An Identification Problem for Partially Observed Infinite Dimensional Linear

Stochastic Systems

W. Grecksch Martin-Luther-Universität Halle-Wittenberg FB Mathematik und Informatik Institut für Optimierung und Stochastik D-06099 Halle/Saale,Germany

C. Tudor University of Bucharest and CIMAT Faculty of Mathematics Str. Academiei 14 70109 Bucharest, Romania

Abstract

In the paper an identification problem is considered for partially observed autonomous infinite dimensional stochastic systems with additive noise.

We formulate the problem as an optimization problem with appropriate objective functionals for finite and infinite time.

Key words: Identification, optimization, partially observed, infinite dimensional stochastic systems

AMS subject classification: 93E12, 60H15

1. Introduction

The identification problem for several classes of Itô equations with partial or complete observation has been considered by many authors (see Ahmed and Radaideh [1] and the references therein). Usually, the technique used for systems with partial observation involves the Zakai equation, which is a stochastic partial differential equation. For partially observed finite dimensional linear Itô equations in [1] is presented a formulation of the identification problem as a deterministic optimization problem with a natural objective functional, without using stochastic partial differential equations. In the present paper we consider an infinite dimensional partially observed Itô equation with additive noise and formulate the identification problem for the dispersion in the signal process and for the drift and dispersion in the observed process by means of a minimization problem. We consider finite and infinite time. For the finite time we take the same form of the objective function as in [1]. In the long run we propose an objective functional of average type which is more appropriate for such a situation. In both cases the existence of the optimum (which generally it is not unique) is shown. Next M_{+}(m × p) denote the set of m × p matrices τ of real numbers with ττ^{∗} > 0 ( ∗ denote the transpose or adjoint of a matrix or operator). Let Z, U be separable Hilbert spaces. The Hilbert space of all Hilbert-Schmidt operators on Z is denoted by N (Z) and the Hilbert-Schmidt norm (resp. Hilbert-Schmidt inner product) is denoted by ∥.∥_{2} (resp. ⟨., .⟩_{2}). N_{S}^{+}(Z)) is the subspace of N (Z) consisting from self-adjoint and nonnegative operators. If A, B are linear operators then we write A ≤ B if ⟨Ax, x⟩ ≤ ⟨Bx, x⟩ for every x. Also we denote by L(U, Z) the space of bounded linear operators from U into Z endowed with the uniform topology. The trace of an operator Γ is denoted by Tr(Γ).

2. The Identification Problem as an Optimization Problem

We consider an arbitrary linear stochastic system governed by

\{ \begin{matrix}dX(t) = & AX(t)dt + σdw(t), & X(0) = X_{0}, t ≥ 0 \\ dy(t) = & HX(t)dt + σ_{0}dw_{0}(t), & y(0) = 0, t ≥ 0, \end{matrix}  (2.1)

where {X(t)}_{t≥0} is an Z-valued signal process and {y(t)}_{t≥0} is an R^{m}-valued observation process. The operator A is the infinitesimal generator of a C_{0}-semigroup {S(t)}_{t≥0} on Z , σ ∈ L(Z, U), H ∈ L(Z, R^{m}), σ_{0} ∈ M_{+}(m × p) and {w(t)}_{t≥0} (resp. {w_{0}(t)}_{t≥0}) is a Wiener process with values in U and covariance operator W (resp. with values in R^{p}). We shall assume that w, w_{0} are independent. The precise meaning of (2.1) is the following:

\{ \begin{matrix}X(t) = & S(t)X_{0} + ∫_{0}^{t} S(t − s)σdw(s), & t ≥ 0 \\ y(t) = & ∫_{0}^{t} HX(s)ds + σ_{0}w_{0}(t), & t ≥ 0, \end{matrix}  (2.2)

i.e., X is so called mild solution of the first equation in (2.1). Roughly speaking the identification problem for (2.1), in the finite interval [0, T] , is the following: estimate the unknown parameters

π = (σ, H, σ_{0}) ∈ Σ_{0}, Σ_{0} ⊂ Σ = L(Z, U) × L(Z, R^{m}) × M_{+}(m × p),

given the observation {y(t)}_{0≤t≤T} and the knowledge of the mean X̄_{0} = E(X_{0}) and the covariance P_{0} = C_{X_{0}} (P_{0} is a nonnegative self-adjoint and nuclear operator).

Next we formulate the identification problem as a minimization problem. In order to justify this we use the Kalman filter. Let X̂_{π} be the optimal filter of X given the observation {y(t)}_{0≤t≤T}. Then it is known (see [2], [4], [7]) that X̂_{π} is given by the mild solution of

\{ \begin{matrix}dX̂_{π}(t) & = AX̂_{π}(t)dt + K_{π}(t)H^{∗}(σ_{0}σ_{0}^{∗})^{−1}dη_{π}(t) & 0 ≤ t ≤ T \\ X̂_{π}(0) & = X̄_{0}, \end{matrix}  (2.3)

where η_{π} is a Wiener process (called the innovation process) given by

η_{π}(t) = y(t) − ∫_{0}^{t} HX̂_{π}(s)ds, 0 ≤ t ≤ T,  (2.4)

and K_{π} represents the covariance operator of the error process X_{π} − X̂_{π} and is the unique mild solution of the Riccati equation

K̇_{π} = A^{∗}K_{π} + K_{π}A + σW σ^{∗} − K_{π}H^{∗}(σ_{0}σ_{0}^{∗})^{−1}HK_{π}, K_{π}(0) = P_{0},  (2.5)

i.e., of the integral equation

K_{π}(t)x = S(t)P_{0}S^{∗}(t)x + ∫_{0}^{t} S(t − s) [σW σ^{∗}−
−K_{π}(s)H^{∗}(σ_{0}σ_{0}^{∗})^{−1}HK_{π}(s)] S^{∗}(t − s)xds, x ∈ Z, 0 ≤ t ≤ T.  (2.6)

Remark 2.1 By Theorem 2 of [5] K_{π} is also the unique solution in the space C(R_{+}, N_{S}^{+}(Z)) of the equation

K_{π}(t) = S(t)P_{0}S^{∗}(t) + ∫_{0}^{t} S(t − s) [σW σ^{∗}−

−K_{π}(s)H^{∗}(σ_{0}σ_{0}^{∗})^{−1}HK_{π}(s)] S^{∗}(t − s)ds, t ≥ 0.  (2.7)

(the integral in (2.7) is in the Bochner’s sense on the space N (Z)).

For the next use we define

R = σ_{0}σ_{0}^{∗}, N = σW σ^{∗}, X̄_{π}(t) = E [X_{π}(t)] ,

e_{π}(t) = X̂_{π}(t) − X̄_{π}(t), P_{π}(t) = C_{X(t)}, K̃_{π}(t) = C_{e_{π}(t)}, t ≥ 0.

Then P_{π} is mild solution of the Riccati equation

Ṗ_{π} = A^{∗}P_{π} + P_{π}A + N, P_{π}(0) = P_{0},  (2.8)

i.e.,

P_{π}(t) = S(t)P_{0}S^{∗}(t) + ∫_{0}^{t} S(t − s)NS^{∗}(t − s)ds, t ≥ 0.  (2.9)

Also, we have

\{ \begin{matrix}de_{π}(t) = & [A − K_{π}(t)H^{∗}R^{−1}H] e_{π}(t)dt+ \\ & K_{π}(t)H^{∗}R^{−1} [dy(t) − HX̄_{π}(t)dt] , t ≥ 0 \\ e_{π}(0) = & 0 \end{matrix}  (2.10)

or equivalently

e_{π}(t) = ∫_{0}^{t} S_{A−K_{π}H^{∗}R^{−1}H}(t, s)K_{π}(s)H^{∗}R^{−1} [dy(s) − HX̄_{π}(s)ds] , t ≥ 0, (2.11)

where S_{A−K_{π}H^{∗}R^{−1}H}(t, s) is the perturbation of S by −K_{π}H^{∗}R^{−1}H. It is easily seen that for the true parameter π

K̃_{π} + K_{π} − P_{π} = 0,  (2.12)

and this justifies the formulation of our identification problem as the following minimization problem:

(FIP) \{ \begin{matrix}min ; π∈∑_{0} J_{π}(T) \\ subject to (2.5), (2.8) and K̃_{π} (t) = C_{e_{π}(t)}, \end{matrix}

where the objective functional is given by

J_{π}(T) = ∫_{0}^{T} ∥K̃_{π}(t) + K_{π}(t) − P_{π}(t)∥_{2}^{2} dt,  (2.13)

and e_{π} is given by (2.10). In long run a more appropriate formulation of the identification problem is

(SIP) \{ \begin{matrix}min ; π∈∑_{0} J_{π}^{∞} \\ subject to (2.5), (2.8) and K̃_{π} (t) = C_{e_{π}(t)}, \end{matrix}

where

J_{π}^{∞} = \overline{lim_{T →∞}} \frac{1}{T} ∫_{0}^{T} ∥[K̃_{π}(t) + K_{π}(t) − P_{π}(t)] C∥_{2}^{2} dt,  (2.14)

and C ∈ N (Z).

3. Existence of Optimum: Finite Horizon

We need the following Lemma 3.1 Let {α_{n}(t)}_{0≤t≤T} , n = 1, 2, ..., {α(t)}_{0≤t≤T} be point progressively measurable processes with values in L(U, Z) and let {β_{n}(t)}_{0≤t≤T} , n = 1, 2, ..., {β(t)}_{0≤t≤T} be progressively measurable processes with values in Z. Define the processes {X_{n}(t)}_{0≤t≤T} , n = 1, 2, ..., {X(t)}_{0≤t≤T} by

X_{n}(t) = ∫_{0}^{t} α_{n}(s)dw(s) + ∫_{0}^{t} β_{n}(s)ds, 0 ≤ t ≤ T,  (3.1)

X(t) = ∫_{0}^{t} α(s)dw(s) + ∫_{0}^{t} β(s)ds, 0 ≤ t ≤ T,  (3.2)

and let C_{X_{n}(t)}, C_{X(t)} be the covariance operators of X_{n}(t), X(t). Assume that

E [∫_{0}^{T} ∥γ(t)∥^{2} dt] < ∞, for γ = α_{n}, α, β_{n}, β,  (3.3)

lim_{n→∞} E [∫_{0}^{T} ∥α_{n}(t) − α(t)∥^{2} dt] = 0,  (3.4)

lim_{n→∞} E [∫_{0}^{T} ∥β_{n}(t) − β(t)∥^{2} dt] = 0.  (3.5)

Then we have

lim_{n→∞} E [∥X_{n}(T) − X(T)∥^{2}] = 0,  (3.6)
lim_{n→∞} ∥C_{X_{n}(T)} − C_{X(T)}∥_{2} = 0 .  (3.7)

Proof The relation (3.6) follows from the inequality

E [∥X_{n}(T) − X(T)∥^{2}] ≤ 2 (T r W + T) ×

× {E [∫_{0}^{T} ∥α_{n}(t) − α(t)∥^{2} dt] + E [∫_{0}^{T} ∥β_{n}(t) − β(t)∥^{2} dt]}

and from (3.4), ( 3.5). Next we have to prove that

∥C_{X_{n}(T)}∥_{2} ⟶ ∥C_{X(T)}∥_{2} ,  (3.8)

C_{X_{n}(T)} ⟶ C_{X(T)} weakly in N (Z).  (3.9)

Let {e_{m}}_{m} be an orthonormal basis in Z and ε > 0. Then

∥C_{X_{n}(T)}∥_{2}^{2} = ∑_{m} ∥C_{X_{n}(T)}e_{m}∥^{2} = ∑_{m} ∑_{k} ⟨C_{X_{n}(T)}e_{m}, e_{k}⟩^{2}

= ∑_{m} ∑_{k} {E [⟨X_{n}(T) − E (X_{n}(T)) , e_{m}⟩ ⟨X_{n}(T) − E (X_{n}(T)) , e_{k}⟩]}^{2} ,

and thus

|∥C_{X_{n}(T)}∥_{2}^{2} − ∥C_{X(T)}∥_{2}^{2}|

≤ ∑_{m} ∑_{k} |E [⟨X_{n}(T) − E (X_{n}(T)) , e_{m}⟩ ⟨X_{n}(T) − E (X_{n}(T)) , e_{k}⟩

− ⟨X(T) − E (X(T)) , e_{m}⟩ ⟨X(T) − E (X(T)) , e_{k}⟩] ×

× [⟨X_{n}(T) − E (X_{n}(T)) , e_{m}⟩ ⟨X_{n}(T) − E (X_{n}(T)) , e_{k}⟩

+ ⟨X(T) − E (X(T)) , e_{m}⟩ ⟨X(T) − E (X(T)) , e_{k}⟩]|

≤ \frac{ε}{2} ∑_{m} ∑_{k} |E [⟨X_{n}(T) − E (X_{n}(T)) , e_{m}⟩ ⟨X_{n}(T) − E (X_{n}(T)) , e_{k}⟩

+ ⟨X(T) − E (X(T)) , e_{m}⟩ ⟨X(T) − E (X(T)) , e_{k}⟩]|^{2}

+\frac{1}{2ε} ∑_{m} ∑_{k} |E [⟨X_{n}(T) − E (X_{n}(T)) , e_{m}⟩ ⟨X_{n}(T) − E (X_{n}(T)) , e_{k}⟩

− ⟨X(T) − E (X(T)) , e_{m}⟩ ⟨X(T) − E (X(T)) , e_{k}⟩]|^{2} = \frac{ε}{2}I_{n} + \frac{1}{2ε}J_{n}

for every ϵ > 0. Further, an elementary calculation gives.

I_{n} ≤ 2 ∑_{m} ∑_{k} |E [⟨X_{n}(T) − E (X_{n}(T)) , e_{m}⟩ ⟨X_{n}(T) − E (X_{n}(T)) , e_{k}⟩]|^{2}

+2 ∑_{m} ∑_{k} |E [⟨X(T) − E (X(T)) , e_{m}⟩ ⟨X(T) − E (X(T)) , e_{k}⟩]|^{2}

≤ 2 ∑_{m} ∑_{k} E [⟨X_{n}(T) − E (X_{n}(T)) , e_{m}⟩^{2}] E [⟨X_{n}(T) − E (X_{n}(T)) , e_{k}⟩^{2}]

+2 ∑_{m} ∑_{k} E [⟨X(T) − E (X(T)) , e_{m}⟩^{2}] E [⟨X(T) − E (X(T)) , e_{k}⟩^{2}]

= 2 (E [∥X_{n}(T) − E (X_{n}(T))∥^{2}])^{2} + 2 (E [∥X(T) − E (X(T))∥^{2}])^{2}

≤ 2E ([∥X_{n}(T)∥^{2}])^{2} + 2 (E [∥X(T)∥^{2}])^{2} ≤ C where C is a positive constant independent of n. Next

J_{n} ≤ 2 ∑_{m} ∑_{k} |E [⟨(X_{n}(T) − E (X_{n}(T))) − (X(T) − E (X(T))) , e_{m}⟩ ×

× ⟨X_{n}(T) − E (X_{n}(T)) , e_{k}⟩]|^{2} + 2 ∑_{m} ∑_{k} |E [⟨X(T) − E (X(T)) , e_{m}⟩ ×

× ⟨(X_{n}(T) − E (X_{n}(T))) − (X(T) − E (X(T))) , e_{k}⟩]|^{2} = 2 (J_{n,1} + J_{n,2}) ,

J_{n,1} ≤ ∑_{m} ∑_{k} E [⟨(X_{n}(T) − E (X_{n}(T))) − (X(T) − E (X(T))) , e_{m}⟩^{2}] ×

×E [⟨X_{n}(T) − E (X_{n}(T)) , e_{k}⟩^{2}] = E [∥(X_{n}(T) − E (X_{n}(T))) −

− (X(T) − E (X(T)))∥^{2}] E [∥X_{n}(T) − E (X_{n}(T))∥^{2}]
≤ C_{1}E [∥X_{n}(T) − X(T)∥^{2}] → 0,

for n → ∞ and similarly we obtain

J_{n,2} ≤ C_{2}E [∥X_{n}(T) − X(T)∥^{2}] → 0

for n → ∞ where C_{1}, C_{2} are positive constants.It follows that (3.8) hold. Now if Γ ∈ N (Z) we have

⟨C_{X_{n}(T)}, Γ⟩_{2} = ∑_{m} ⟨C_{X_{n}(T)}e_{m}, Γe_{m}⟩

= ∑_{m} E [⟨X_{n}(T) − E (X_{n}(T)) , e_{m}⟩ ⟨X_{n}(T) − E (X_{n}(T)) , Γe_{m}⟩] ,

and hence

|⟨C_{X_{n}(T)}, Γ⟩_{2} − ⟨C_{X(T)}, Γ⟩_{2}|

≤ ∑_{m} |E [⟨X_{n}(T) − E (X_{n}(T)) , e_{m}⟩ ⟨X_{n}(T) − E (X_{n}(T)) , Γe_{m}⟩]

−E [⟨X(T) − E (X(T)) , e_{m}⟩ ⟨X(T) − E (X(T)) , Γe_{m}⟩]|

≤ ∑_{m} |E [⟨(X_{n}(T) − E (X_{n}(T))) − (X(T) − E (X(T))) , e_{m}⟩ ×

× ⟨X_{n}(T) − E (X_{n}(T)) , Γe_{m}⟩]| + ∑_{m} |E [⟨X(T) − E (X(T)) , e_{m}⟩ ×

× ⟨(X_{n}(T) − E (X_{n}(T))) − (X(T) − E (X(T))) , Γe_{m}⟩]| = γ_{n,1} + γ_{n,2},

γ_{n,1} ≤ \frac{ε}{2} ∑_{m} E [⟨X_{n}(T) − E (X_{n}(T)) , Γe_{m}⟩^{2}]

+\frac{1}{2ε} ∑_{m} E [⟨(X_{n}(T) − E (X_{n}(T))) − (X(T) − E (X(T))) , e_{m}⟩^{2}]

= \frac{ε}{2}E [∥Γ^{∗} (X_{n}(T) − E (X_{n}(T)))∥^{2}]

+\frac{1}{2ε}E [∥(X_{n}(T) − E (X_{n}(T))) − (X(T) − E (X(T)))∥^{2}]

≤ 2ε ∥Γ∥^{2} E [∥X_{n}(T)∥^{2}] + \frac{2}{ε}E [∥X_{n}(T) − X(T)∥^{2}]

and then by (3.6) we obtain that

\overline{lim_{n→∞}} |⟨C_{X_{n}(T)}, Γ⟩_{2} − ⟨C_{X(T)}, Γ⟩_{2}|≤ C_{1}ε,

and consequently γ_{n,1} → 0 for n → ∞ Similarly it follows that γ_{n,2} → 0 for n → ∞ and it holds (3.9). Then (3.7) follows together with (3.8).

Theorem 3.2 Assume the observation process is a semimartingale of the form

y(t) = ΓV (t) + ∫_{0}^{t} f(s)ds,  (3.10)

where V is a R^{m} -valued Wiener process, Γ is a m × m -matrix and f is a progressively measurable R^{m} -valued process such that

E [∫_{0}^{T} ∥f(s)∥^{2} ds] < ∞.

Then the functional J_{π}(T) is continuous as function of π = (σ, H, σ_{0}) with respect to the strong topology. In particular if ∑_{0} is compact then the (FIP) has at least one solution.

Proof Let σ_{n} → σ strongly, H_{n} → H , σ_{0}^{n} → σ_{0} and denote

π_{n} = (σ_{n}, H_{n}, σ_{0}^{n}) , N_{n} = σ_{n}W σ_{n}^{∗}, R_{n} = σ_{0}^{n} (σ_{0}^{n})^{∗} .

Because of K_{π}H^{∗}(σ_{0}σ_{0}^{∗})^{−1}HK_{π} is a nonnegative operator it follows from (2.6) that

K_{π_{n}}(t) ≤ S(t)P_{0}S^{∗}(t) + ∫_{0}^{t} S(t − s)N_{n}S^{∗}(t − s)ds

and this implies

∥K_{π_{n}}(t)∥_{2} ≤ T r (K_{π_{n}}(t))

≤ T r [S(t)P_{0}S^{∗}(t) + ∫_{0}^{t} S(t − s)N_{n}S^{∗}(t − s)ds]

≤ ∥S(t)∥^{2} T r(P_{0}) + T sup_{0≤t≤T} ∥S(t)∥^{2} ∥σ_{n}∥^{2} T r(W)

and hence

sup_{n} sup_{0≤t≤T} ∥K_{π_{n}}(t)∥_{2} = C_{1} < ∞.  (3.11)

Next, (2.7), (3.11) and Gronwall’s inequality, imply

lim_{n→∞} sup_{0≤t≤T} ∥K_{π_{n}}(t) − K_{π}(t)∥_{2} = 0.  (3.12)

Also, it is easily seen that

lim_{n→∞} sup_{0≤t≤T} ∥P_{π_{n}}(t) − P_{π}(t)∥_{2} = 0.  (3.13)

From (2.10) it follows that e_{π_{n}} satisfies the equation

de_{π_{n}}(t) = [A − K_{π_{n}}(t)H_{n}^{∗}R_{n}^{−1}H_{n}] e_{π_{n}}(t)dt

+K_{π_{n}}(t)H_{n}^{∗}R_{n}^{−1} [dy(t) − H_{n}X̄_{π_{n}}(t)dt] , t ≥ 0,  (3.14)

and we have a similar equation for e_{π} .

Let S_{n}(t, s) (resp. S(t, s) ) be the perturbation of A by

−K_{π_{n}}(t)H_{n}^{∗}R_{n}^{−1}H_{n} (resp. − K_{π}(t)H^{∗}R^{−1}H)

i.e.,

\{ \begin{matrix}S_{n}(t, s) & = S(t − s) − ∫_{0}^{t} S(t − u)K_{π_{n}}(u)H_{n}^{∗}R_{n}^{−1}H_{n}S_{n}(u, s)du \\ S(t, s) & = S(t − s) − ∫_{0}^{t} S(t − u)K_{π}(u)H^{∗}R^{−1}HS(u, s)du. \end{matrix}  (3.15)

Then the solution of (3.14) has the form

e_{π_{n}}(t) = ∫_{0}^{t} S_{n}(t, s)K_{π_{n}}(s)H_{n}^{∗}R_{n}^{−1}dy(s)

− ∫_{0}^{t} S_{n}(t, s)K_{π_{n}}(s)H_{n}^{∗}R_{n}^{−1}H_{n}X̄_{π_{n}}(s)ds,

or equivalently

e_{π_{n}}(t) = ∫_{0}^{t} S_{n}(t, s)K_{π_{n}}(s)H_{n}^{∗}R_{n}^{−1}dV (s)

+ ∫_{0}^{t} S_{n}(t, s)K_{π_{n}}(s)H_{n}^{∗}R_{n}^{−1} [f(s) − H_{n}X̄_{π_{n}}(s)] ds,

and a similar equality holds for e_{π}. Now it is not difficult to check that the hypotheses of Lemma 3.1 are satisfied for the processes e_{π_{n}}, e_{π}, and consequently

lim_{n→∞} ∥K̃_{π_{n}}(t) − K̃_{π}(t)∥_{2} = 0, ∀0 ≤ t ≤ T.  (3.16)

Finally (3.12), (3.13), (3.16) and the dominated convergence theorem imply that J_{π_{n}}(T) → J_{π}(T).

Remark 3.3 In general in (FIP) we may not have uniqueness since X could be the same for different parameters.

4. Existence of Optimum: Infinite Horizon

In the present section we consider (SIP). Recall that the operator A is exponentially stable if there exist constants M, α > 0 such that

∥S(t)∥ ≤ Me^{−αt}, ∀t ≥ 0.  (4.1)

We introduce the following hypotheses: Hypothesis I: A is exponentially stable. Hypothesis II: For every π = (σ, H, σ_{0}) ∈ ∑_{0} the pair (A, H) is exponentially detectable, i.e., there is Λ ∈ L(R^{m}, Z) such that the operator A + ΛH generates an exponentially stable C_{0}− semigroup.

Remark 4.1 Assume the hypotheses I and II are satisfied. Then it is known (see for example [3]) that

lim_{t→∞} K_{π}(t) = K_{π}^{0} strongly,

and K_{π}^{0} ∈ L(Z, Z) is the unique nonnegative self-adjoint solution of the infinite dimensional algebraic Riccati equation

A^{∗}K_{π}^{0} + K_{π}^{0}A + N − K_{π}^{0}H^{∗}R^{−1}HK_{π}^{0} = 0.  (4.2)

Furthermore the operator A − K_{π}^{0}H^{∗}R^{−1}H is exponentially stable. Similarly

lim_{t→∞} P_{π}(t) = P_{π}^{0} strongly,

and P_{π}^{0} ∈ L(Z, Z) be the unique nonnegative self-adjoint solution of the algebraic Riccati equation

A^{∗}P_{π}^{0} + P_{π}^{0}A + N = 0.  (4.3)

Theorem 4.2 Suppose the hypotheses I and II hold . Let {β(t)}_{t∈R} be a extended R^{m}-valued Brownian motion over the entire real line. Then we have:

(i) The process

e_{π}^{∞}(t) = ∫_{−∞}^{t} S(t − s)K_{π}^{0}H^{∗}R^{−1}dβ(s)
:= L^{2} − lim_{n→∞} ∫_{−n}^{t} S(t − s)K_{π}^{0}H^{∗}R^{−1}dβ(s), t ∈ R,  (4.4)

is well defined and stationary.

Moreover

lim_{t→∞} ∥C_{e_{π}^{∞}(t)} − K̃(t)∥_{2} = 0.  (4.5)

(ii) If C ∈ N (Z) and

J_{π}^{∞} = ∥(C_{e_{π}^{∞}(0)} + K_{π}^{0} − P_{π}^{0}) C∥_{2}^{2} ,  (4.6)

then (SIP) is equivalent with

min ; π∈∑_{0} J_{π}^{∞},

subject to (4.2), (4.3) and (4.4). (iii) The functional J_{π}^{∞} is continuous as function of π = (σ, H, σ_{0}) with respect to the strong topology. In particular if ∑_{0} is compact, then (SIP) has at least one solution.

Proof (i) The welldefinedness of e_{π}^{∞}(t) is a consequence of the exponential stability of S . Next, for h ∈ R, consider the new Brownian motion

β^{h}(u) = β(u + h) − β(h), u ∈ R.

If z_{1}, z_{2} ∈ Z, s, t ∈ R, we have

E [⟨e_{π}^{∞}(t + h), z_{1}⟩ ⟨e_{π}^{∞}(s + h), z_{2}⟩]

= E [∫_{−∞}^{t+h} ⟨(S(t + h − u)K_{π}^{0}H^{∗}R^{−1})^{∗} z_{1}, dβ(u)⟩ ×

× ∫_{−∞}^{s+h} ⟨(S(s + h − u)K_{π}^{0}H^{∗}R^{−1})^{∗} z_{2}, dβ(u)⟩]

= E [∫_{−∞}^{t} ⟨(S(t − u)K_{π}^{0}H^{∗}R^{−1})^{∗} z_{1}, dβ^{h}(u)⟩ ×

× ∫_{−∞}^{s} ⟨(S(s − u)K_{π}^{0}H^{∗}R^{−1})^{∗} z_{2}, dβ^{h}(u)⟩]

= E [⟨e_{π}^{∞}(t), z_{1}⟩ ⟨e_{π}^{∞}(s), z_{2}⟩] .

Define

Λ = (H^{∗}R^{−1}) (H^{∗}R^{−1})^{∗} .

Then

C_{e_{π}^{∞}(t)} − K̃(t) = ∫_{−∞}^{0} S(t − s)K_{π}^{0}ΛK_{π}^{0}S^{∗}(t − s)ds

+ ∫_{0}^{t} S(t − s) [K_{π}^{0}ΛK_{π}^{0} − K_{π}(s)ΛK_{π}(s)] S^{∗}(t − s)ds.  (4.7)

Since Λ ∈ N (Z) and sup_{t≥0} ∥K_{π}(t)∥ = C_{1} < ∞ we can choose t_{ε} large such that

∥K_{π}(t_{ε})ΛK_{π}(t_{ε}) − K_{π}^{0}ΛK_{π}^{0}∥_{2} ≤ ε.  (4.8)

Now, the stability of S(t) , (4.7), (4.8) yield for t ≥ t_{ε},

∥C_{e_{π}^{∞}(t)} − K̃(t)∥_{2} ≤ M^{2}C_{1}^{2} ∥Λ∥ ∫_{−∞}^{0} e^{−2α(t−s)}ds

+2M^{2}C^{2} ∥Λ∥ ∫_{0}^{t_{ε}} e^{−2α(t−s)}ds + M^{2}ε ∫_{t_{ε}}^{t} e^{−2α(t−s)}ds
≤ C_{2} (e^{−2αt} + ε) , for C_{2} independent of n,

and hence

\overline{lim_{t→∞}} ∥C_{e_{π}^{∞}(t)} − K̃(t)∥_{2} ≤ C_{2}ε.

(ii) (4.22) follows from (4.5), the equality C_{e_{π}^{∞}(t)} = C_{e_{π}^{∞}(0)} (by the stationarity of e_{π}^{∞}), the strong convergence of K_{π}(t) (resp. P_{π}(t)) to K_{π}^{0} (resp. P_{π}^{0} ) as t → ∞ and since C ∈ N (Z) . (iii) Let σ_{n} → σ strongly, H_{n} → H , σ_{0}^{n} → σ_{0} and denote

π_{n} = (σ_{n}, H_{n}, σ_{0}^{n}) , N_{n} = σ_{n}W σ_{n}^{∗}, R_{n} = σ_{0}^{n} (σ_{0}^{n})^{∗} ,

Since S is exponentially stable we deduce (as in [6]) that K_{π}^{0} satisfies the equation

K_{π}^{0}z = ∫_{0}^{∞} S(s) [N − K_{π}^{0}H^{∗}R^{−1}HK_{π}^{0}] S^{∗}(s)zds, z ∈ Z.  (4.9)

From (4.9) it follows that

K_{π}^{0} ≤ ∫_{0}^{∞} S(s)NS^{∗}(s)ds  (4.10)

and thus

∥K_{π}^{0}∥_{2} ≤ \frac{M}{2α} ∥N∥_{2} .  (4.11)

Let K_{π}(t, η) be the solution of (2.5) with the initial condition η . Define

θ = ∫_{0}^{∞} S(s)NS^{∗}(s)ds, θ_{n} = ∫_{0}^{∞} S(s)N_{n}S^{∗}(s)ds.

From (4.10) and the monotonicity of K_{π}(t, η) as function of η (see [3]or [6]) we have

0 ≤ K_{π}^{0} − K_{π}(t, θ) ≤ S(t)θS^{∗}(t),  (4.12)
0 ≤ K_{π_{n}}^{0} − K_{π_{n}}(t, θ_{n}) ≤ S(t)θ_{n}S^{∗}(t).  (4.13)

Then, from (4.12), (4.13), the stability of S we obtain

∥K_{π}^{0} − K_{π}(t, θ)∥_{2} ≤ M^{2} ∥θ∥_{2} e^{−2αt}, ∀t ≥ 0,  (4.14)
∥K_{π_{n}}^{0} − K_{π_{n}}(t, θ_{n})∥_{2} ≤ M^{2} ∥θ_{n}∥_{2} e^{−2αt} ≤ M_{1}e^{−αt}, ∀t ≥ 0,  (4.15)

for some constant M_{1} > 0 . Next we have

∥K_{π^{0}} − K_{π_{n}}^{0}∥_{2} ≤ ∥K_{π_{n}}^{0} − K_{π_{n}}(t, θ_{n})∥_{2} + ∥K_{π_{n}}(t, θ_{n}) − K_{π_{n}}(t, θ)∥_{2}

+ ∥K_{π_{n}}(t, θ) − K_{π}(t, θ)∥_{2} + ∥K_{π}(t, θ) − K_{π^{0}}∥_{2} .  (4.16)

For ε > 0 , by (4.14), (4.15) we can choose t_{ε} sufficiently large such that

∥K_{π}(t_{ε}, θ) − K_{π^{0}}∥_{2} ≤ ε, ∥K_{π_{n}}^{0} − K_{π_{n}}(t_{ε}, θ_{n})∥_{2} ≤ ε for all n.

As in the proof of Theorem 3.2 it follows

lim_{n→∞} ∥K_{π_{n}}(t_{ε}, θ_{n}) − K_{π_{n}}(t_{ε}, θ)∥_{2} = 0,

lim_{n→∞} ∥K_{π_{n}}(t_{ε}, θ) − K_{π}(t_{ε}, θ)∥_{2} = 0.

Now using the previous facts in (4.16) we deduce that

lim_{n→∞} ∥K_{π_{n}}^{0} − K_{π}^{0}∥_{2} = 0.  (4.17)

Also we have

lim_{n→∞} ∥P_{π_{n}}^{0} − P_{π}^{0}∥_{2} = 0.

Finally, from (4.17) we also obtain,

lim_{n→∞} ∥C_{e_{πn}^{∞}(0)} − C_{e_{π}^{∞}(0)}∥_{2} = 0.

Remark 4.3 (a) For (SIP) the mean X̄_{0} and the covariance P_{0} are not relevant.

(b) If we assume the observation process y of the form

y(t) = ΓV (t) + ∫_{0}^{t} f(s)ds, t ≥ 0,  (4.18)

where V is a R^{m} -valued Wiener process, Γ is a nonsingular m × m -matrix and f is a progressively measurable R^{m} -valued process such that

sup_{t≥0} E [∥f(t)∥^{2}] < ∞,

then in the definition of e_{π}^{∞} we can take

β(s) = V (−s) for s ≤ 0,

and then we have

e_{π}^{∞}(0) = ∫_{−∞}^{0} S(−s)K_{π}^{0}H^{∗}R^{−1}dV (−s)
= ∫_{0}^{∞} S(s)K_{π}^{0}H^{∗}R^{−1}Γ^{−1} [dy(s) + f(s)ds] .  (4.19)

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