∈ [0,t]}, can be obtained by a method similar to the classical method for the standard Brownian noise. Following this method, one uses the existence and properties of the fundamental martingale for the fractional Brownian motion and introduces the process Z as follows: t t ∗ −1 (2.6) Zt = kt (s)B (s) dYs. 0 The process Zt is a P-semimartingale. It is proved in [3] that the σ-algebras ξ,Z F = σ {ξ; Zs,s∈ [0,t]} and Yt coincide for all t ∈ [0,T]uptoP-null sets. The following theorem contains equations for the optimal filter of a special semimartingale. Theorem 2.1 (Theorem 2 [3]). Let ξ =(ξt,t ∈ [0,T]) be an ((Ft), P)-semimartingale given by t ξt = ξ0 + βs ds + mt,t∈ [0,T], 0 where E[ξ2] < ∞, E T β2 dt < ∞,andm =(m ,t ∈ [0,T]) is a square integrable 0 0 t t F ∗ t ∈ (( t), P)-martingale such that m, V t = 0 λs dψH (s), t [0,T]. Then the process π(ξ)=πt(ξ), t ∈ [0,T], satisfies the stochastic differential equation t t πt(ξ)=π(ξ0)+ πs(β) ds + [πs(λ)+πs(ξq(X)) − πs(ξ)πs(q(X))] dvs, (2.7) 0 0 t ∈ [0,T], where t d ∗ −1 ∈ (2.8) qt(X)= kt (s)B (s)a(s, Xs) ds, t [0,T], dψH (t) 0 and t (2.9) vt = Zt − πs(q(X)) dψH (s). 0 The process v plays the same role as the innovation process in the classical model. The process v is a Gaussian (Yt)-martingale with the correlation function ψH .Each Y square integrable ( t)-martingale Mt with respect to the measure P with M0 =0can be represented as M = t P dv , t ∈ [0,T], where P =(P ,t∈ [0,T]) is a (Y )-adapted t 0 s s t t T 2 ∞ process such that E 0 Pt dψH (t) < . We assume that the noises V 1,...,VN in the definition of the process X depend ∗ ∗ i j on each other in the sense that we know the functions ψi,j (t)=V ,V t,where ∗ i V , i =1,...,N, are the corresponding fundamental martingales for Vi, i =1,...,N, i represented in the form of (2.3) with Hi instead of H for any V in the definition of ∗ ∈ 1 kt (s). We assume that ψi,j C [0,T], i =1,...,N. We seek a solution represented 2 as the optimal filter for the functional φ(Xt) such that φ ∈ C [0,T]. If the noises in the definition (1.1) of the input signal X are standard Brownian motions, then the Itô formula applied to the representation of the semimartingale φ(Xt) and Theorem 2.1 solve the filtration problem under consideration (see Chapter 5 of [3]).

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