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 UDC 519.21

S. V. POSASHKOV

Abstract. We consider the problem of the optimal filtration for systems with the being a multivariate fractional . We partially solve the prob- lem 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 clas- sical 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

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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 vari- ables 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 sto- chastic 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

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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 prop- erties 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 semimartin- gale.

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 Itˆo 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]). On the other hand, X

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is not a semimartingale. To avoid this problem we apply an approach of [4] based on the representation of the integral with respect to the fractional Brownian motion in terms of the integral of the corresponding kernel with respect to the fundamental martingale of this fractional Brownian motion. Lemma 2.2 (Lemma4[4]). Let V be a fractional Brownian motion with the Hurst parameter H ∈ (1/2, 1) and let V ∗ be the corresponding Gaussian martingale given by the integral transform (2.3).LetB ∈ L2 and let the function KB(t, s) be defined by  H H t B − H−1/2 − H−3/2 ≤ ≤ ≤ KH (t, s)=H(2H 1) B(r)r (r s) dr, 0 s t T. s Then, for t ∈ [0,T], the function KB(t, ·) belongs to the space L2([0,t],µ), H

µ(A)= dψH (s),A⊂ [0,t], A and   t t B ∗ ∈ (2.10) B(s) dVs = KH (t, s) dVs ,t[0,T], P -a.s. 0 0 t ≤ ≤ Using Lemma 2.2 we introduce a family of semimartingales Xs,0 s t, as follows:   s N s ∗ (2.11) Xt = η + a(u, X ) du + Kbi (t, u) dV i ,s∈ [0,t]. s u Hi u 0 i=1 0 t t ∈ It is obvious that Xt = Xt.TheItˆo formula implies that the process φ(Xs), s [0,t], is a semimartingale represented as follows:   s N s  ∗ (2.12) φ(Xt)=φ(η)+ L (φ(Xt )) du + φ (Xt )Kbi (t, u) dV i ,s∈ [0,s], s u u u Hi u 0 i=1 0

where Lu(φ(·)) is such that  N   bi bj dψi,j (u) Lu(φ(·)) = a(u, Xu)φ (·)+ φ (·) K (t, u) K (t, u) . Hi Hj du i,j=1 The derivatives dψ (u) i,j ,i,j=1,...,N, i= j, du are well defined by assumption. Moreover, it follows from (2.5) that

dψH (u) (2 − 2Hi)Γ(3/2 − Hi) − i = u1 2Hi du 2HiΓ(3 − 2Hi)Γ(Hi +1/2) for i = j. Applying Theorem 2.1 to the semimartingale φ(Xt) we obtain the equation for the optimal filter π (φ(Xt)): s  s t t πs(φ(X )) = π(φ(η)) + πu Lu(φ(X )) du  0 (2.13) s  t t + πu φ(X )q − πu φ(X ) πu(q) dvu. 0 t Putting s = t and using the equality X = Xt,wegettheequationforthefilterπt(φ(X)): t  t t πt(φ(X)) = π(φ(η)) + πs Ls(φ(X )) ds (2.14)  0 t  t t + πs φ(X )q − πs φ(X ) πs(q) dvs. 0

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It is proved in [3] that there exists a probability measure P˜ such that the fractional Brownian motion V˜ considered with respect to the measure P˜ generates the same fil- tration as the process Y and is independent of the input process X. The process Y considered with respect to this measure satisfies the equation  t Yt = ξ + B(s) dV˜s,t∈ [0,T]. 0 This result can be derived from a Girsanov type theorem for the fractional Brownian motion (see, for example, [5]). The relation between P˜ and the measure P for which Y satisfies system (1.1) is as follows: ˜ (2.15) P =ΛT P, where    t t − 1 2 ∈ (2.16) Λt =exp qs(X) dZs qs (X) dψH (s) ,t[0,T]. 0 2 0 t ˜ Then σ(φ(X )) := E[φ(Xt)Λt|Yt] is called the nonnormalized filter. Its name is ex- t plained by the dependence between σ(φ(X )) and the optimal filter πt(φ(X)) known as the Kallianpur–Striebel formula:

σt(φ(X)) (2.17) πt(φ(X)) = . σt(1) ˜ The equation for Λ˜t = σt(1) = E[Λt|Yt] is obtained in [3]; namely,  t (2.18) Λ˜ t := 1 + Λ˜ sπs(q) dZs,t∈ [0,T], P -a.s. 0 Using this equation and (2.13) we obtain an analog of the classical Zakai equation for the nonnormalized filter σt in the case of several fractional Brownian noises distorting the input signal.

Theorem 2.3. Let σt be a nonnormalized filter for the model (1.1).Then   t t t t (2.19) σt(φ(X)) = σ0(φ(η))+ σs Ls(φ(X )) ds+ σs φ(X )q dZs,t∈ [0,T], 0 0 for any twice continuously differentiable function φ. Proof. The Kallianpur–Striebel formula implies that

t t (2.20) σs φ(X ) = Λ˜ sπs φ(X ) . t Representation (2.13) for the optimal filter πs(φ(X )), equation (2.18), and the Itˆofor- mula for differentials imply that   dσ φ(Xt) = Λ˜ dπ φ(Xt) + π φ(Xt) dΛ˜ + d Λ˜,π φ(Xt) s s s s  s s = Λ˜ π L φ(Xt) ds + Λ˜ π φ(Xtq) − π φ(Xt) π (q) dv s s s s s s s s (2.21) + π φ(Xt) Λ˜ π (q) dv + π φ(Xt) Λ˜ π2(q) dψ (s) s  s s s s s s H + Λ˜ π (q) π φ(Xt)q − π φ(Xt)π (q) dψ (s) s s s s s H t t = Λ˜ sπs Lsφ(X ) dt + Λ˜ sπs φ(X )q [dvs + πs(q) dψH (s)] . Now we apply (2.20) and (2.9) again and get

t t t (2.22) dσs φ(X ) = σs Ls(φ(X )) ds + σs φ(X )q dZt. Putting s = t and integrating both sides of equality (2.22) we obtain (2.19). 

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i Remark 2.4. If some constants Hi are equal to 1/2, then the corresponding processes V ∗ are standard Brownian motions and we put V i ≡ V i in this case. The integral terms do not change for these observations V i, and the proof is similar to that of Theorem 2.3. Equations (2.14) and (2.19) imply that the solution of the problem of filtration requires an equation to determine πt(q). We consider this problem in the next section for linear systems and obtain a system of equations for the optimal filter. 2.3. The optimal filtration in Gaussian linear systems with fractional Brow- nian noises. We assume in this section that the input signal Xt, t ∈ [0,T], and the process Yt, t ∈ [0,T], are such that   dX = a(t)X dt + N b (t) dV i,t∈ [0,T],X = η, (2.23) t t i=1 i t 0 dYt = A(t)Xt dt + B(t) dVt,t∈ [0,T],Y0 = ξ.

Similarly to the general case (see system of equations (1.1)), Vi, i =1,...,N,andV are fractional Brownian motions with the Hurst parameters Hi ∈ [1/2, 1), i =1,...,N, and H ∈ (1/2, 1). We also assume that, for all i =1,...,N, the processes Vi and V are independent, while V , i =1,...,N, depend on each other in the sense that we i ∗ ∗ ∗ i j i know the functions ψi,j (t)=V ,V t,whereV , i =1,...,N, are corresponding fundamental martingales for Vi, i =1,...,N, given by (2.3). The coefficients A, B, a, and bi, i =1,...,N, are bounded nonrandom functions acting from [0,T]toR.The coefficient B does not vanish and B−1 is a bounded function. The pair (η, ξ)doesnot depend on (V1,...,VN ,V)andπ0 = E[η|ξ] has the Gaussian distribution with mean m0 and variance γ0. Then the system (2.23) has a unique solution with a Gaussian distribution. Nevertheless, it is neither a Markov process nor a semimartingale. The problem is to filtrate values of X at the moment t by using values Y observed up to the moment t. The solution of this problem is the conditional distribution of Xt with respect to the σ-algebra Yt = σ{Ys,s∈ [0,t]} generated by the process Y . Since the processes in (2.23) are Gaussian, the distribution of the optimal Gaussian filter is Gaussian, too. To completely define the filter one needs to know the mean value 2 πt(X) and the variance γXX(t)=E(Xt − πt(X)) of errors of filtration. To obtain necessary equations we apply the technique proposed in [4]. By ζ we denote the two-dimensional process (X, q) where X is defined in system (2.23), while q is given by (2.8). Our current aim is to get an equation for the mean value πt(ζ) and covariance  matrix γζζ(t)=E(ζt − πt(ζ))(ζt − πt(ζ)) . The processes ζ and Y are jointly Gaussian; however, ζ is not a semimartingale. As in the preceding section we apply Lemma 2.2 to solve the problem. First we introduce the following nonrandom functions:  d t A p(t, s)= k∗(r) (r) dr, dψ (t) t B H s t d ∗ A Q(i, t, s)= k (r)Kbi (r, s) (r) dr. t Hi dψH (t) s B ∈ t t t  ∈ Given a fixed t [0,T] we define a Gaussian semimartingale ζs =((Xs,qs) , s [0,t]), as follows:  N  s s ∗ (2.24) Xt = η + a(u)X du + Kbi (t, u) dV i , s u Hi u 0 i=1 0  N  s s ∗ t i (2.25) qs = p(t, 0)η + a(u)p(t, u)Xu du + Q(i, t, u) dVu . 0 i=1 0

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Consider the following family of 2 × 2 covariance matrices:

t − t −  ≤ ≤ ≤ (2.26) Γζζ(t, s)=E ζs πs(ζ ) (ζs πs(ζ)) , 0 s t T. Let ε˜ =(0, 1);˜p(t, u)=(1,p(t, u)); (2.27)  ˜b(i, t, u)= Kbi (t, u),Q(i, t, u) , 0 ≤ u ≤ t; Hi and  10 (2.28)a ˜(t, u)=a(u)(˜p(t, u); 0)¯ = a(u) , 0 ≤ u ≤ t. p(t, u)0 We need the following auxiliary result. Lemma 2.5. Let x, y,andz be Gaussian random variables such that

(2.29) E[x|Ys]=0, E[y|Ys]=0, E[z|Ys]=0 P -a.s. for some s ∈ [0,T].Then

E[xyz|Ys]=0 P -a.s.

Proof. Since x, y,andz are Gaussian and the σ-algebra Ys is generated by the Gaussian process Y , condition (2.29) implies that x, y,andz do not depend on Ys. Therefore the product xyz does not depend on Ys either. Thus the conditional expectation E[xyz|Ys] is equal to the unconditional mean value E[xyz]. To evaluate E[xyz] we use the following properties of Gaussian distributions: y = ρx + α, z = x + β, where ρ =cov(x, y),  =cov(x, z), and α and β do not depend on x.Wehave E[xyz]=E[x(ρx + α)z]=E[ρx2z]+E[xαz]=E[ρx2(x + β)] + E[αx(x + β)] = E[ρx3]+E[ρx2β]+E[x2α]+E[xαβ] = E[ρx]+E[β] E[ρx2]+E[α] E[x2]+E[x] E[ρ]=0, since E[x]=0,E[x3]=0,E[α] = 0, and E[β]=0. 

Now we obtain the equation for the filtration of the process ζ. Theorem 2.6. Let (X, Y ) be a solution of system (2.23) and let q and Z be the pro- cesses defined by (2.8) and (2.6), respectively. Then the conditional expectation πt(ζ) and covariance matrix γζζ(t) of the errors of filtration satisfy the equations   t t (2.30) πt(ζ)=˜p(t, 0)m0 + a(s)˜p(t, s)πs(X) ds + Γζζ(t, s)˜ε [dZs − πs(q) dψH (s)] , 0 0 (2.31) γζζ(t)=Γζζ(t, t),

where the matrix Γζζ(t, s), 0 ≤ s ≤ t ≤ T , is such that  s     Γζζ(t, s)=˜p(t, 0)˜p (s, 0)γ0 + a˜(t, u)Γζζ(s, u)+Γζζ(t, u)˜a (s, u) du 0  N s ˜ ˜ (2.32) + b(i, t, u)b (j, s, u) dψi,j (u) 0 i,j=1 s −   Γζζ(t, u)˜εε˜ Γζζ(s, u) dψH (u). 0

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Proof. For all t, r ∈ [0,T], the vector (ζt,ζr,Y) is Gaussian. Thus the conditional co- t r variance for (ζ ,ζ ) with respect to Ys, s ∈ [0,t∧ r], is nonrandom. It is obvious that t ζt = ζ. t To derive equation (2.30) we use (2.27) and (2.28) and represent the semimartingale ζs in the vector form

 N  s s ∗ t ˜ i (2.33) ζs =˜p(t, 0)m0 + a(u)˜p(t, u)Xu du + b(i, t, u) dVu . 0 i=1 0

Applying Theorem 2.1 we get   s s  t t t (2.34) πs(ζ )=˜p(t, 0)m0 + a(u)˜p(t, u)πu(X) du + πu(ζ q) − πu(ζ )πu(q) dvu, 0 0

since the noises in the input signal and those in the observation are independent. It follows from definition (2.26) of the matrix Γζζ(t, s)that   s s t (2.35) πs(ζ )=˜p(t, 0)m0 + a(u)˜p(t, u)πu(X) du + Γζζ(t, u)˜εdvu. 0 0

t Since ζt = ζt, equation (2.30) follows. t r  To derive the equation for the covariance matrix Γζζ(t, s)wenotethatζ (ζ ) is a semimartingale on the interval [0,t∧ r] and rewrite equality (2.33) as follows:

 N  s s ∗ t ˜ i (2.36) ζs =˜p(t, 0)m0 + a˜(t, u)ζu du + b(i, t, u) dVu . 0 i=1 0

The Itˆo formula gives us the representation for ζt(ζr):  s  t r   2 r  t   ζs(ζs ) =˜p(t, 0)˜p (r, 0)η + a˜(t, u)ζu(ζu) + ζu(ζu) a˜ (r, u) du 0 N    s ∗ ˜ r  t ˜ i (2.37) + b(i, t, u)(ζu) + ζ(u)b (i, r, u) dVu i=1 0  N s ˜ ˜ + b(i, t, u)b (j, r, u) dψi,j (u). i,j=1 0

Applying Theorem 2.1, we get  s  t r   2 r  t   πs(ζ (ζ ) )=˜p(t, 0)˜p (t, 0) E η |ξ + a˜(t, u)πu(ζ(ζ ) )+πs(ζ ζ )˜a (r, u) du 0  N s ˜ ˜ (2.38) + b(i, t, u)b (j, r, u)dψ(i,j)(u) 0 i,j =1 s r  r  + [πu(ζ(ζ ) q) − πu(ζ(ζ ) )πu(q)] dvu. 0

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t r  For all fixed t and r, the process πs(ζ )πt(ζ ) ,0≤ s ≤ r ∧ t, is a semimartingale. t r  Applying the Itˆo formula we get the representation for πs(ζ )πt(ζ ) :

π(ζt)π(ζr) =˜p(t, 0)˜p(r, 0)m2  0 s  r  t   + a˜(t, u)πu(ζ)πu(ζ ) + πu(ζ )πu(ζ )˜a (r, u) du 0 (2.39) s r    + Γζζ(t, u)˜επu(ζ ) + πu(ζ )˜ε Γζζ(r, u) dvu 0 s   + Γζζ(t, u)˜εε˜ Γζζ(r, u) dψH (u). 0 Using (2.38) and (2.39) we obtain

Γ(t, r, s)=π (ζt(ζr)) − π (ζt)π (ζr) s s  t s   =˜p(t, 0)˜p (r, 0)γ0 + [˜a(t, u)Γ(u, r, u)+Γ(t, u, u)˜a (r, u)] du 0  N s ˜ ˜ + b(i, t, u)b (j, r, u) dψi,j (u) 0 (2.40) i,j =1 s −   Γζζ(t, u)˜εε˜ Γζζ(r, u) dψH (u) 0 s r  r  r  + πu(ζ(ζ ) q) − πu(ζ(ζ ) )πu(q) − Γζζ(t, u)˜επu(ζ ) 0    − πu(ζ )˜ε Γζζ(r, u) dvu,

t −  r − r  ∈ ∧ where Γ(t, r, s)=E(ζs πs(ζ ))(ζs πs(ζ )) , s [0,t r]. The definitions of π and Γ together with Lemma 2.5 imply that

r  r  r  πu ζ(ζ ) q − πu ζ(ζ ) πu(q) − Γζζ(t, u)˜επu(ζ ) (2.41)   − πu(ζ )˜ε Γζζ(r, u)=0 P-a.s, 0 ≤ u ≤ s,

since all the processes are Gaussian. Putting s = r in (2.40) we make sure that equa-  tion (2.32) holds, since Γζζ(t, s)=Γ(t, s, s)=Γ(s, t, s). 

It is obvious that πt(X) is the first component of the vector πt(ξ), while γXX(t)isthe (1, 1)-entry of the matrix γζζ(t).

i Remark 2.7. Similarly to the general case, the terms corresponding to V with Hi =1/2 do not change in the definition of the process Xt.

3. Concluding remarks We considered the problem of filtration for systems governed by multivariate frac- tional Brownian noises. We derived the equation for the optimal filter πt(φ(X)) and nonnormalized filter σt(φ(X)) for the nonlinear case. The complete solution of the problem of filtration requires an equation for the pro- cess πt(q). This equation is obtained for linear systems. In the linear case, we completely solved the problem of filtration in the sense that we derived the system of equations for the expectation πt(X) and covariance γXX(t) of the errors of filtration.

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Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine Received 2/JUL/2004

Translated by V. SEMENOV

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