Asymptotically optimal filtering in linear systems with fractional Brownian noises

Statistics & Operations Research Transactions SORT 28 (2) July-December 2004, 177-190 Statistics & Operations Research Transactions Asymptotically optimal filtering in linear systems with fractional Brownian noises⋆ M. L. Kleptsyna∗, A. Le Breton∗∗, M. Viot∗∗∗ ∗Universit´e du Maine,∗∗,∗∗∗ Universit´e J. Fourier. Abstract In this paper, the filtering problem is revisited in the basic Gaussian homogeneous linear system driven by fractional Brownian motions. We exhibit a simple approximate filter which is asymptotically optimal in the sense that, when the observation time tends to infinity, the variance of the corresponding filtering error converges to the same limit as for the exact optimal filter. MSC: Primary 60G15, 60G35. Secondary 62M20, 93E11 Keywords: fractional Brownian motion, homogeneous linear system, optimal filtering, filtering error, asymptotic variance 1 Introduction Several contributions have been already reported around filtering problems concerning models where the driving processes are fractional Brownian motions (fBm’s for short) : see Kleptsyna et al. (2000) for a rather general approach and further references. The specific case of a homogeneous linear system has been investigated in Kleptsyna and Le Breton (2002) where explicit closed form equations are derived both for the optimal ⋆Author’s research was supported by the Grant INTAS 99-00559. She is also associated with the Institute of Information Transmission Problems, Moscow, Russia. ∗Laboratoire de Statistique et Processus/Universit´ e du Maine. Av. Olivier Messiaen, 72085 Le Mans, Cedex 9, France. e-mail: [email protected]. ∗∗ Address for correspondence: Fax: 33 476 631 263. His research was supported by the Project IDOPT, CNRSUJF-INPG-INRIA. Laboratoire de Mod´ elisation et Calcul/Universit´ e J. Fourier. BP 53, 38041 Grenoble Cedex 9, France. e-mail: [email protected]. ∗∗∗ Laboratoire de Mod´ elisation et Calcul/Universit´ e J. Fourier. BP 53, 38041 Grenoble Cedex 9, France. e-mail: [email protected]. Received: October 2003 Accepted: May 2004 178 Asymptotically optimal filtering in linear systems with fractional Brownian noises filter and the variance of the filtering error. Moreover, therein it is shown that this filter is asymptotically stable in the sense that the variance of the filtering error converges to a finite limit as the observation time tends to infinity. Here our aim is to exhibit a simple approximate filter which has the same asymptotic behaviour as the optimal one. Let us fix this more precisely. As in Kleptsyna and Le Breton (2002), we deal with real-valued processes X= (Xt,t≥0) and Y=(Yt,t≥0), representing the signal and the observation respectively, governed by the following homogeneous linear system of stochastic differential equations interpreted as integral equations :        dXt=θXtdt +dVH t,t≥0,X0=0, dYt=µXtdt +dWH t,t≥0,Y0=0.(1.1) Here VH=(VH t,t≥0) and WH=(WH t,t≥0) are independent normalized fBm’s with the same Hurst parameter Hin [1 2,1) and the coefficients θand µ,0 are fixed real constants. The system (1.1) has a uniquely defined solution process (X,Y) which is Gaussian. Supposing that only Yis observed but one wishes to know X, the classical problem of filtering the signal Xat time tfrom the observation of Yup to time toccurs. The solution to this problem is the conditional distribution of Xtgiven {Ys,0≤s≤t}, which of course is Gaussian. Then, it is completely determined by the conditional mean πt(X)=IE(Xt/{Ys,0≤s≤t}), which we shall call the exact optimal filter, and the variance γXX (t)=IE(Xt−πt(X))2of the filtering error. In Kleptsyna and Le Breton (2002), a system of Volterra type integral equations for these characteristics is provided and the following stability property of the filter is also shown : lim t→+∞γXX (t)=γH, where the constant γHis given by γH=Γ(2H+1) 2(θ2+µ2)H£1+pθ2+µ2+θ pθ2+µ2−θsinπH¤.(1.2) In the classical case H=1 2where the noises are standard Brownian motions, the system of filtering equations reduces to the well-known Kalman-Bucy system (see, e.g., Davis (1977) and Liptser and Shiryaev (1978)) and the asymptotic variance of the filtering error is γ1 2 =µ−2[pθ2+µ2+θ]. In that case, substituting the constant γ1 2for the function γXX (t) in the Kalman-Bucy system, one gets the simpler filtering equation dπ∗ t(X)=−qθ2+µ2π∗ t(X)dt +µγ1 2dYt;π∗ 0(X)=0,(1.3) which generates the filter π∗ t(X)=µγ1 2Zt 0e−√θ2+µ2(t−s)dYs.(1.4) M. L. Kleptsyna, A. Le Breton, M. Viot 179 It turns out that π∗ t(X) is an asymptotically optimal filter in the sense that the variance IE(Xt−π∗ t(X))2of the corresponding filtering error converges to γ1 2as tgoes to infinity. Observe that actually, in this case, the asymptotic optimality in filtering is achieved in the class of filters which can be represented as Rt 0φ(t−s)dYs. In the present paper, we show that this still holds for H>1 2and we identify in this class a filter for which the variance of the filtering error converges to γH. The paper is organized as follows. At first in Section 2, we fix some notations and preliminaries; in particular we associate to the problem under study an equivalent deterministic control problem. Then, our main result is stated and proved in Section 3 by exploiting the solution of this auxiliary problem which belongs to a family of infinite time horizon deterministic control problems which are investigated in Section 4. 2 Preliminaries Fractional Brownian motion. Here, for some H∈[1 2,1), BH=(BH t,t≥0) is a normalized fractional Brownian motion with Hurst parameter H. This means that BH is a Gaussian process with continuous paths such that BH 0=0, IEBH t=0 and IEBH sBH t=1 2[s2H+t2H−|s−t|2H],s,t≥0.(2.1) Of course the fBm reduces to the standard Brownian motion when H=1 2. For H,1 2, the fBm is outside the world of semimartingales but a theory of stochastic integration with respect to fBm has been developed (see, e.g., Decreusefond and ¨ Ust¨ unel (1999) or Duncan et al. (2000)). Actually the case of deterministic integrands, which is sufficient for the purpose of the present paper, is easy to handle (see, e.g., Norros et al. (1999)). In particular, for a stochastic integral St=Zt 0g(t−s)dBH s,(2.2) we can evaluate IES2 t=             Zt 0g2(s)ds if H=1 2, H(2H−1)Zt 0Zt 0g(s)g(r)|s−r|2H−2dsdr if H∈(1 2,1). In the second case, exploiting the representation |s−r|2H−2=1 B(H−1 2,2−2H)Z+∞ s∨r(τ−s)H−3 2(τ−r)H−3 2dτ , where B(., .) denotes the Beta function, it is easy to check that we can rewrite IES2 t=H(2H−1) B(H−1 2,2−2H)Z+∞ 0{Zs∧t 0g(r)(s−r)H−3 2dr}2ds. 180 Asymptotically optimal filtering in linear systems with fractional Brownian noises Therefore, we have also for all H∈[1 2,1) lim t→+∞IES2 t=2HΓ(3 2−H) Γ(H+1 2)Γ(2 −2H)Z+∞ 0eg2(s)ds,(2.3) where eg(s)=d ds Zs 0g(r)(s−r)H−1 2dr ,(2.4) and Γis the Gamma function. Actually, the connection (2.4) can be inverted by g(s)=1 B(H+1 2,3 2−H) d ds Zs 0eg(r)(s−r)1 2−Hdr .(2.5) Filtering errors. As announced in Section 1, in the system (1.1), we shall concentrate on filters which take the form πφ t(X)=Zt 0φ(t−s)dYs. From the first equation in (1.1), we have Xt=eθtZt 0e−θsdVH s, and, taking into account the second one, we get πφ t(X)=µZt 0φ(t−s)eθs{Zs 0e−θudVH u}ds +Zt 0φ(t−s)dWH s. Hence, it comes that πφ t(X)=µZt 0{Zt uφ(t−s)eθsds}e−θudVH u+Zt 0φ(t−s)dWH s, or πφ t(X)=µZt 0{Zt−u 0φ(w)e−θwdw}eθ(t−u)dVH u+Zt 0φ(t−s)dWH s. Finally, the filtering error corresponding to the filter πφ t(X) can be written as Xt−πφ t(X)=Zt 0eθ(t−s){1−µZt−s 0φ(w)e−θwdw}dVH s−Zt 0φ(t−s)dWH s, or equivalently Xt−πφ t(X)=Zt 0Zφ(t−s)dVH s−Zt 0φ(t−s)dWH s,(2.6) M. L. Kleptsyna, A. Le Breton, M. Viot 181 where the function Zφis defined from φ,Zφ=Zsay, by Z(τ)=eθτ{1−µZτ 0φ(w)e−θwdw}. Notice that Zis governed by the differential equation ˙ Z(τ)=θZ(τ)−µφ(τ); Z(0) =1.(2.7) Asymptotic variance of filtering errors. Now, starting from (2.6), according to the identities (2.2)-(2.4) with (Z,VH) and (φ, WH) in place of (g,BH) and due to the independence of VHand WH, we get that the asymptotic variance of the filtering error corresponding to the filter πφ t(X), i.e., lim t→+∞IE(Xt−πφ t(X))2=J(φ),(2.8) is given by J(φ)=2HΓ(3 2−H) Γ(H+1 2)Γ(2 −2H)Z+∞ 0{e Z2(s)+e φ2(s)}ds,(2.9) where, for Zlinked to φby (2.7), e Z(s)=d ds Zs 0Z(r)(s−r)H−1 2dr ;e φ(s)=d ds Zs 0φ(r)(s−r)H−1 2dr .(2.10) Actually, it is readily seen from (2.7) and (2.10) that the dynamics which links e Zto e φis nothing but e Z(t)=θZt 0e Z(s)ds −µZt 0e φ(s)ds +tH−1 2.(2.11) Notice that of course if H=1 2, and hence e φ≡φand e Z≡Z, equation (2.11) is nothing but equation (2.7) written in integral form and if H>1 2, then (2.11) can be rewritten as ˙ e Z(t)=θe Z(s)−µe φ(s)ds +(H−1 2)tH−3 2;e Z(0) =0. Due to the limiting property (2.8), our guess is that in order to define an asymptotically optimal filter π∗ t(X), one may take π∗ t(X)=πφ∗ t(X) where the function φ∗corresponds through (2.5) to an optimal control e φ∗in the control problem : min e φe J(e φ) subject to (2.11),(2.12) with the performance criterion e J(e φ)=J(φ) defined by (2.9). The concerned infinite time horizon deterministic control problem (2.12) belongs to the class of control problems which are solved in Section 4. Their solutions make us able to formulate and prove our main result. 182 Asymptotically optimal filtering in linear systems with fractional Brownian noises 3 Asymptotically optimal filtering At first, let us discuss the case when H=1 2. Here, in the control problem studied in Section 4, we must take x=1, K≡0, a=θ,b=−µand q=r=1. Hence, applying Theorem 4.1 (see also the particular case 4.1), it comes that the optimal control in (2.12) is φ∗(t)=µγ1 2e−√θ2+µ2t, where γ1 2 =pθ2+µ2+θ µ2, is the value of the optimal cost. This means nothing but that, as claimed in Section 1, an asymptotically optimal filter is π∗ t(X)=πφ∗ t(X) given by (1.4). Now, we turn to the case H∈(1 2,1) where we can prove the following statement which provides also an asymptotically optimal filter : Theorem 3.1 Define the function V∗by V∗(t)=H−1 2 B(H+1 2,3 2−H)Z+∞ 0e−√θ2+µ2tττH−1 2 τ+1dτ , t>0.(3.1) Let the pair of functions (φ∗,Z∗)be defined by            φ∗(t)=θ+pθ2+µ2 µ[Z∗(t)+V∗(t)], ˙ Z∗(t)=θZ∗(t)−µφ∗(t); Z∗(0) =1. (3.2) Then the filter π∗ t(X)=Zt 0φ∗(t−s)dYs, is asymptotically optimal, i.e., lim t→+∞IE(Xt−π∗ t(X))2=γH, where γHis given by (1.2). Proof. For H∈(1 2,1), in the control problem studied in Section 4, we must take x=0, K(t)=(H−1 2)tH−3 2,a=θ,b=−µand q=r=2HΓ(3 2−H) Γ(H+1 2)Γ(2 −2H). Hence, applying Theorem 4.1 (see also the particular case 4.2), we get that the following pair (e φ∗,e Z∗) is optimal in the control problem (2.12) :           e φ∗(t)=θ+pθ2+µ2 µ[e Z∗(t)+e V∗(t)], ˙ e Z∗(t)=θe Z∗(t)−µe φ∗(t)+(H−1 2)tH−3 2;e Z∗(0) =0, M. L. Kleptsyna, A. Le Breton, M. Viot 183 where e V∗(t)=(H−1 2)Z+∞ 0e−√θ2+µ2r(t+r)H−3 2dr .(3.3) Moreover, it is easy to check that the optimal cost in (2.12) is e J(e φ∗)=γHwhere γH is given by (1.2). Hence, it is clear that to define an asymptotically optimal filter by π∗ t(X)=πφ∗ t(X) we can take the second component φ∗of the triple (V∗, φ∗,Z∗) which corresponds through (2.5) to the triple (e V∗,e φ∗,e Z∗). It is easy to check that φ∗is defined by (3.2) where V∗corresponds through(2.5) to e V∗and so, finally, we have just to identify V∗. From (3.3), we compute Zt 0(t−s)1 2−He V∗(s)ds =(H−1 2)Zt 0(t−s)1 2−H{Z+∞ 0e−√θ2+µ2r(s+r)H−3 2dr}ds =(H−1 2)Z+∞ 0e−√θ2+µ2r{Zt 0(t−s)1 2−H(s+r)H−3 2ds}dr =(H−1 2)Z+∞ 0e−√θ2+µ2r{Zt t+r 0v1 2−H(1 −v)H−3 2dv}dr. Observing that actually d dt Zt t+r 0v1 2−H(1 −v)H−3 2dv =t1 2−HrH−1 2 t+r, it follows that Zt 0(t−s)1 2−He V∗(s)ds =(H−1 2)Z+∞ 0e−√θ2+µ2r{Zt 0 u1 2−HrH−1 2 u+rdu}dr =(H−1 2)Zt 0{Z+∞ 0e−√θ2+µ2ru1 2−HrH−1 2 u+rdr}du =(H−1 2)Zt 0{Z+∞ 0e−√θ2+µ2uττH−1 2 τ+1dτ}du. From (2.5), we see that this means exactly that V∗is given by (3.1). ¤ Remark 3.1 (a) Observe that from (3.1) we have also ˙ V∗(t)=−qθ2+µ2H−1 2 B(H+1 2,3 2−H)Z+∞ 0e−√θ2+µ2tττH+1 2 τ+1dτ , t>0. Then, splitting the integral into two terms corresponding to the decomposition of τH+1 2 as the difference [τH+1 2+τH−1 2]−τH−1 2, one may easily check that V∗is actually the solution of the differential equation ˙ V∗(t)=qθ2+µ2V∗(t)−βH(H−1 2)t−1 2−H; lim t→+∞V∗(t)=0, where βH=(θ2+µ2)1−2H 4 Γ(3 2−H). 184 Asymptotically optimal filtering in linear systems with fractional Brownian noises Since R+∞ 0V∗(t)dt =1, it means also that that V∗is the solution of the integral equation V∗(t)=Zt 0qθ2+µ2V∗(s)ds +βHt1 2−H−1.(3.4) (b) Let us emphasize that, similarly to the case H=1 2where the filter π∗ t(X) can be generated by the approximate Kalman-Bucy algorithm (1.3), a recursive scheme can be also provided for the asymptotically optimal filter in the case H∈(1 2,1). At first, we observe that due to the first equation in (3.2) we can write π∗ t(X)=µγ1 2[Z∗ t+V∗ t],(3.5) where Z∗ t=Zt 0Z∗(t−s)dYs;V∗ t=Zt 0V∗(t−s)dYs. Since the function Z∗is differentiable, we have Z∗ t=Z∗(0)Yt+Zt 0{Zs 0 ˙ Z∗(s−r)dYr}ds. Hence, due to the second equation in (3.2), the process Z∗is generated from Yby the equation Z∗ t=θZt 0Z∗ sds −µZt 0π∗ s(X)ds +Yt,(3.6) Now, using equation (3.4), we can write V∗ t=Zt 0ψ(t−s)dYs+βHZt 0(t−s)1 2−HdYs, where the function ψsatisfies ˙ ψ(t)=qθ2+µ2V∗(t); ψ(0) =−1. Consequently, we get that Zt 0ψ(t−s)dYs=ψ(0)Yt+Zt 0{Zs 0 ˙ ψ(s−r)dYr}ds =qθ2+µ2Zt 0V∗ sds −Yt. Finally, the following equation holds for V∗ t: V∗ t=qθ2+µ2Zt 0V∗ sds +Zt 0[βH(t−s)1 2−H−1]dYs.(3.7) M. L. Kleptsyna, A. Le Breton, M. Viot 185 The system (3.5)-(3.7) provides a closed-form recursion which generates the filter π∗ t(X) from the observation process Y. It is readily seen that when H=1 2, and hence V∗≡0 and π∗ t(X)=µγ1 2Z∗ t, this system reduces to the single equation π∗ t(X)=−qθ2+µ2Zt 0π∗ s(X)ds +µγ1 2Yt, which is nothing but equation (1.3). (c) Suppose that H>1 2but one does as if the noises were standard Brownian motions and hence uses the filter generated by the approximate Kalman-Bucy algorithm (1.3), i.e., the filter eπt(X)=µγ1 2Zt 0e−√θ2+µ2(t−s)dYs. Then it can be checked that the corresponding asymptotic variance of the filtering error limt→+∞IE(Xt−eπt(X))2is the constant eγH=Γ(2H+1) (θ2+µ2)H−1 2 γ1 2. Moreover the consequent loss of performance with respect to the asymptotically optimal filter can be evaluated by eγH−γH=Γ(2H+1) 2(θ2+µ2)Hµ2γ2 1 2(1 −sinπH). Let us observe that, for fixed parameters θand µ, the asymptotic relative efficiency γH eγH = 1+µ2γ2 1 2sinπH 1+µ2γ2 1 2 , ofeπt(X) decreases as Hincreases in (1 2,1). 4 About optimal control problems Given a function K=(K(t),t≥0) and constants aand b, we consider the state dynamics ˙ Xt=aXt+bUt+K(t),t≥0; X0=x,(4.1) where the control U=(Ut,t≥0) can be chosen in order to drive the state X=(Xt,t≥ 0). Let Abe the class of measurable functions U, called admissible controls, such that the corresponding differential equation (4.1) has a unique solution X. Given constants q>0 and r>0, we define the performance criterion Jby J(U)=Z+∞ 0[qX2 t+rU2 t]dt .(4.2)